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

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This occurs if and only if 2 is a primitive root modulo t with t prime. Examplesincludet = 5, 11, 13, 19, 29, 37, 53, 59, 61, 67, 83, 101, . . . .By Artin’s Conjecture [1], there are infinitely many primes p for which 2 is a primitiveroot (i.e. where 2 is a generator of the multiplicative group of GF(p)). Artin’s Conjectureincludes the assertion that every prime q is a primitive root — i.e. a generator of themultiplicative group — modulo p, for infinitively many primes p. This has been proved forall 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 = 2is an exception. Hence, we believe there are infinitely many irreducible polynomials ofthis kind which never divide trinomials.Definition 1. The t th cyclotomic polynomial is given byΦ t (x) = ∏ d|t(x t/d − 1) µ(d) (2.1)where µ(d) is the Möbius mu-function.For any odd integer t > 3, letΦ t (x) = f 1 (x)f 2 (x) . . . f r (x)be the factorization of the t th cyclotomic polynomial into irreducible factors over GF(2).12
It is known that all the f i (x)’s have the same degree (say, n) and the same primitivityt. These factors are all the irreducible polynomials having primitivity t.Theorem 6. If any one of the f i (x)’s divides a trinomial, then all r of the f i (x)’s dividetrinomials.Proof. Collectively, the roots of the polynomialsf 1 (x), f 2 (x), . . . , f r (x)are all the powersα, α 2 , α 3 , . . . , α t−1 ,of a single root α of Φ t (x), which can be taken to be a root of any one of the polynomialsf i (x). Also, the roots of α t = 1 always form a cyclic group under multiplication. If α isa primitive root of α t = 1, then every other primitive root will be a power of α. Supposef i (x) divides the trinomial x m + x a + 1. Thenα m + α a + 1 = 0,where we selected α to be a root of f i (x). For any other polynomial f j (x) from the setof divisors of Φ t (x), let one of its roots be β = α u , withGCD(t, u) = 1 (i.e. rt + su = 1 for some r, s).13
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 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

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

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