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Irreducible polynomials which divide trinomials over GF(2)

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Irreducible polynomials which divide trinomials over GF(2)

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Content IRRED U C IB LE POLYNOM IALS W HICH DIVID E TRINOM IALS O V ER GF(2) by Pey-Feng Lee A D issertation Presented to the FACULTY OF T H E GRAD UA TE SCHOOL U N IV ERSITY OF SO U TH ERN CALIFORNIA In Partial Fulfillment of the Requirem ents for the Degree D O C TO R OF PH ILO SO PH Y (ELECTRICA L EN GIN EERIN G ) May 2005 Copyright 2005 Pey-Feng Lee Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. UMI Number: 3180443 INFORMATION TO USERS The quality of this reproduction is dependent upon the quality of the copy submitted. Broken or indistinct print, colored or poor quality illustrations and photographs, print bleed-through, substandard margins, and improper alignment can adversely affect reproduction. In the unlikely event that the author did not send a complete manuscript and there are missing pages, these will be noted. Also, if unauthorized copyright material had to be removed, a note will indicate the deletion. ® UMI UMI Microform 3180443 Copyright 2005 by ProQuest Information and Learning Company. All rights reserved. This microform edition is protected against unauthorized copying under Title 17, United States Code. ProQuest Information and Learning Company 300 North Zeeb Road P.O. Box 1346 Ann Arbor, Ml 48106-1346 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. D edication Dedicated with love to my mother, my wife, and my daughters. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Acknowledgem ents I would like to express my enormous gratitude and appreciation to Professor Solomon W. Golomb, the chairman of my dissertation committee, for his consistent encouragement, concern, and guidance throughout my graduate studies at USC. I have benefited greatly from his extensive knowledge, remarkable experience, and ample resources. My deep appreciation is also given to Professor William C. Lindsey and Professor Robert Guralnick for taking their precious time and efforts to serve on both my guidance and dissertation committees, and for their constructive comments and suggestions. Also, I would like to thank Professor Charles L. Weber and Professor Alan E. Willner for serving on my guidance committee. A special acknowledgment is due to Professor Lloyd R. Welch for providing the clever testing criterion and a sample program. 1 also wish to thank Professor John Brillhart, University of Arizona, and Professor Ram Murty, Queen’s University, for patiently reply ing to my e-mails. Also, 1 would like to thank all the faculty and students at EE/Systems and Mathematics for offering many invaluable classes and discussions, and I am grateful iii Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. to Milly Montenegro, Mayumi Thrasher, and Gerrielyn Ramos for their capable admin istrative help. Last but not least, I wish to acknowledge my wife Yuhsuan, who was awaiting our second baby during the time of my preparation of this thesis, for her continuing support, patience, and understanding. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Table of Contents D edication ii Acknow ledgem ents iii List o f Tables vi List o f Figures vii A bstract viii 1 Introduction 1 1.1 M otivations............................................................................................................. 1 1.2 Outline of the t h e s i s ............................................................................................. 5 2 Theorem s and R esults 7 2.1 Basic Theorems and R esu lts................................................................................ 7 2.2 Further Theorems and Results .......................................................................... 16 3 T he M ultiplicative M odule 30 3.1 Introduction............................................................................................................. 30 3.2 Prime Generators of the Multiplicative Module M ....................................... 31 3.3 Composite Generators of the Multiplicative Module M ................................ 34 4 Som e R elated Problem s 39 4.1 Generalized A rtin’s C o n je c tu re .......................................................................... 39 4.2 Generalized TZZ and BGL Conjectures .......................................................... 50 5 C onclusion 55 Bibliography 57 v Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. List of Tables 2.1 Frequency distribution of the index r of the first 1,000,000 odd primes (1 < r < 1 0 0 ) .......................................................................................................... 28 3.1 Prime generators of the multiplicative module M (non-Mersenne primes) 34 3.2 Composite generators of the multiplicative module M (g | < f? n ( 2 ) ) ....... 37 3.3 Composite generators of the multiplicative module M (g | $n(2)$2ra(2)) • 37 3.4 Factors of 4>n(2) versus members of the multiplicative module M (2 < n < 61) .................................................................................................................... 38 4.1 Numerical results of F2 (r) for the first 1,000,000 odd primes (1 < r < 100) 44 4.2 Numerical results of Fs(r), Fs(r), Fr(r) of the first 1,000,000 odd primes (1 < r < 1 0 0 ) .......................................................................................................... 46 4.3 Collections of possible indices r of the first 100,000 odd primes (2 < a < 100) 48 vi Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. List of Figures 1.1 The n-stage binary linear feedback shift register Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. A bstract Shift-register sequences, also known as pseudorandom sequences, or pseudonoise se quences, have played increasingly important roles in many important applications. The simplest linear feedback shift registers to generate binary sequences involve only two taps, which corresponds to a trinomial over GF(2). It is therefore of interest to know which irreducible polynomials / ( x) divide trinomials over GF(2), since the output sequences corresponding to f ( x ) can be obtained from a two-tap linear feedback shift register (with a suitable initial state) if and only if f ( x ) divides some trinomial t(x) = xm + xa + 1 over GF(2). In this thesis we develop the theory of irreducible polynomials which do, or do not, divide trinomials over GF(2). Abundant theorems and results are presented relating to the primitivity t and the index r, which is the number of irreducible factors of the tth cyclotomic polynomial 3>t(.x) over GF(2). The set of all positive (odd) integers t such that the irreducible polynomials of odd primitivity t > 1 divide trinomials over GF(2) form a multiplicative module M, which is closed with respect to multiplication by odd viii Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. numbers. The set G of generators of M is quite sparse, and its members seem related to numbers of the form $ n(2). The distribution of the values of r among the set of all odd primes p leads to a generalization of Artin’s Conjecture concerning primitive roots modulo p. We generalized the conjectures of Blake, Gao and Lambert, and of Tromp, Zhang and Zhao, and proved the generalization of the Blake, Gao and Lambert conjecture. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Chapter 1 Introduction 1.1 M otivations Linear feedback shift register sequences have been used in a variety of important applica tions, such as CDMA (Code Division Multiple Access) communications, spread spectrum communications, CW radar, bit error rate measurements, error correcting codes, and stream ciphers. There are two main advantages for using linear feedback shift register sequences: they are extremely fast and easy to implement both in hardware and software, and they can be readily analyzed using algebraic techniques. A thorough introduction to the theory of shift register sequences is in the book by Golomb [5]. A typical n-stage binary linear feedback shift register (known as a Fibonacci config uration) is illustrated in Figure 1.1. Both the feedback coefficients Cl, C2, . ■ ■ , Cn Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. and the initial state a _ i,a _ 2). .. ,a. of the shift register are elements of GF(2). During each unit of time, the contents of each stage i shifts to next stage i — 1 synchronously for 1 < i < n — 1, and the new content of stage n — 1 is computed as the sum (modulo 2), i=1 where the ct's indicate whether the contents of stage n — i is to be be included in the sum or not, and {aj } = {a0)ai;«2,«3) • • •} is the output sequence generated by this device. stage n-1 a-2 n-3 0 output sequeirce Figure 1.1: The n-stage binary linear feedback shift register Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. The polynomial n f(x) = l + ^2ciXl (1.1) i— 1 associated with the shift register in Figure 1.1 is call the characteristic