Integers free of small prime factors in arithmetic progressions

Xuan, Ti Zuo

doi:10.1017/s0027763000007212

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Cited by 6 publications

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“…holds uniformly in the range 1 ≤ u ≤ 4y 1/2 / log y, y ≥ 2, with c 1 a constant. And Xuan [12] (Corollary 1) showed that if D 2 > 0 is fixed and (a, q) = 1, then…”

Section: Preliminary Lemmasmentioning

confidence: 99%

On sums of sparse prime subsets

Cui

Xue

2013

Acta. Math. Sin.-English Ser.

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“…holds uniformly in the range 1 ≤ u ≤ 4y 1/2 / log y, y ≥ 2, with c 1 a constant. And Xuan [12] (Corollary 1) showed that if D 2 > 0 is fixed and (a, q) = 1, then…”

Section: Preliminary Lemmasmentioning

confidence: 99%

On sums of sparse prime subsets

Cui

Xue

2013

Acta. Math. Sin.-English Ser.

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Limitations to equidistribution in arithmetic progressions

Savalia

Vatwani

2023

Res Math Sci

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In a recent paper, Bringmann, Craig, Ono, and the author showed that the number of t-hooks (t ≥ 2) among all partitions of n is not always asymptotically equidistributed on congruence classes a (mod b). In this short note, we clarify the situation of t = 1, i.e. all hook lengths, and show that this case does give asymptotic equidistribution, closing the story of the distribution properties of t-hooks on congruence classes.

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Two algorithms to find primes in patterns

Sorenson

Webster

2020

Math. Comp.

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Let k ≥ 1 be an integer, and let P = (f1(x), . . . , f k (x)) be k admissible linear polynomials over the integers, or the pattern. We present two algorithms that find all integers x where max {fi(x)} ≤ n and all the fi(x) are prime.• Our first algorithm takes at most OP (n/(log log n) k ) arithmetic operations using O(k √ n) space. • Our second algorithm takes slightly more time, OP (n/(log log n) k−1 ) arithmetic operations, but uses only exp O(log n/ log log n) space. This result is unconditional for k > 6; for 2 < k ≤ 6, the proof of its running time, but not the correctness of its output, relies on an unproven but reasonable conjecture due to Bach and Huelsbergen. We are unaware of any previous complexity results for this problem beyond the use of a prime sieve.We also implemented several parallel versions of our second algorithm to show it is viable in practice. In particular, we found some new Cunningham chains of length 15, and we found all quadruplet primes up to 10 17 .

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Integers free of small prime factors in arithmetic progressions

Abstract: Abstract. For real x ≥ y ≥ 2 and positive integers a, q, let Φ(x, y; a, q) denote the number of positive integers ≤ x, free of prime factors ≤ y and satisfy-

Cited by 6 publications

References 14 publications

On sums of sparse prime subsets

On sums of sparse prime subsets

Limitations to equidistribution in arithmetic progressions

Two algorithms to find primes in patterns

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