A lower bound for the least prime in an Arithmetic progression

Li, Junxian; Pratt, Kyle; Shakan, George

doi:10.1093/qmath/hax001

Cited by 14 publications

(11 citation statements)

References 20 publications

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“…For large x, computations of [12, section 3.4] suggest that τ is strictly greater than one; with this in mind, we expect at most finitely many exceptions to inequality (14) for any fixed q. In section 3 we will compare this heuristic prediction with results of computations.…”

Section: 2mentioning

confidence: 94%

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On the nth record gap between primes in an arithmetic progression

Kourbatov¹

2018

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Let q > r ≥ 1 be coprime integers. Let R(n, q, r) be the nth record gap between primes in the arithmetic progression r, r + q, r + 2q, . . . , and denote by N q,r (x) the number of such records observed below x. For x → ∞, we heuristically argue that if the limit of N q,r (x)/ log x exists, then the limit is 2. We also conjecture that R(n, q, r) = O q (n 2 ). Numerical evidence supports the conjectural a.s. upper bound R(n, q, r) < ϕ(q)n 2 + (n + 2)q log 2 q.The median (over r) of R(n, q, r) grows like a quadratic function of n; so do the mean and quartile points of R(n, q, r). For fixed values of q ≥ 200 and n ≈ 10, the distribution of R(n, q, r) is skewed to the right and close to both Gumbel and lognormal distributions; however, the skewness appears to slowly decrease as n increases. The existence of a limiting distribution of R(n, q, r) is an open question.Mathematics Subject Classification: 11N05, 11N13, 11N56

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Section: 2mentioning

confidence: 94%

“…The above is valid for large n and x. To make estimate (12) applicable to moderate n, we add a semi-empirical correction term of size O q (n) (motivated in part by heuristics of [14]):…”

Section: The Nth Record Gap R(n Q R)mentioning

confidence: 99%

On the nth record gap between primes in an arithmetic progression

Kourbatov¹

2018

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“…Both heuristic probabilistic arguments and numerical evidence indicate that P(k) satisfies the much sharper bound P(k) = O(𝜑(k) log 2 k) = O(k log 2 k), where 𝜑 stands for Euler's totient function. More precisely, based on such evidence [38], Li, Pratt and Shakan conjecture that…”

Section: The Smallest Prime Number In An Arithmetic Progressionmentioning

confidence: 99%

Polynomial Multiplication over Finite Fields in Time \( O(n \log n \)

Harvey

Hoeven

2022

J. ACM

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Assuming a widely believed hypothesis concerning the least prime in an arithmetic progression, we show that polynomials of degree less than \( n \) over a finite field \( \mathbb {F}_q \) with \( q \) elements can be multiplied in time \( O (n \log q \log (n \log q)) \) , uniformly in \( q \) . Under the same hypothesis, we show how to multiply two \( n \) -bit integers in time \( O (n \log n) \) ; this algorithm is somewhat simpler than the unconditional algorithm from the companion paper [ 22 ]. Our results hold in the Turing machine model with a finite number of tapes.

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“…GRH implies that L ď 2 `ε for every ε ą 0 and it is known that L ď 5, see [33]. Furthermore, one cannot have L ă 1, see [26] for accurate lower bounds. Theorem 1.5 shows that the analogue of the Linnik constant for polynomials of given degree is at most 1 `ε for every ε ą 0.…”

Section: Introductionmentioning

confidence: 99%