polynomial of the shift register. In fact, the characteristic polynomial f(x) is even independent of the initial condition. If a is a root of an irreducible polynomial f(x) of degree n over GF(2), then the primitivity t of a (and also of f(x) ) is defined as the smallest positive integer t such that o^ — l (or f( x ) | (xt - 1).) As shown in [5], the output sequence of a binary linear feedback shift register is periodic, and if f(x) is irreducible, the primitivity t of f(x) will always be a divisor of 2n — 1. When t = 2n — 1, f(x) is called a primitive polynomial over GF(2), which corresponds to a maximum-length sequence (or m-sequence for short). It is a fact that every primitive polynomial is also irreducible over GF(2), but the converse is not true. The number of primitive polynomials of degree n over GF(2) is given by H n)=a ^ i> 1 p . * , for each n > 1; and the number of irreducible polynomials of degree n over GF(2) is given by (1.3) d\n 3 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. for each n > 1. (Here < f> (n) is the Euler phi-function, and p(n) is the Mobius mu- function.) Generally, different initial states of the shift register may give rise to different output sequences. If the corresponding polynomial f{x) is irreducible, the period of the shift register sequence does not depend on the initial state, except for the initial condition all 0’s. Specifically, if f(x) is a primitive polynomial, all the non-zero initial states of the shift register will generate the very same m-sequence except for phase shifts. For certain values of n, there are primitive trinomials xn + xa + 1 of degree n. These correspond to shift registers in which only two taps are involved in the feedback modulo 2 adder. This is the simplest way to generate an m-sequence, which is why primitive (and irreducible) trinomials have been of special interest among the set of all primitive (and irreducible) polynomials. Tables of primitive (and irreducible) trinomials can be found in [5] [10] [13] [14]. It is easy to show that there are irreducible (though not primitive) trinomials over GF(2) for infinitely many different degrees n. It is also conjectured (and highly likely) but still unproved that there are primitive trinomials for infinitely many degrees n. However, by a theorem of R. Swan [11], there are infinitely many degrees n (including all multiples of 8) for which there are no irreducible trinomials, and a fortiori no primitive trinomials, over GF(2). When a primitive trinomial of degree n does not exist, an almost primitive trinomial may be used as an alternative. (A polynomial p(x) of degree n is almost primitive if p(x) ^ 0 and p(x) has a primitive factor of degree > n / 2.) Algorithms for finding almost primitive trinomials can be found in [3]. Moreover, as shown in [9], the moments of the 4 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. partial-period correlation of an m-sequence are related to the number of trinomials of bounded degree (determined by the particular partial period under consideration) that the characteristic polynomial of the m-sequence divides. It is therefore of interest to know which irreducible polynomials f(x) divide trinomials over GF(2), since the output sequence corresponding to f(x) can be obtained from a two- tap linear feedback shift register (with a suitable initial state) if and only if f ( x ) divides some trinomial t(x) = xm + xa + 1 over GF(2). 1.2 O utline of the thesis In this thesis we develop the theory of which irreducible polynomials do, or do not, divide trinomials over GF(2). In Chapter 2, useful theorems and results are presented relating to the primitivity t and the index r. An irreducible polynomial f( x ) either divides infinitely many trinomials, or divides no trinomials. If f(x) divides trinomials, f(x) must divide some trinomial of degree less than its primitivity t. We prove that every primitive polynomial divides trinomials, and also present a whole family of irreducible polynomials xt_1 + x*-2 + ... + x + 1 which never divide trinomials. Once f(x) divides any trinomial, then all the polynomials having the same primitivity t divide trinomials. Then a clever criterion is introduced to determine whether a given irreducible polynomial f(x) divides trinomials, for every odd primitivity t > 1. Further theorems and results are presented in terms of the index r, which is the number of irreducible factors of the tth cyclotomic polynomial 5 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. $t(x ) over GF(2). We show how the index r contributes to determining whether a given irreducible polynomial f(x) of primitivity t divides trinomials when t is a prime number. In Chapter 3, the multiplicative module M is introduced, which is the set of positive (odd) integers t such that the irreducible polynomials of odd primitivity t > 1 divide trinomials over GF(2). The set G of generators of M consists of certain prime and composite values. Then computational results are presented. These results show that the set G is quite sparse, and its members seem related to numbers of the form < h n(2). In Chapter 4, A rtin’s Conjecture [1] concerning primitive roots is discussed. The distribution of the values of r for members of the set of all odd primes p leads to a generalization of Artin’s Conjecture concerning primitive roots modulo p. Then we generalize the conjecture of Blake, Gao and Lambert (BGL) [2], and of Tromp, Zhang and Zhao (TZZ) [12]. We prove the former generalization, and ascertain that the latter generalization is true if the original TZZ conjecture is true. Finally, this thesis is summarized, and possible future research work is presented, in the last chapter. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Chapter 2 Theorem s and R esults In this chapter, we primarily focus on the question of which irreducible polynomials divide (any) trinomials, and which ones never divide trinomials, over GF(2). The main theorems and results are presented, relating to the primitivity t and the index r. 2.1 B asic Theorem s and R esults In what follows, we assume that f( x ) is an irreducible polynomial of degree n > 1 over GF(2) having a root a and primitivity t, which means that a* = 1 (i.e. that / ( x) divides xl — 1), where t is the smallest positive integer with this property. The following facts, many of which depend on the primitivity t of fix), have been established. Lem m a 1. fix) divides h(x) if and only if every root a of fix) is also a root of h(x). Proof. Suppose h(x) = f(x)q(x). Then h{a) = f(a)q(a) = 0 • q(a) = 0 . 7 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Conversely, if h(a ) = 0, we can divide h(x) by f( x ) to get h(x) = f(x)q(x) + r(x), where deg(r(:r)) < deg{f{x)). Then 0 = /i(a) = f(a)q(a) + r(a) = 0 • q(a)) + r(a) = r(a), from which r(x) also has a as a root. This contradicts the choice of the irreducible polynomial f(x) as the lowest degree polynomial (the minimal polynomial) having a as a root. Hence r(x) = 0 and f(x) divides h(x). □ Theorem 1. f(x) divides some trinomial if and only if there exist distinct positive integers i and j with a1 + = 1. Proof. By the previous Lemma, the trinomial h(x) = x l + x - > + 1 is divisible by f(x) if and only if h(a) — a1 + oP + 1 = 0, i.e. a 1 + aJ = 1. □ Theorem 2. If f(x) divides any trinomial, then f(x) divides infinitely many trinomials. Proof. If fix) divides a trinomial xm + xa + 1, we have am + aa + 1 = 0 . 8 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Since also a* = 1, we get a m + st + a a+ rt + j _ = Q for all positive integers r and s, from which f(x) divides x m + st + x a+ rt + L □ Theorem 3. If f(x) divides any trinomials, then f(x) divides some trinomial of degree < t. Proof. If f(x) divides xm + xa + 1, we have am + a ° + 1 = 0. Since a1 = 1, this gives am' + aa' + 1 = 0 where m' = m(mod t ) and a' = a(mod t), from which we can pick m! and a' on the range from 0 to t - 1. Then f(x) must divides some trinomial xm' + xa' + 1 of degree < t. □ Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Hence, if / ( x) divides no trinomial of degree < t, then f(x) will never divide any trinomials. This provides a finite decision procedure for whether a given irreducible polynomial ever divides trinomials. Also, an irreducible polynomial f(x) either divides infinitely many trinomials, or divides no trinomials. In the following, we show that every primitive polynomial divides infinitely many trinomials, and present a whole family of irreducible polynomials xt^i + x L ~2 + ... + x + 1 which divide no trinomials. T h eo rem 4. If f(x) is a primitive polynomial of degree n (i.e. if t = 2n — 1), then f(x) divides trinomials. Proof. Since a is a root of f(x), the powers 1, a 1, a 2, a 3, .. . , a t_1 are all distinct, and constitute all the non-zero elements of the field GF(2n). Hence, for a li i,0 < * < t, 1 + a1 = a3 for some j ^ i, 0 < j < t. Thus, f{x) divides x1 + x^ + 1 for each such pair (i,j). □ In fact, when f(x) is a primitive polynomial, it divides exactly (t — l)/2 trinomials of degree less than t. Since the