Irreducible Polynomials Which Divide Trinomials Over GF(2). - The ...

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Table 3.1: Prime generators of the multiplicative module M (non-Mersenne primes)generators patterns73 Φ 9 (2)121,369 Φ 39 (2)/79178,481 Φ 23 (2)/47262,657 Φ 27 (2)599,479 Φ 33 (2)3.3 Composite Generators of the Multiplicative Module MIf g ∈ G is composite, then (by the definition of G) no prime factor of g is in G. <strong>The</strong>smallest composite g ∈ G is 85 = 5 · 17 , where 85 ∈ G but 5 ∉ G and 17 ∉ G. Allthe eight irreducible factors of Φ 85 (x) divide trinomials, even though none of them is atrinomial. By complete computer search, there are ten composite elements of G up tot ≤ 1, 000, 000. <strong>The</strong>se 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}.34
Among these seventeen composite elements of G known so far, most of them areeither divisors of Φ n (2) or of Φ n (2)Φ 2n (2) for various values of n. For example,2, 047 = 23 · 89 =Φ 11 (2),8, 727, 391 = 71 · 122, 921 =Φ 35 (2),14, 709, 241 = 631 · 23, 311 =Φ 45 (2)are of the form Φ n (2);486, 737 = 233 · 2, 089 =Φ 29 (2)/1, 103,2, 304, 167 = 1, 103 · 2, 089 =Φ 29 (2)/233,23, 828, 017 = 11, 119 · 2, 143 =Φ 51 (2)/103are of the form Φ n (2)/c, where c is a prime factor of Φ n (2);85 = 5 · 17 =Φ 4 (2)Φ 8 (2),3, 133 = 13 · 241 =Φ 12 (2)Φ 24 (2),4, 369 = 17 · 257 =Φ 8 (2)Φ 16 (2),49, 981 = 151 · 331 =Φ 15 (2)Φ 30 (2),140, 911 = 43 · (29 · 113) =Φ 14 (2)Φ 28 (2),15, 732, 721 = 241 · (97 · 673) =Φ 24 (2)Φ 48 (2)are of the form Φ n (2)Φ 2n (2);35
Page 2: IRREDUCIBLE POLYNOMIALS WHICH DIVID
Page 6 and 7: Table of ContentsDedicationAcknowle
Page 8 and 9: List of Figures1.1 The n-stage bina
Page 10 and 11: numbers. The set G of generators of
Page 12 and 13: and the initial statea −1 , a −
Page 14 and 15: for each n ≥ 1.(Here φ(n) is the
Page 16 and 17: Φ t (x) over GF(2). We show how th
Page 18 and 19: Conversely, if h(α) = 0, we can di
Page 20 and 21: Hence, if f(x) divides no trinomial
Page 22 and 23: This occurs if and only if 2 is a p
Page 24 and 25: Then for some s, 1 ≤ s ≤ t −
Page 26 and 27: This computationally useful criteri
Page 28 and 29: ut3(p − 1)/2r < pfor all r > 1, w
Page 30 and 31: Corollary 1. Let p > 3 be a prime a
Page 32 and 33: Note that when r = 2 and p = 7, nei
Page 34 and 35: Thent 1 (x)t ∗ 1(x)t 3 (x)t ∗ 3
Page 36 and 37: We may therefore reduce all exponen
Page 38 and 39: Table 2.1: Frequency distribution o
Page 40 and 41: Chapter 3The Multiplicative ModuleI
Page 42 and 43: It was mentioned that among the eig
Page 46 and 47: 60, 787 = 89 · 683 =Φ 11 (2)Φ 22
Page 48 and 49: Table 3.4: Factors of Φ n (2) vers
Page 50 and 51: A remarkable quantitative version (
Page 52 and 53: and⎧⎪⎨F 7 (r) ∼ =⎪⎩3 if
Page 54 and 55: Table 4.1: Numerical results of F 2
Page 56 and 57: Table 4.2: Numerical results of F 3
Page 58 and 59: Table 4.3: Collections of possible
Page 60 and 61: 4.2 Generalized TZZ and BGL Conject
Page 62 and 63: In fact, the following is known [2]
Page 64 and 65: Since GCD(2 n − 1, v) = 1, η j
Page 66 and 67: conjectures of Blake, Gao and Lambe
Page 68: [13] N. Zierler and J. Brillhart, O

Irreducible Polynomials Which Divide Trinomials Over GF(2). - The ...

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