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

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Table of ContentsDedicationAcknowledgementsList of TablesList of FiguresAbstractiiiiiviviiviii1 Introduction 11.1 Motivations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Outline of the thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 Theorems and Results 72.1 Basic Theorems and Results . . . . . . . . . . . . . . . . . . . . . . . . . . 72.2 Further Theorems and Results . . . . . . . . . . . . . . . . . . . . . . . . 163 The Multiplicative Module 303.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 303.2 Prime Generators of the Multiplicative Module M . . . . . . . . . . . . . 313.3 Composite Generators of the Multiplicative Module M . . . . . . . . . . . 344 Some Related Problems 394.1 Generalized Artin’s Conjecture . . . . . . . . . . . . . . . . . . . . . . . . 394.2 Generalized TZZ and BGL Conjectures . . . . . . . . . . . . . . . . . . . 505 Conclusion 55Bibliography 57v
List of Tables2.1 Frequency distribution of the index r of the first 1, 000, 000 odd primes(1 ≤ r ≤ 100) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 283.1 Prime generators of the multiplicative module M (non-Mersenne primes) 343.2 Composite generators of the multiplicative module M (g | Φ n (2)) . . . . . 373.3 Composite generators of the multiplicative module M (g | Φ n (2)Φ 2n (2)) . 373.4 Factors of Φ n (2) versus members of the multiplicative module M (2 ≤n ≤ 61) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 384.1 Numerical results of F 2 (r) for the first 1, 000, 000 odd primes (1 ≤ r ≤ 100) 444.2 Numerical results of F 3 (r), F 5 (r), F 7 (r) of the first 1, 000, 000 odd primes(1 ≤ r ≤ 100) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 464.3 Collections of possible indices r of the first 100, 000 odd primes (2 ≤ a ≤ 100) 48vi
Page 2: IRREDUCIBLE POLYNOMIALS WHICH DIVID
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 44 and 45: Table 3.1: Prime generators of the
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
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