primitive polynomials precisely correspond to m-sequences, 10 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. this theorem says that every m-sequence can be obtained from a two-tap linear shift register. This is a very simple and efficient way to generate an m-sequence. Theorem 5. For odd t > 3, if f(x) = ^ = a;4-1 + x + ... + x + 1 is irreducible, then f{x) divides no trinomials. Proof. The lowest degree polynomial having the root a of f(x) as a root is the irreducible polynomial f(x) = ~ ry = X4-1 + X4-2 + . . . + X + 1, (;X - 1) which has t > 3 terms. Suppose f(x) divides some trinomial xm + xa + 1, so that am + a “ + 1 = 0. Then am' + aa' + 1 = 0, where m ' and a' are m and a reduced modulo t, respectively, and are less than t. Thus a is a root of xm > + xa' + 1, a trinomial of degree < t — 1. But the only polynomial of degree < t — 1 with a as a root is its minimal polynomial, the f-term irreducible polynomial f(x) of degree t — 1. □ 11 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. This occurs if and only if 2 is a primitive root modulo t with t prime. Examples include t = 5,11,13,19,29,37,53,59,61,67,83,101,.... By A rtin’s Conjecture [1], there are infinitely many primes p for which 2 is a primitive root (i.e. where 2 is a generator of the multiplicative group of GF(p)). A rtin’s Conjecture includes the assertion that every prime q is a primitive root — i.e. a generator of the multiplicative group — modulo p, for infinitively many primes p. This has been proved for all primes q assuming the Generalized Riemann Hypothesis; and without that hypothesis, for all primes q with at most two exceptions. It is therefore extremely unlikely that q = 2 is an exception. Hence, we believe there are infinitely many irreducible polynomials of this kind which never divide trinomials. D efinition 1. The tth cyclotomic polynomial is given by $ t(x ) = - l f W (2.1) d\t where ji{d) is the Mobius mu-function. For any odd integer t > 3, let $t(x) = fi{x)f 2 (x)... f r(x) be the factorization of the tth cyclotomic polynomial into irreducible factors over GF(2). 12 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. It is known that all the fi(x)’s have the same degree (say, n) and the same primitivity t. These factors are all the irreducible polynomials having primitivity t. Theorem 6. If any one of the /i(x )’s divides a trinomial, then all r of the /j(x )’s divide trinomials. Proof. Collectively, the roots of the polynomials ■ ■ ■ , fr(x) are all the powers a, a 2, a 3, . .. , a t_1, of a single root a of ®t(x), which can be taken to be a root of any one of the polynomials fi(x). Also, the roots of cF = 1 always form a cyclic group under multiplication. If a is a primitive root of of — 1, then every other primitive root will be a power of a. Suppose fi(x) divides the trinomial xm + xa + 1. Then O L m + aa + 1 = 0 , where we selected a to be a root of fi(x). For any other polynomial fj(x) from the set of divisors of 4b (a-') j let one of its roots be (i — au. with GCD(t, u) — 1 (i.e. rt + su — 1 for some r, s). 13 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Then for some s, 1 < s < t — 1, we have a = /?s, from which (/?s)m + (/T)a + l = (3sm + !3sa + l = 0, whereby fj(x) divides the trinomial x + xsa + 1 . □ Specifically, if any one of the /j(x )’s is already a trinomial, then all the /*(x)’s di vide trinomials. The theorem provides that for any odd t, either all the fi(x )’s divide trinomials or none divides trinomials. Next, a clever criterion for testing whether an irreducible polynomial divides trinomials is the following. Theorem 7 (W elch’s C riterion). For any odd integer t. the irreducible polynomials of primitivity t divide trinomials if and only if GCD( 1 + x l,l + (1 + x)1) has degree greater than 1 . Proof. Let ct(x) = (n°t necessary the tth cyclotomic polynomial). Then ( 1 + x4 ) = ( 1 + x)ct(x), and (! + ( ! + x)4) = xct (l + x). 14 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Thus, except for possible linear factors, GCD{ 1 + x \ 1 + (1 + a:)*) = GCD(ct(x), ct (l + x)). Let a{x) = j - — y = • • • fr(x ) (:X - l ) be the factorization of ct(x) into irreducible factors. Then the roots of fi(%), f2(x), ■ ■ ■ , fr(x) collectively are a, a 2, a 3, . .. , a *-1 where a ^ 1 and a1 = 1. Thus, the roots of the irreducible factors of ct( 1 + x) are Hence, the GCD in question has degree > 1 if and only if one of the roots (say 1 + a3) from cj( 1 + x) equals one of the roots (say a1) from ct{x) (i.e. 1 + a? = a1). This is the precise condition that a factor of ct(x) with a as a root divides the trinomial xl + x] + 1 . □ 15 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. This computationally useful criterion, due to L.R. Welch, determines whether the irreducible polynomials of primitivity t divide trinomials, for any odd integer t > 3, without directly identifying which irreducible polynomial divides which trinomial. 2.2 Further Theorem s and R esults When the primitivity t is a prime p. let % (x ) = _ jy = h ( x ) f2(x) ... f rOr) be the factorization of the pth cyclotomic polynomial into irreducible factors over GF(2). Here the index r = 4>{p)/n = (p — 1 )/n is the number of irreducible factors of Qp(x) over GF(2), and (p — 1 )/r is the order of 2 in the multiplicative group modulo p. Again, all the /i(x )’s have the same degree (say, n) and the same primitivity p, and they are all the irreducible polynomials having primitivity p over GF(2). Let a be a root of The following results relate to the index r. D efinition 2. Let f(x) be an irreducible polynomial of degree n. T he reciprocal of fix) is defined as f { x ) = x nf{±). 16 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. When f*(x) = f{x), then we say f(x) is self-reciprocal (i.e. if a is a root of /(x ), then a - 1 is also a root of /(*))• Lem m a 2. For prime values p > 3 with ^ ( x ) = ^,~i) = h ( x ) h ( x ) ■ ■ ■ fr{x) as a product of r > 1 irreducible polynomials, if any of the /*(x)’s is self-reciprocal, then fi(x) cannot be a trinomial. Proof. Since fi(x) is self-reciprocal, we have If ft(x) is a trinomial, it must be x (p-i)/r + x (p~l)/2r + which divides x ^ - m r + 1? whereby a 3(p-l)/2r = 17 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. but 3(p — l ) / 2 r 1 , which contradicts p being the smallest positive exponent with oP = 1 . □ T heorem 8. For prime p > 3, if any of the /,;(a:)’s is self-reciprocal, then none of the fi(xy s divide trinomials. Proof. Since fi(x) is self-reciprocal, it cannot be a trinomial by the previous Lemma. Suppose fi(x) divides some trinomial. WLOG, fi(x) divides a trinomial U(x) = x rn + xa + 1 with 1 < a < to < p. Write U{x) = xm + xa + 1 = fi{x)gi(x). Then t*(x) = x m + xm~a + 1 = fi(x)g*(x) where g*(x) = xdgi{i ) and d = degree^(x)). Thus ti(x) + t*(x) = xm~a + xa = fi(x){gi(x) + g*(x)) and fi(x) divides x m - a + x a = x a ^ + x \m - 2 a \^ 18 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. where 0 < |m — 2a\ < p, which contradicts a, the root of fi(x), having primitivity p, unless to — 2a = 0. In this case, 9*{x)=gi{x) and t*(x) = U{x), so that U{x) = xm + xm/2 + 1, which divides x 3”V2 + 1 , so that a S m /2 = 1 } whereas also dP = 1. This requires a \p— 3 m /2| i, but since (p — l)/r < m 1 , we have 3(p — l)/2 r < 3m /2 < 3p/2, from which 0 < \p - 3m /2| < p, contradicting a having primitivity p. □ 19 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Corollary 1. Let p > 3 be a prime and 3 > p(x) = ^ - l ) = h ( x )h {x ) ■ • • fr(x ) be a product of r irreducible polynomials. If r > 1 is an odd number, then the fi(x)’s divide no trinomials. Proof. When r > 1 is odd, then at least one of the fi(x)’s is self-reciprocal. The result follows from the theorem. □ Hence, if the index r of the pth cyclotomic polynomial < & p(x) is any odd number, then all the irreducible factors of $ p(x) divide no trinomials, except for the trinomial x2 + x + 1 with r — 1 and p — 3, which is already a trinomial. When r is an even number, it no longer guarantees that at least one of the /(.i;)’s is self-reciprocal. But there are only two cases: either at least one of the f ( x f s is self-reciprocal, or none of them is self-reciprocal. In the former case, all the / ( a;)’s divide no trinomials by the above theorem. In the latter case, the polynomials appear in pairs (i.e. if a is a root of fi(x), then a~l is a root of f*(x)). Then the fi(x)’s may or may not divide trinomials. Theorem 9. Let p > 7 be a prime and Qp(x) = = fi(x)f 2 (x) be a product of two irreducible polynomials (i.e. r = 2). Then the /j(x )’s divide no trinomials. Proof. If either one of the fi(x f s is self-reciprocal, then the /i(x )’s divide no trinomials by the previous theorem. Otherwise, the fi{xf s form a reciprocal pair. If a is a root of 20 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. then a 1 is a root of ^ (a ;). Suppose f\(x) divides some trinomial t\(x) (including the case that f\{x) itself is a trinomial). Then we can write ti(x) = xm + x a + 1 = fi(x)gi(x ), with 1 < a < m < p, and replacing a by a - 1 for all roots of t\(x), we