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

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A remarkable quantitative version (for each such a) of this conjecture is: Let π a (x)be the number of primes less than or equal to x for which a is a primitive root.<strong>The</strong>n π a (x) is asymptotic toC a x, as x → ∞,ln xwhere C a (Artin’s constant) is defined by:C a =∞∑ 1[1 −p(p − 1) ] = 0.3739558136...p=2In 1967, Hooley [8] proved both Artin’s conjecture and the asymptotic formula forπ a (x) subject to the assumption of the Generalized Riemann Hypothesis (GRH). In 1984,R. Gupta and M. R. Murty [6] proved, without any hypothesis, that in any set of thirteenprime numbers, Artin’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 isknown, so far, for which Artin’s Conjecture has been proved unconditionally.WithC a = Ca,1we generalize to C r a, where a has order (p−1)/r modulo p. When a = 2, by observing thefirst 1, 000, 000 odd primes and their corresponding indices r, we propose the followingresults:1. All C r 2 > 0, where 2 has order (p − 1)/r modulo p. 40
2. If d|m, then C d 2 > Cm 2 .3. C r 2 = F 2(r)r·φ(r) C1 2 , where⎧⎪⎨F 2 (r) ∼ =⎪⎩3 if 8 divides r1.5 if 2 divides r but 4 does not divide r1 otherwiseand φ(r) is the Euler’s phi-function.<strong>The</strong> numerical results relating to C2 r and F 2(r) (1 ≤ r ≤ 100) for the first 1, 000, 000odd primes are tabulated in Table 4.1.It is interesting that the values ofF 2 (r) = r · φ(r) · C r 2/C 1 2seem limited to a set of only three members. A similar situation happens when a = 3,5, and 7, where⎧⎪⎨F 3 (r) ∼ =⎪⎩3.2 if 12 divides r1.6 if 2 divides r but 6 does not divide r1.2 if 4 divides r but 12 does not divide r1 otherwise⎧⎪⎨F 5 (r) ∼ =⎪⎩3 if 10 divides r1.4 if 2 divides r but 10 does not divide r0 if 5 divides r but 10 does not divide r1 otherwise41
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 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 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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