get f 2{x)gt(x) = tt(x) = xm + xm- a + l. Then the product ti{x)t\{x) =fi(x)f 2 (x)gi(x)gi(x) = ^P{x)gi{x)gl(x) =x2m + x2m- a + xm+a + xm + xa + xm~a + 1 , and this 7-nomial must have both a and a - 1 as roots, which must satisfy both ap = 1 and a~p = 1. We may therefore reduce all exponents in t\ {x)t\(x) modulo p, to get an at most 7-term polynomial of degree < p, but which has all p — 1 roots of ^-pix) as roots, contradicting the fact that the only (non-zero) polynomial of degree 7 terms. □ 21 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Note that when r — 2 and p = 7, neither of the /i(x )’s is self-reciprocal, where $ 7 (x) = x6 + x5 + x4 + x 3 + x2 + x + 1 = (x 3 + x2 + 1 ) (x 3 + x + 1 ) has only 7 terms, and the two factors of ^ t{x) are already trinomials. Theorem 10. Let p be a prime and < 'I )p(.x) = j ^ i ) ~ / i (x')/2 (x ).h(x ) f^i^) be a product of four irreducible polynomials (i.e. r = 4). Then the /*(*)’s divide no trinomials. Proof. If any one of the fi( x f s is self-reciprocal, then the /*(&)’s divide no trinomials by Theorem 8 . Otherwise, the /j(x )’s come in pairs. Let a be a root of ,fi(x) and a - 1 be a root of f 2 (x). Similarly, let (3 = au be a root of fs(x) and (3~x be a root of f 4 (x). Suppose fi(x) divides some trinomial ti(x) (including the case that fi(x) itself is a trinomial). Then we can write ti(x) = xm + xa + 1 = f\(x)g\(x), with 1 < a < m < p. Replacing a by o r 1 for all roots of p (x), we get f 2 (x)g{(x) = t\{x) = xm + xm- a + 1 . 22 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Then the product ti(x)tl(x) =fi(x)f 2 (x)gi(x)g{(x) = x 2m + x 2 m - a + x m + a + xm + / + x m - a + 1 is a 7-term polynomial which has both a and a - 1 as roots. Similarly, suppose f 3(x) divides some trinomial t3(x) (including the case that f 3{x) itself is a trinomial). Then we can write t3(x) = xk + xb + 1 = f 3{x)g3(x), with 1 < b < k < p. Replacing /3 by / ? — 1 for all roots of t3(x), we get f 4 (x)g3(x) = t* 3(x) = xk + x k~h + 1 . Then the product h{x)t*3{x) = h{x)fA{x)g3{x)gl{x) =x2k + x2k~b + x k+b + x k + xb + xk~b + 1 is also a 7-term polynomial which has both (3 and (3~l as roots. 23 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Then ti(x)tl(x)t3(x)tl(x) = /i (x)f2 (x)f3(x)f4(x)g1 (x)gf (x)g3(x)gl (x) =%(x)gi{x)gt(x)g3(x)g3(x) has at most 49 terms, and has cn , a -1 ,/3 and /3- 1 as roots, satisfying tiP = 1, a~p = 1, fP = 1 and (3~p = 1. We may therefore reduce all exponents in ti(x)tl(x)t3(x)tl(x) modulo p, to get an at most 49-term polynomial of degree < p, but which has all p — 1 roots of < & p(x) as roots, contradicting the fact that the only (non-zero) polynomial of degree p(x) as roots is Tpfx) itself, which has p > 49 terms. Since the first $pix ) = ^x _ iy = fi(x)f2(x)f3{x)U(x), with four irreducible factors, happens at p = 113 > 49, the result follows. □ Theorem 11. Let p be a prime and 4 > p( . 7;) = = fi(x )f 2 (x ) ■ ■ ■ f r ( x ) be a product of r irreducible polynomials. If the index r is any even number, then the ,/i(x)\s divide no trinomials if p > 7r/2. Proof. If any one of the fi(x)’s is self-reciprocal, then the /i(x )’s divide no trinomials by the above theorem. Otherwise, the /j(x )’s appear in pairs. By using similar arguments, let cni be a root of fc(x) and a f 1 be a root of /i+i(x) with 1 < i < r, where i is 24 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. odd. Suppose fi(x) divides some trinomial ti{x) (including the case that fi(x) itself is a trinomial). Then we can write ti(x) = xm + xa + 1 = fi(x)gi( x), with 1 < a < m < p. Replacing on by on~1 for all roots of U(x), we get f i+i(x)g*(x) = t*(x) = x m + xm~a + 1 . Then the product ti(x)t*(x) =fi{x)fi+i{x)gi{x)g*{x) =x2m + x2m~a + xm+a + xm + xa + xm~a + 1 is a 7-term polynomial which has both a, and a~l as roots. Therefore, tl{x)t\(x)h{x)tl{x) . ..tr-l{x)t*r_x{x) = h{x)f2{x)h{x)U{x).. ■ fr(x)g1(x)gl(x)g3(x)g^(x). . . gr-i(x)g*-i(x) =®P{x)gi(x)gl(x)g3(x)gz{x). . .gT-i(x)g*-i{x) has at most 7r / 2 terms, and has ccj’s and a ^ ’s as roots, satisfying a? — 1 and a~p — 1 for every odd integer i < r. 25 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. We may therefore reduce all exponents in ti(x)ti(x)t3(x)tl(x) ... U - i i x ^ ^ x ) modulo p, to get an at most 7r/2-term polynomial of degree < p, but which has all p — 1 roots of § p(x) as roots, contradicting the fact that the only (non-zero) polynomial of degree 7% 2 terms. □ The above theorem also provides a finite decision procedure for whether a given irreducible polynomial of prime primitivity p divides trinomials, in terms of the index r. Among the first 1,000,000 odd primes (i.e. 3 to 15,485,867), about 98.9% have their index r < 1 0 0 ; 95.7% have r < 24; and 93.5% have r < 16. As predicted by Artin’s Conjecture, about 37.4% have r = 1. We also found that about 28.1% have r = 2; 6 .6 % have r — 3; 4.7% have r = 4; 1.9% have r = 5; 5.0% have r = 6 ; 0.9% have r = 7; 3.5% have r — 8 ; 0.7% have r = 9; and 1.4% have r = 10. (The probabilistic argument used in A rtin’s Conjecture for r = 1 can be modified for these larger values of r.) The frequency distribution of the index r ( l < r < 1 0 0 ) of the first 1 , 0 0 0 , 0 0 0 odd primes is presented in Table 2.1. According to the index r (1 < r < 16), we summarize the test results for whether or not the irreducible polynomials of prime primitivity p divide trinomials as follows: 1. When r — 1, fi(x) divides trinomials only at p = 3. 2. When r = 2, the /i( s )’s divide trinomials only at p = 7. 26 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 3. When r = 4, the /*(x)’s divide no trinomials. 4. When r = 6 , the /,;(.x)’s divide trinomials only at p = 31. 5. When r = 8 , the /,(x )’s divide trinomials only at p = 73. 6 . When r = 10, the /j(a;)’s divide no trinomials. 7. When r = 12, the fi(x)'s divide no trinomials. 8 . When r = 14, the /j(x )’s divide no trinomials. 9. When r = 16, the /i(x )’s divide no trinomials. 10. When r > 1 is an odd number, the fi(x)’s divide no trinomials. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Table 2.1: Frequency distribution of the index r of the first 1 , 0 0 0 , 0 0 0 odd primes ( 1 < r < 100) r - 1 0 0 k -2 0 0 k -300k -400k -500k -600k -700k -800k -900k - 1 0 0 0 k f{r) 1 37470 37345 37503 37399 37394 37351 37243 37461 37398 37460 374024 2 28059 28111 28022 28050 28084 28004 28169 28125 27956 28212 280792 3 6633 6677 6595 6642 6641 6633 6713 6648 6699 6583 66464 4 4680 4645 4712 4710 4602 4752 4679 4666 4628 4571 46645 5 1858 1883 1876 1914 1888 1981 1944 1888 1915 1838 18985 6 5015 4971 5102 4936 4945 5095 4963 4973 5003 4953 49956 7 893 899 892 890 883 887 870 857 857 884 8812 8 3525 3515 3433 3502 3493 3472 3534 3599 3504 3552 35129 9 744 731 748 733 719 722 755 737 767 739 7395 1 0 1382 1416 1468 1448 1430 1404 1396 1381 1376 1416 14117 1 1 350 322 324 345 309 294 356 355 367 331 3353 1 2 813 826 823 870 799 796 820 841 824 891 8303 13 242 241 228 228 280 239 233 230 239 257 2417 14 695 630 660 676 643 702 646 662 684 693 6691 15 316 339 327 348 354 328 333 318 352 361 3376 16 883 849 860 922 883 897 947 846 843 840 8770 17 139 145 145 140 125 136 148 128 135 108 1349 18 538 621 529 507 538 564 544 530 556 539 5466 19 95 126 1 1 1 98 1 2 1 108 1 0 0 108 91 103 1061 2 0 244 247 243 254 258 234 232 229 247 246 2434 2 1 176 170 167 163 168 158 159 139 157 164 1621 2 2 259 274 261 260 255 256 228 254 256 228 2531 23 87 97 83 6 8 75 6 8 73 84 75 73 783 24 608 653 636 614 626 596 588 634 620 639 6214 25 83 73 69 72 59 73 79 74 62 75 719 26 173 154 167 187 174 174 184 197 178 174 1762 27 8 8 74 82 85 8 8 8 6 89 80 82 8 8 842 28 1 1 1 116 1 0 2 108 1 1 2 114 1 1 1 118 1 1 0 114 1116 29 50 37 35 48 62 46 31 40 38 44 431 30 255 246 214 259 259 248 243 237 271 244 2476 31 43 41 50 46 40 43 36 31 48 40 418 32 2 2 1 2 2 0 219 208 217 229 199 207 227 226 2173 33 64 63 65 69 60 56 74 61 64 46 622 34 81 1 0 1 96 1 0 1 118 1 0 1 95 1 0 1 105 96 995 35 46 46 51 47 45 45 34 41 49 41 445 36 73 91 75 96 93 89 93 81 90 89 870 37 32 2 2 29 26 36 24 35 32 34 25 295 38 78 97 83 90 82 76 81 87 92 81 847 39 45 52 38 38 45 43 44 46 39 44 434 40 176 179 170 140 181 171 176 194 179 191 1757 41 2 2 24 29 17 30 2 1 25 2 0 27 2 2 237 42 109 1 1 1 118 117 129 135 126 126 1 1 0 1 1 1 1192 43 18 29 17 2 1 31 2 2 14 26 23 23 224 44 41 52 45 43 40 44 43 47 34 55 444 45 41 38 33 26 43 39 34 41 44 32 371 46 51 45 52 48 58 57 53 47 59 6 6 536 47 2 1 17 1 2 17 2 0 19 2 0 14 2 2 17 179 48 145 143 162 125 149 166 158 153 173 157 1531 49 25 1 1 13 13 27 15 17 17 17 27 182 50 58 65 67 59 6 6 54 63 41 62 49 584 28 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Table 2.1 (Continued...) r - 1 0 0 k -2 0 0 k -300k -400k -500k 51 2 2 30 25 2 1 2 2 52 38 35 39 29 23 53 1 1 1 1 2 1 1 2 1 0 54 62 62 60 6 6 6 6 55 15 16 19 2 0 25 56 78 75 1 0 0 90 94 57 1 1 15 14 27 25 58 40 33 28 29 36 59 4 1 1 15 1 0 1 2 60 38 38 35 38 45 61 13 1 2 1 1 6 15 62 25 38 28 28 2 2 63 18 19 18 15 19 64 56 65 48 50 61 65 14 16 9 1 1 1 0 6 6 53 32 44 40 56 67 14 14 8 5 9 6 8 1 1 16 17 17 9 69 7 1 2 13 15 15 70 35 33 34 41 2 1 71 5 8 2 13 8 72 70 58 63 57 58 73 6 8 1 1 7 4 74 27 2 2 26 24 27 75 18 14 1 1 1 0 2 0 76 1 1 1 1 13 7 1 1 77 3 1 0 1 0 8 5 78 40 32 32 31 29 79 5 6 7 7 7 80 47 40 50 44 34 81 7 4 5 6 8 82 1 1 1 0 14 1 2 18 83 4 7 8 1 1 5 84 15 2 0 15 2 1 15 85 3 1 0 6 5 3 8 6 17 23 13 15 26 87 7 6 5 8 9 8 8 34 32 30 31 29 89 3 4 7 4 7 90 27 28 35 25 2 2 91 3 6 7 5 1 2 92 2 0 8 13 8 1 1 93 1 2 7 4 9 8 94 7 17 1 0 1 0 1 0 95 4 2 7 3 5 96 42 45 40 34 40 97 0 3 5 3 6 98 1 2 19 1 2 17 9 99 1 6 1 0 4 1 2 1 0 0 7 8 9 6 9 )0 k -700k -800k -900k - 1 0 0 0 k f{r) 23 31 28 23 29 254 32 27 25 36 38 322 13 17 15 1 2 13 135 46 60 47 52 52 573 18 18 17 1 2 23 183 71 85 72 92 76 833 18 25 2 2 8 15 180 38 37 39 36 42 358 1 0 1 2 1 1 7 14 106 34 42 43 38 41 392 1 0 15 13 13 9 117 26 34 28 28 28 285 14 14 19 13 2 2 171 38 42 70 44 46 520 1 2 5 1 0 1 2 8 107 53 47 50 55 46 476 6 16 6 1 2 13 103 14 15 13 24 18 154 9 15 1 2 1 0 1 1 119 40 37 30 37 38 346 4 8 6 13 8 75 73 70 73 61 54 637 3 1 1 5 5 5 65 19 2 0 16 29 26 236 16 9 1 0 13 15 136 1 2 18 1 1 15 1 0 119 1 1 8 9 1 0 5 79 35 33 26 36 2 1 315 1 0 4 1 1 3 7 67 42 40 43 60 37 437 13 6 4 8 7 6 8 2 0 13 16 13 13 140 1 0 8 8 2 3 6 6 17 23 18 19 2 1 184 1 0 5 4 7 5 58 13 9 16 16 14 162 13 13 16 1 2 9 98 35 28 35 27 29 310 5 6 1 1 7 7 61 24 34 30 29 24 278 3 8 4 3 6 57 4 1 1 16 6 14 1 1 1 4 8 1 0 1 0 1 0 82 17 13 15 6 1 1 116 3 6 4 8 6 48 46 41 47 37 39 411 5 4 6 2 8 42 1 1 13 19 17 1 2 141 1 0 1 0 6 6 4 69 14 9 1 1 13 1 0 96 29 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Chapter 3 The M ultiplicative M odule In this chapter, the multiplicative module M is introduced, which is the set of positive (odd) integers t such that the irreducible polynomials of odd primitivity t > 1 divide trinomials over GF(2). The computational results are presented, and the set C J of gen erators of M is discussed. 3.1 Introduction It is a simple fact that if the irreducible polynomials of primitivity t divide trinomials, then the irreducible polynomials of primitivity mt also divide trinomials for every odd integer m > 1. Therefore, let M be the set of positive (odd) integers t such that the irreducible polynomials of odd primitivity t > 1 divide trinomials. Then, in view of the closure property, we call M a multiplicative module. That is, for every t £ M, we also have mt £ M for every odd integer m > 1. An element g of M is a generator of M if and only if g £ M but no proper factor h of g is in M. Let G be the subset of M consisting of the generators of M. From Theorem 30 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 4 in Chapter 2, the polynomials of primitivity t = 2” — 1 divide trinomials for every integer n > 1. Hence each of these numbers {3, 7, 15, 31, 63, 127, 255, 511, 1023, ...} is in M, and each has (at least) one factor in G. Prom the computational results, the set G of the generators of M consists of both prime and composite values, and these numbers also suggest some interesting patterns. 3.2 Prim e Generators of the M ultiplicative M odule M Clearly, all the Mersenne primes (2n — 1 being prime) are members of G. These include {3; 7; 31; 127; 8,191; 131,071; 524,287; 2,147,483,647; ...}. Aside from the Mersenne primes, there are other primes in G. The first non-Mersenne- prime generator of M is 73, corresponding to eight irreducible polynomials of degree 9 and primitivity t = 73, which do divide trinomials. (In fact, two of these eight irreducible polynomials are already trinomials; therefore all eight of them must divide trinomials by Theorem 6 .) By complete computer search for all odd primes t < 3,000,000, only five other prime elements of G (not Mersenne primes) exist: (73; 121,369; 178,481; 262,657; 599,479}. 31 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. It was mentioned that among the eight irreducible factors of $ 7 3 (x), two of them are already trinomials. However, none of the irreducible factors of 4 > t (x) are trinomials for t = 121,369 or 178,481 or 262,657 or 599,479. It is not necessary that if the irreducible factors of 4 > t(x) divide trinomials, then at least one of the factors has to be a trinomial. Let 4 > n(2) denote the nth cyclotomic polynomial evaluated at 2. All the elements of G currently known can be expressed fairly simply in terms of the numbers 4 > n(2). The Mersenne primes are precisely the numbers 4 > „ (2) when n is prime and 2n — 1 is prime. Of the other five known prime numbers in G, three are values of 4 > n(2): 73 = $ 9 (2 ), 262,657= $ 2 7 (2 ), and 599,479 = $ 3 3 (2 ). This suggests the possibility that whenever 4>„(2) is prime, where n is an odd prime, then 4 > n(2) < E G. Prom [4], which lists the factorizations of 2" - 1 for all odd integers n < 1 , 2 0 0 , the first counterexample occurs at 151 = $ 1 5 (2 ), which is not in G, and the next non-Mersenne-prime case 4,432,676,798,593 = 4 > 4 9 (2) is too large to test by current methods. 32 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Besides 73 = $ g (2 ), 2 6 2 ,6 5 7 = $ 2 7( 2 ), and 5 9 9 ,4 7 9 = $ 3 3 (2 ), the other two cases result from dividing $ „ ( 2 ) by a small prime factor: 12 1 ,3 6 9 = $ 3g (2)/79, and 178,481 = 4>23(2 )/4 7 . These two are both instances where 4>„(2) has two prime factors, and t is the much larger of these two factors. While this may be a good way to look for likely values of f, it is not a reliable indicator. For example, $ 3 5 (2 ) = 7 1 - 1 2 2 ,9 2 1 , but t = 12 2 ,9 2 1 is not in G, and the next non-Mersenne-prime cases, $ 3 7( 2 ) = 2 2 3 ' 1 ,6 1 6 ,3 1 8 , 1 7 7 and $ 4 1 (2) = 1 3 , 3 6 7 - 1 6 4 , 5 1 1 ,3 5 3 , are also too large to test by current methods. The five known prime generators which are not Mersenne primes axe listed in Table 3.1 along with their patterns. 33 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Table 3.1: Prime generators of the multiplicative module M (non-Mersenne primes) generators patterns 73 $9(2) 121,369 $39(2)/79 178,481 $ 2 3 (2)/47 262,657 $ 2 7 (2 ) 599,479 $ 3 3 (2 ) 3.3 C om posite G enerators of the M ultiplicative M odule M If g G G is composite, then (by the definition of G) no prime factor of g is in G. The smallest composite g € G is 85 = 5 • 17 , where 85 6 G but 5 G and 17 G. All the eight irreducible factors of $85 {x) divide trinomials, even though none of them is a trinomial. By complete computer search, there are ten composite elements of G up to t < 1,000,000. These are: {85; 2,047; 3,133; 4,369; 11,275; 49,981; 60,787; 76,627; 140,911; 486,737}. Seven other larger composite elements of G are currently known: {1,826,203; 2,304,167; 2,528,921; 8,727,391; 14,709,241; 15,732,721; 23,828,017}. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Among these seventeen composite elements of G known so far, most of them are either divisors of < F U(2) or of 4 > n(2) 2 n(2) for various values of n. For example, 2,047 = 23-89 = $ n (2 ), 8, 727,391 = 71 ■ 122,921 = $ 3 5 (2 ), 14,709,241 = 631 • 23,311 = $ 4 5 (2 ) are of the form < f > n(2 ); 486, 737 = 233 • 2,089 = $ 2 9 (2)/l, 103, 2,304,167 = 1,103 ■ 2,089 = $ 29(2)/233, 23,828,017 = 11,119 • 2,143 = $ 5i(2)/103 axe of the form $ „ ( 2 )/c, where c is a prime factor of 4?„(2); 85 = 5 • 17 = $ 4(2)$8(2), 3,133 = 13-241 = $ 12(2) $ 24(2), 4,369 = 17-257 = $ 8(2)$ie(2), 49,981 = 151 • 331 = $ is(2 )$ 3 o(2), 140,911 = 43 ■ (29 • 113) = $ 1 4 (2)$ 2 8 (2), 15,732,721 = 241 • (97 • 673) = $ 24(2) $ 48(2) are of the form $ „ ( 2 )$ 2n (2 ); 35 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 60,787 = 89 • 683 = $ n (2 ) $ 22(2)/23, 76,627 = 19 • (37 • 109) = $ i 8(2) $ 36(2) / 3, 1,826,203 = 337 • 5,419 = $ 2 1 (2)$ 4 2(2)/7, 2,528,921 = 41 • 61,681 = $ 2 0 (2)$ 4 0 (2)/5 are of the form $ n(2) 2n(2)/c, where c is a prime factor of 4 > n(2). The only exception so far, which is neither a divisor of < 5 n(2) nor of < hn(2 ) 4 > 2n(2 ), is 11,275 = 11 • (5 ■ 41) • 5 = $io(2)$2o(2)$ 5 (2). All these composite generators of M are listed in Table 3.2 and Table 3.3, respectively, according to their patterns, which also suggests testing the following kinds of numbers for membership in G. 1. If 4 > n(2) is prime and both of 4 > n(2) and 4 > 2 rj(2) ^ M, then $ n(2)4>2n(2) € G, though this is not true for n = 1 0 . 2. If $ n(2) is composite and $ 2n(2) ^ M , then $ n(2)$ 2n(2)/c G G, where c is a prime factor of 4>n(2). Also, it seems possible that $ 4n(2) ^ M for all integers n > 0 (for all n up to 15 this has been verified); and Trl(2 ) ^ M implies $ 2«(2) ^ M (for all n up to 20 this has been verified). The factorization of 4 > n(2) up to n = 61, along with its membership in M, is listed in Table 3.4. 36 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Table 3.2: Composite generators of the multiplicative module M (g | $n(2)) generators factors patterns 2,047 23-89 $ n ( 2 ) 486,737 233• 2,089 $29(2)/l,103 2,304,167 1,103-2,089 $29 (2)/233 8,727,391 71•122,921 $ 3 5 (2 ) 14,709,241 631•23,311 $45(2) 23,828,017 11,119-2,143 $5i(2)/103 Table 3.3: Composite generators of the multiplicative module M (g | $ n(2)$ 2 n(2)) generators factors patterns 85 5-17 $ 4 (2 )$ 8 (2 ) 4,369 17 • 257 $ 8(2) $ i 6(2) 60,787 89 • 683 $ i i (2) $ 22(2 )/2 3 3,133 13 ■ 241 $ 1 2 (2 )$ 2 4 (2 ) 140,911 43 • (29 • 113) $ u (2) $ 28(2) 49,981 151 • 331 $ 15(2) $ 30(2) 76,627 19 ■ (37 • 109) $ 1 8 (2)$36 (2)/3 2,528,921 41-61,681 $ 20(2) $ 4o(2 ) /5 1,826,203 337-5,419 $ 2 1 (2)$ 4 2 (2)/7 15,732,721 241 • (97 ■ 673) $ 24(2) $ 48(2) note: 11,275= 11 • (5 ■ 41) • 5 = $io(2)$2o(2)$5(2) G G. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Table 3.4: Factors of 4 ?n(2) versus members of the multiplicative module M (2 < n < n factorization of < / > n(2 ) <M2) e M'l n factorization of < j> n(2 ) < f> n (2) G M'i 2 3 Yes 32 65,537 No 3 7 Yes 33 599,479 Yes 4 5 No 34 43,691 No 5 31 Yes 35 71-122,921 Yes 6 3 Yes 36 37 • 109 No 7 127 Yes 37 223-616,318,177 Yes 8 17 No 38 174,763 No 9 73 Yes 39 79 • 121,369 Yes 1 0 1 1 No 40 61,681 No 1 1 23-89 Yes 41 164,511,353-13,367 Yes 1 2 13 No 42 5419 No 13 8,191 Yes 43 431-2,099,863-9,719 Yes 14 43 No 44 397-2,113 No 15 151 No 45 631-23,311 Yes 16 257 No 46 2, 796,203 No 17 131,071 Yes 47 13,264,529-2,351-4,513 Yes 18 3 • 19 Yes 48 97 • 673 No 19 524,287 Yes 49 4,432, 676, 798, 593 Unknown 2 0 5-41 No 50 251-4,051 No 2 1 7-337 Yes 51 103-11,119-2,143 Yes 2 2 683 No 52 53-157-1,613 No 23 47 • 178,481 Yes 53 69,431-20,394,401-6,361 Yes 24 241 No 54 3-87,211 Yes 25 601-1,801 No 55 881-3,191-201,961 Unknown 26 2,731 No 56 15,790,321 No 27 262,657 Yes 57 1,212,847-32,377 Unknown 28 29-113 No 58 59 • 3,033,169 Unknown 29 233-1,103-2,089 Yes 59 3,203,431,780,337-179,951 Yes 30 331 No 60 61•1,321 No 31 2,147,483,647 Yes 61 2,305,843,009,213,693,951 Yes Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Chapter 4 Some R elated Problem s In this chapter, some related problems are discussed. The distribution of the values of r for members of the set of all odd primes p leads to a generalization of Artin’s Conjecture concerning primitive roots modulo p. We also generalize the conjectures of Blake, Gao and Lambert (BGL) and of Tromp, Zhang and Zhao (TZZ), and prove the former generalization. 4.1 G eneralized A rtin’s Conjecture Conjecture 1 (A rtin’s P rim itive R oot C onjecture). Every nonzero integer a not equal to — 1 or a square number is a primitive root modulo p for infinitely many primes p, with a proposed density for the set of such primes p for given a which is always a rational multiple of a constant Ca known as Artin’s Constant. 39 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. A remarkable quantitative version (for each such a) of this conjecture is: Let na(x) be the number of primes less than or equal to x for which a is a primitive root. Then ira(x) is asymptotic to In 1967, Hooley [8] proved both Artin’s conjecture and the asymptotic formula for R. Gupta and M. R. Murty [6] proved, without any hypothesis, that in any set of thirteen prime numbers, A rtin’s conjecture is true for at least one of them. In 1986 Heath-Brown [7 ] refined the result to show that in any set of three prime numbers, such as {2,3,5}, Artin’s conjecture is true for at least one of them. However, no specific value of a is known, so far, for which Artin’s Conjecture has been proved unconditionally. we generalize to Cr a, where a has order (p — l)/r modulo p. When a = 2, by observing the first 1,000,000 odd primes and their corresponding indices r, we propose the following results: 1. All C 2 > 0, where 2 has order (p — l ) / r modulo p. In a; as x — > oo, where Ca (Artin’s constant) is defined by: ] = 0.3739558136. 7r0(x) subject to the assumption of the Generalized Riemann Hypothesis (GRH). In 1984, W ith 40 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 2 . If d\m, then C f > C f . 3- C2 = V$F)Ch where F2{r) £ 3 if 8 divides r 1.5 if 2 divides r but 4 does not divide r 1 otherwise and 4>{r) is the Euler’s phi-function. The numerical results relating to C\ and F2(r) (1 < r < 100) for the first 1,000,000 odd primes are tabulated in Table 4.1. It is interesting that the values of F2(r) = r ■ <p(r) ■ CljC\ seem limited to a set of only three members. A similar situation happens when a — 3, 5, and 7, where Fs(r) = 3.2 if 12 divides r 1 . 6 if 2 divides r but 6 does not divide r 1.2 if 4 divides r but 12 does not divide r 1 otherwise Fs(r) £ * 3 if 10 divides r 1.4 if 2 divides r but 10 does not divide r 0 if 5 divides r but 10 does not divide r 1 otherwise 41 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. and 3 if 28 divides r -PV(^) — 1.4 if 2 divides r but 7 does not divide r 1 otherwise. \ The numerical results relating to Fs(r), F^(r), and F^{r) (1 < r < 100) for the first 1,000,000 odd primes are tabulated in Table 4.2. Similarly, let T ra(x) be the number of primes less than or equal to x for which a has order (p — 1 )/r modulo p. Then na(x) is asymptotic to Cr ax as x — » oo. In i It is noteworthy that not all of the C£ > 0. For example, when r is an odd multiple of 5, then Cl ~ 0. In Table 4.3, we list all integers a (2 < a < 100) and their corresponding indices r (1 < r < 256) of the first 100,000 odd primes. Then we have the following results: 1. If a is a perfect square, all the indices must be even numbers. 2. If a — aiGt2 is not a perfect square, and a — 4m + 1 ( m e N), then all the indices occur except for the odd multiples of a\. 3. Otherwise, all the indices occur. Hence, it seems reasonable to generalize Artin’s conjecture for primitive roots as follows: 42 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. C o n jectu re 2 (G eneralized A rtin ’s C o n jectu re). Every nonzero integer a not equal to — 1 or a square number has each of the orders (p— 1), (p — 1)/2, and (p— l)/3 , modulo p, for infinitely many primes p. 43 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Table 4.1: Numerical results of ThW for the first 1,000,000 odd primes (1 < r < 100) r /O ’) cr ,/c$ l/r • (j> {r) F2(r) 1 374024 1.000000000 1.000000000 1.000000000 2 280792 .7507325733 .5000000000 1.501465147 3 66464 .1776998267 .1666666667 1.066198960 4 46645 .1247112485 .1250000000 .9976899880 5 18985 .5075877484e-l .5000000000e-l 1.015175497 6 49956 .1335636216 .8333333333e-l 1.602763459 7 8812 .2355998546e-l .2380952381e-l .9895193893 8 35129 .9392178042e-l ,3125000000e-l 3.005496973 9 7395 ,1977145852e-l .1851851852e-l 1.067658760 1 0 14117 .3774356726e-l .2500000000e-l 1.509742690 1 1 3353 .8964665369e-2 .9090909091e-2 .9861131906 1 2 8303 .2219911022e-l .2083333333e-l 1.065557291 13 2417 .6462152161e-2 .6410256410e-2 1.008095737 14 6691 .1788922636e-l .1190476190e-l 1.502695015 15 3376 ,9026158749e-2 .8333333333e-2 1.083139050 16 8770 .2344769320e-l .7812500000e-2 3.001304730 17 1349 .3606720424e-2 ,3676470588e-2 .9810279554 18 5466 .1461403546e-l .9259259259e-2 1.578315830 19 1061 .2836716360e-2 ,2923976608e-2 .9701569952 2 0 2434 .6507603790e-2 .6250000000e-2 1.041216606 2 1 1621 .4333946485e-2 .3968253968e-2 1.092154514 2 2 2531 ,6766945437e-2 ,4545454545e-2 1.488727996 23 783 .2093448549e-2 .1976284585e-2 1.059284966 24 6214 .1661390713e-l .5208333333e-2 3.189870169 25 719 ,1922336535e-2 .2 0 0 0 0 0 0 0 0 0 e- 2 .9611682675 26 1762 ,4710927641e-2 ,3205128205e-2 1.469809424 27 842 ,2251192437e-2 .2057613169e-2 1.094079524 28 1116 .2983765748e-2 ,2976190476e-2 1.002545291 29 431 .1152332471e-2 .1231527094e-2 .9356939661 30 2476 .6619896049e-2 ,4166666667e-2 1.588775052 31 418 .1117575343e-2 . 1075268817e-2 1.039345069 32 2173 .5809787607e-2 .1953125000e-2 2.974611255 33 622 .1662994888e-2 .1515151515e-2 1.097576626 34 995 ,2660257096e-2 ,1838235294e-2 1.447179860 35 445 .1189763224e-2 .1190476190e-2 .9994011086 36 870 .2326053943e-2 ,2314814815e-2 1.004855303 37 295 .7887194405e-3 .7507507508e-3 1.050574295 38 847 .2264560563e-2 .1461988304e-2 1.548959425 39 434 .1160353346e-2 ,1068376068e-2 1.086090732 40 1757 .4697559515e-2 .1562500000e-2 3.006438090 41 237 ,6336491776e-3 .6097560976e-3 1.039184651 42 1192 .3186961265e-2 ,1984126984e-2 1.606228478 43 224 .5988920497e-3 .5537098560e-3 1.081599042 44 444 .1187089599e-2 ,1136363636e-2 1.044638847 45 371 .9919149573e-3 .9259259259e-3 1.071268154 46 536 .1433063119e-2 .9881422925e-3 1.450259876 47 179 .4785789147e-3 .4625346901e-3 1.034687614 48 1531 .4093320215e-2 ,1302083333e-2 3.143669926 49 182 .4865997904e-3 ,4859086492e-3 1.001422369 50 584 .1561397130e-2 .1 0 0 0 0 0 0 0 0 0 e- 2 1.561397130 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Table 4.1 (Continued...) r f(r) c y c i 1 jr • 4> {r) F2(r) 51 254 .6791008064e-3 .6127450980e-3 1.108292516 52 322 ,8609073215e-3 .8012820513e-3 1.074412337 53 135 .3609394050e-3 .3628447025e-3 .9947490001 54 573 .1531987252e-2 .1028806584e-2 1.489091609 55 183 .4892734156e-3 ,4545454545e-3 1.076401514 56 833 .2227129810e-2 . 7440476190e-3 2.993262465 57 180 .4812525399e-3 .4873294347e-3 .9875302119 58 358 .9571578294e-3 .6157635468e-3 1.554424315 59 106 .2834042735e-3 .2922267680e-3 .9698094238 60 392 ,1048061087e-2 ,1041666667e-2 1.006138643 61 117 .3128141510e-3 .2732240437e-3 1.144899793 62 285 .7619831882e-3 .5376344086e-3 1.417288730 63 171 ,4571899129e-3 ,4409171076e-3 1.036906722 64 520 .1390285115e-2 .4882812500e-3 2.847303916 65 107 .2860778987e-3 .3205128205e-3 .8925630440 6 6 476 .1272645606e-2 .7575757576e-3 1.679892200 67 103 ,2753833979e-3 .2261420172e-3 1.217745385 6 8 154 ,4117382842e-3 .4595588235e-3 .8959425065 69 119 .3181614014e-3 .3293807642e-3 .9659380145 70 346 .9250743268e-3 .5952380952e-3 1.554124869 71 75 ,2005218916e-3 .2012072435e-3 .9965938011 72 637 ,1703099266e-2 .5787037037e-3 2.942955532 73 65 .1737856394e-3 ,1902587519e-3 .9134173207 74 236 .6309755524e-3 ,3753753754e-3 1.680918871 75 136 ,3636130302e-3 .3333333333e-3 1.090839091 76 119 .3181614014e-3 .3654970760e-3 .8704895943 77 79 .2112163925e-3 .2164502164e-3 .9758197336 78 315 ,8421919449e-3 .5341880342e-3 1.576583321 79 67 ,1791328899e-3 .1622849724e-3 1.103816868 80 437 .1168374222e-2 .3906250000e-3 2.991038008 81 6 8 .1818065151e-3 ,2286236854e-3 .7952216971 82 140 ,3743075311e-3 .3048780488e-3 1.227728702 83 6 6 .1764592646e-3 .1469291801e-3 1.200981755 84 184 ,4919470408e-3 ,4960317460e-3 .9917652343 85 58 .1550702629e-3 .1838235294e-3 .8435822302 8 6 162 ,4331272859e-3 .2768549280e-3 1.564455757 87 98 ,2620152717e-3 .2052545156e-3 1.276538404 8 8 310 .8288238188e-3 .2840909091e-3 2.917459842 89 61 . 1630911385e-3 .1276813075e-3 1.277329796 90 278 .7432678117e-3 .4629629630e-3 1.605458473 91 57 ,1523966376e-3 .1526251526e-3 .9985027697 92 1 1 1 .2967723996e-3 .2470355731e-3 1.201334674 93 82 .2192372682e-3 . 1792114695e-3 1.223343957 94 116 .3101405257e-3 .2312673450e-3 1.341047633 95 48 .1283340107e-3 .1461988304e-3 .8778046332 96 411 .1098859966e-2 ,3255208333e-3 3.375697816 97 42 .1122922593e-3 .1073883162e-3 1.045665518 98 141 ,3769811563e-3 .2429543246e-3 1.551654439 99 69 .1844801403e-3 .1683501684e-3 1.095812033 1 0 0 96 .2566680213e-3 .2500000000e-3 1.026672085 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Table 4.2: Numerical results of F3(r), F$(r), Fj(r) of the first 1,000,000 odd primes (1 < r < 100) r f3{r) F3(r) f5{r) * 5 (r) M r) F7(r) 1 373959 1.000000000 393818 1.000000000 374120 1.000000000 2 299639 1.602523271 265842 1.350075416 282791 1.511766278 3 66531 1.067459267 69966 1.065964481 66417 1.065171602 4 56131 1.200794739 66341 1.347647898 68468 1.464086390 5 18912 1.011447779 0 0. 18934 1.012188603 6 33183 1.064811918 47125 1.435942492 50079 1.606297445 7 8941 1.004179603 9407 1.003240075 8918 1.001165401 8 14044 1.201757412 16786 1.363960002 17135 1.465626002 9 7421 1.071598758 7778 1.066512958 7369 1.063631990 1 0 14976 1.601886838 28461 2.890776958 14333 1.532449482 1 1 3417 1.005110186 3654 1.020623740 3387 .9958569443 1 2 24992 3.207881078 11818 1.440421718 12150 1.558858121 13 2394 .9986763255 2458 .9736680396 2459 1.025350155 14 7060 1.585842299 6267 1.336729150 4572 1.026536941 15 3287 1.054768036 0 0. 3444 1.104672298 16 3379 1.156575988 4147 1.347871352 4221 1.444156955 17 1443 1.049569605 1455 1.004931212 1349 .9807762216 18 3638 1.050660634 5289 1.450446653 5625 1.623810542 19 1116 1.020625256 1135 .9856583497 1086 .9927616808 2 0 2808 1.201415128 7109 2.888237714 3439 1.470758046 2 1 1628 1.097061443 1689 1.080773352 1568 1.056174490 2 2 2724 1.602528620 2335 1.304409651 2504 1.472468727 23 738 .9985800582 779 1.000903971 703 .9508125734 24 6258 3.213015331 2911 1.419213952 2945 1.511386721 25 740 .9894132780 0 0. 720 .9622580990 26 1920 1.601886838 1679 1.330177899 1827 1.523639474 27 762 .9903010759 875 1.079813518 834 1.083406394 28 1287 1.156362061 1609 1.372776257 3325 2.986207634 29 462 1.003168796 510 1.051551731 470 1.020100502 30 1625 1.042895077 4997 3.045264564 2540 1.629423714 31 411 1.022117398 431 1.017805179 417 1.036592537 32 920 1.259603325 1030 1.339095725 1083 1.482134074 33 606 1.069529013 622 1.042410454 629 1.109643965 34 1136 1.652544798 1034 1.428314602 1064 1.547139955 35 479 1.075946828 0 0. 445 .9991446599 36 2757 3.184905297 1260 1.382161303 1363 1.573869346 37 279 .9937666963 304 1.028211001 286 1.018261520 38 865 1.582152054 791 1.373842740 844 1.543077087 39 395 .9886645327 476 1.131324623 402 1.005752166 40 664 1.136381261 1719 2.793574697 847 1.448946862 41 225 .9867391880 236 .9827890039 226 .9906981716 42 753 1.014849221 1173 1.501180749 816 1.099283653 43 2 2 1 1.067298822 205 .9401043121 229 1.105458142 44 485 1.141301587 608 1.358597119 601 1.413664066 45 384 1.108998580 0 0. 392 1.131615524 46 584 1.580408547 541 1.390215785 589 1.593253502 47 184 1.063774371 178 .9771925103 186 1.074874372 48 1541 3.164753356 715 1.394349675 805 1.652517909 49 2 1 1 1.161191467 2 0 1 1.050378601 175 .9626590399 50 643 1.719439832 1089 2.765236734 599 1.601090559 46 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Table 4.2 (Continued...) r h(T) *3(r) M r) F5(r) M r) F7(r) 51 230 1.003746400 246 1.019435374 243 1.060023522 52 363 1.211426921 424 1.343646050 449 1.497786807 53 1 1 0 .8106771061 148 1.035727163 1 2 1 .8913610605 54 428 1.112464201 552 1.362416142 605 1.571848605 55 183 1.076588610 0 0. 172 1.011440180 56 315 1.132102717 364 1.242238801 851 3.057158131 57 187 1.026112488 2 1 0 1.094211031 195 1.069549877 58 378 1.641548939 312 1.286604472 352 1.527980327 59 107 .9791287279 117 1.016647284 123 1.125056132 60 1248 3.203773675 1241 3.025153750 603 1.547311023 61 97 .9493554106 94 .8736015113 123 1.203303753 62 304 1.512037416 294 1.388560198 296 1.471613386 63 205 1.243291377 160 .9214408686 175 1.060889554 64 232 1.270556398 263 1.367697769 273 1.494450978 65 1 2 0 1.001179274 0 0. 115 .9590505719 6 6 303 1.069529012 438 1.468089320 427 1.506575430 67 8 8 1.040584663 87 .9768827223 83 .9810381695 6 8 185 1.076481647 235 1.298467820 271 1.576221533 69 131 1.063528355 136 1.048443697 142 1.152336149 70 340 1.527440174 655 2.794184116 229 1.028333155 71 95 1.262571565 92 1.161043934 61 .8103549662 72 680 3.142162643 314 1.377773489 344 1.588880573 73 72 1.011961204 73 .9742774582 81 1.137966428 74 230 1.638468388 193 1.305557389 207 1.473986956 75 147 1.179273664 0 0. 133 1.066502727 76 159 1.163293302 178 1.236632150 178 1.301742756 77 74 .9142178691 76 .8915793590 80 .9879183152 78 203 1.016196963 300 1.426039440 320 1.601197477 79 60 .9886645327 63 .9857497625 64 1.054121672 80 181 1.239066315 429 2.788699348 230 1.573826580 81 91 1.064378716 90 .9995987997 72 .8417833851 82 160 1.403362401 161 1.340923980 174 1.525499839 83 50 .9099928068 53 .9159510235 67 1.218865605 84 597 3.218406295 271 1.387280419 604 3.254741794 85 63 .9164641042 0 0. 76 1.105099968 8 6 152 1.468139555 145 1.329903661 164 1.583363627 87 99 1.289788453 94 1.162892503 69 .8985566128 8 8 136 1.280140336 146 1.304968285 153 1.439538116 89 48 1.005286675 64 1.272790984 46 .9629851378 90 195 1.126326683 559 3.065984795 256 1.478028440 91 51 .8935525022 56 .9316791006 56 .9807334548 92 125 1.353089510 1 2 0 1.233463174 125 1.352507217 93 58 .8654424682 90 1.275208345 80 1.193200043 94 140 1.618787086 119 1.306583244 130 1.502512563 95 48 .8779572090 0 0. 44 .8044477707 96 389 3.195558872 196 1.528909293 179 1.469817171 97 33 .8217371419 41 .9694630513 43 1.070287607 98 139 1.529911033 126 1.316892575 1 0 1 1.111183578 99 67 1.064234314 77 1.161399428 64 1.016144552 1 0 0 1 1 1 1.187295934 281 2.854110274 130 1.389928365 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Table 4.3: Collections of possible indices r of the first 1 0 0 ,000 odd primes ( 2 < a < 1 0 0 ) a factorization possible indices % of the least r 2 2 All .37470 3 3 All .37391 4 2 2 Even .56202 5 5 All but odd multiples of 5. .39347 6 2 -3 All .37367 7 7 All .37487 8 2 3 All .22534 9 32 Even .59924 1 0 2- 5 All .37523 1 1 1 1 All .37541 1 2 22 • 3 All .37559 13 13 All but odd multiples of 13. .37677 14 2 -7 All .37429 15 3 -5 All .37471 16 2 4 Even .37453 17 17 All but odd multiples of 17. .37578 18 2 • 32 All .37495 19 19 All .37497 2 0 22 • 5 All but odd multiples of 5. .39288 2 1 3 -7 All but odd multiples of 21. .37272 2 2 2 - 1 1 All .37682 23 23 All .37364 24 23 ■ 3 All .37498 25 52 Even .57151 26 2-13 All .37476 27 33 All but { 4, 8 , 16, 20, 28, ... }. .22436 28 22 • 7 All .37410 29 29 All but odd multiples of 29. .37338 30 2 - 3 - 5 All .37516 31 31 All .37536 32 2 5 All .29563 33 3-11 All but odd multiples of 33. .37380 34 2 • 17 All .37426 35 5 -7 All .37295 36 22 • 32 Even .56082 37 37 All but odd multiples of 37. .37510 38 2-19 All .37495 39 3-13 All .37420 40 23 • 5 All .37448 41 41 All but odd multiples of 41. .37562 42 2-3-7 All .37440 43 43 All .37396 44 2 2 • 1 1 All .37509 45 32 • 5 All but odd multiples of 5. .39469 46 2-23 All .37416 47 47 All .37404 48 24 • 3 All .37376 49 72 Even .56647 50 2 - 52 All .37500 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Table 4.3 (Continued...) a factorization possible indices r % of the least r 51 3-17 All .37443 52 22 • 13 All but odd multiples of 13. .37644 53 53 All but odd multiples of 53. .37444 54 2 • 33 All .37448 55 5-11 All .37472 56 23 • 7 All .37284 57 3-19 All but odd multiples of 57. .37388 58 2-29 All .37487 59 59 All .37520 60 22 ■ 3 • 5 All .37455 61 61 All but odd multiples of 61. .37604 62 2-31 All .37381 63 32 • 7 All .37522 64 2 6 Even .33755 65 5-13 All but odd multiples of 65. .37365 6 6 2-3-11 All .37418 67 67 All .37543 6 8 22 • 17 All but odd multiples of 17. .37581 69 3-23 All but odd multiples of 69. .37432 70 2-5-7 All .37258 71 71 All .37533 72 23 • 32 All .37533 73 73 All but odd multiples of 73. .37452 74 2-37 All .37337 75 3 • 52 All .37466 76 22 • 19 All .37389 77 7-11 All but odd multiples of 77. .37452 78 2-3-13 All .37375 79 79 All .37406 80 24 • 5 All but odd multiples of 5. .39384 81 34 Even .37483 82 2 ■ 41 All .37316 83 83 All .37314 84 22 • 3 • 7 All but odd multiples of 2 1 . .37246 85 5-17 All but odd multiples of 85. .37474 8 6 2-43 All .37592 87 3-29 All .37420 8 8 2 3 • 1 1 All .37601 89 89 All but odd multiples of 89. .37388 90 2 • 32 • 5 All .37461 91 7-13 All .37486 92 22 • 23 All .37347 93 3-31 All but odd multiples of 93. .37516 94 2-47 All .37455 95 5-19 All .37612 96 25 • 3 All .37456 97 97 All but odd multiples of 97. .37519 98 2 • 72 All .37445 99 32 ■ 11 All .37435 1 0 0 22 • 52 Even .56268 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 4.2 Generalized TZZ and BGL Conjectures In the generation of pseudorandom sequences, one may only require that GCD(xm + xk + 1, x 2 n “ 1 + 1) contain a primitive factor of degree n over GF(2). In 1994, Tromp, Zhang and Zhao [12] conjectured that GCD(xm + x k + 1. x2’l~1 + 1) is a primitive polynomial of each degree n over GF(2) for some integers m, k, and verified that the answer is yes for n < 171. In 1996, Blake, Gao and Lambert [ 2 ] extended the computational results for n up to 500, and also slightly relaxed the condition and asked: given an integer n, do there exist integers m,k such that GCD{xm + xk + 1 , .r2 ” " 1 + 1) contains a primitive factor of degree n over GF(2)? Using Theorem 4, the latter question can be easily generalized and answered. Theorem 12 (G eneralized BGL C onjecture). Every primitive polynomial f(x) divides some trinomial xm + xk + 1 for every k, 1 < k < 2 n — 1, with m ^ k, 1 < k < to < 2 n - 1. Proof. If ct is a root of f(x), then the powers 1, a 1, a 2, • ■ ■ , a 2n~2 50 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. axe all distinct, and constitute all the non-zero elements of the field GF(2T l). Hence, for all k, 1 < k < 2n — 1, 1 + x k = xm for some m ^ k, l < k < m < 2 n — 1. Thus, f(x) divides x m + x k + 1 for each such pair (k, m ). □ Hence, given an integer n, there do exist integers m, k such that GCD(xm + x k + 1, x2" - 1 + 1) contains a primitive factor of degree n over GF(2). Furthermore, there exist (2r a _ 1 — 1) such (k , m) pairs for any given primitive polynomial of degree n, and this is also true for every primitive polynomial of degree n over GF(2 ). Though Tromp, Zhang and Zhao’s question is not yet answered, it is reasonable to strengthen their conjecture as follows. Conjecture 3 (G eneralized TZZ C onjecture). Every primitive polynomial f(x) divides some trinomials xm + xk + 1 for 1 < k < m < 2n — 1 such that GCD(xm + x k + I ,* 2”- 1 + 1) = fix). 51 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. In fact, the following is known [2], T h e o rem 13. If there is a primitive polynomial /(x) of degree n for which a trinomial xm + xk + 1 exists with l< f c < m < 2 n - l and GCD(xm + x k + 1, x2n~l + 1) = /(x), then this is true for every primitive polynomial of the same degree n. Proof. Suppose that f(x) is a primitive polynomial of degree n and GCD(xm + xk + 1, x 2 " - 1 + 1) = f(x). Let a be a root of /(x) and [3 be a root of g(x), where g(x) is any primitive polynomial of degree n over GF(2). Then (3 — au for some integer u with GCD(2n — l,tt) = 1, as both a and (3 are primitive elements in GF(2n). Let v be an integer such that uv = 1 (modulo 2" — 1). Then a — (3V, and GCD(xmv + xkv + 1, X 2 " - 1 + 1) - g{x), from which p m v + p k v + 1 = a m u v + a kuv + l = a ™ + a k + I = 0] that is, (3 is a root of xmv + xkv + 1. We see that g{x) | (xmv + xkv + l ). 52 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Therefore it suffices to show that (xmv + xkv + 1 )/g(x) has no roots in GF(2n). Consider the roots of xmv + x kv + 1 and xm + x k + 1 in some extension field of GF(2). Since n— 1 f(x)= _ a2i)’ i=0 we have n — 1 Xm + Xk + 1 = Y\_{x - o ? ) - 7) *=0 7 where every 7 in the second product is not in GF(2). Hence n — 1 x mv + x kv + 1 = _ a 2i) _ 7)_ t= 0 7 Obviously, every root of xv — 7 is not in GF(2n) when 7 is not in GF(2 '1 ). The only roots of xmv + x kv + 1 which are in GF(2n) come from x v — a 2' , 0 < i < n — 1. Let rj be a primitive vth root of unity over GF(2). Then xv — o ? = xv — 2 = — /32 if). j = 0 53 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Since GCD(2n - l , v ) = 1, e GF(2n) iff v \ j. This means that /32V (£ GF(2n) for 1 < j < v — 1- Hence xv — o? has only one root, i.e. /32’ in GF(2n) for 1 < i < n — 1. As n — 1 g i x ) = 0 21), i=0 we see that (xmv + xkv + 1 )/g(x) has no roots in GF(2"), which completes the proof. □ 54 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Chapter 5 Conclusion The theory of the irreducible polynomials which divide trinomials over GF(2) has an interesting structure. All the polynomials of (odd) primitivity t are the r irreducible factors of $t(x) over GF(2), and either all or none of them divide trinomials. W hether it is all or none depends to a considerable extent on r. (For all odd t > 3 and all odd r, the answer is none. For r = 4,10,12,14, and 16, the answer is none for all odd primes t. There is only one prime value of t. for each of r — 2,6, and 8, for which the answer is all rather than none. For each even r > 16, there is only a finite range, t < 7r/2, for prime values of t where the answer might be all.) The odd values of t > 1 such that polynomials of primitivity t divide trinomials form a multiplicative module M, which is closed with respect to multiplication by odd numbers. The set G of generators of M is quite sparse, and its members seem related to numbers of the form < & n(2). The distribution of the values of r among the set of all odd primes p leads to a gener alization of Artin’s Conjecture concerning primitive roots modulo p. We generalized the 55 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. conjectures of Blake, Gao and Lambert (BGL), and of Tromp, Zhang and Zhao (TZZ). We proved the former generalization, and ascertained that the latter generalization is true if the original TZZ conjecture is true. 56 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Bibliography [1 ] E. Artin, The Collected Papers of Emil Artin, (ed. by Serge Lang and John T. Tate), Addison-Wesley 1965; Springer-Verlag 1982, viii-ix. [2] I. F. Blake, S. Gao and R. Lambert, Construction and distribution problems for ir reducible trinomials over finite fields, in Applications of Finite Fields (D. Gollmann, ed.), Oxford, Clarendon Press, 1996, 19-32. [3 ] R. P. Brent and P. Zimmermann, Algorithms for Finding Almost Irreducible and Almost Primitive Trinomials, Proceedings of a Conference in Honor of Professor H.C. Williams, The Fields Institute, May 2003. [4 ] J. Brillhart, D.H. Lehmer, J.L. Selfridge, B. Tuckerman, and S.S. Wagstaff, Jr., Factorizations of 5n±l(b = 2, 3, 5, 6, 7, 10, 11, 12) up to high powers, Contemporary Mathematics, Third Edition, Vol. 22, American Math. Soc. 2003. [5] S.W. Golomb, Shift Register Sequences, Holden-Day, Inc. 1967; Second Edition, Aegean Park Press 1982. [6] R. Gupta and M. R. Murty, A remark on A rtin’s Conjecture, Inventiones Math. 78 (1984), 127-130. [7] D.R. Heath-Brown, Artin’s conjecture for primitive roots, Quart. J. Math. Oxford 37 (1986), 27-38. [8 ] C. Hooley, Artin’s conjecture for primitive roots, J. Reine Angew. Math. 225 (1967), 209-220. [9] J. H. Lindholm, An Analysis of the Pseudo-Randomness Properties of Subsequences of Long m-Sequences, IEEE Transactions on Information Theory, Vol. IT-14, No. 4, July, 1968, 569-576. [10] G. Seroussi, Table of Low-Weight Binary Irreducible Polynomials, Computer Sys tems Laboratory, HPL-98-135, 1998. [11] R.G. Swan, Factorization of Polynomials over Finite Fields, Pacific J. Mathematics 12 (1962), 1099-1106. [12] J. Tromp, L. Zhang and Y. Zhao, Small weight bases for Hamming codes, Theoret ical Computer Science 181(2), 1997, 337-345. 57 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. [13] N. Zierler and J. Brillhart, On Primitive Trinomials (mod 2), Information and Con trol 13, 1968, 541-554. [14] N. Zierler and J. Brillhart, On Primitive Trinomials (mod 2) II, Information and Control 14, 1969, 566-569. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.

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Creator Lee, Pey-Feng (author)

Core Title Irreducible polynomials which divide trinomials over GF(2)

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Publisher University of Southern California (original), University of Southern California. Libraries (digital)

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Advisor Golomb, Solomon (committee chair), Guralnick, Robert (committee member), Lindsey, William C. (committee member)

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