Strong pseudoprimes to twelve prime bases

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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“…This algorithm extends the Atkin-Bernstein prime sieve with our spacesaving wheel sieve. See [27,28,29].…”

Section: Previous Workmentioning

confidence: 99%

“…The very few remaining bits were tested with the base-2 strong pseudoprime test even though we had not sieved all the way to B. We also, then, replaced the use of the pseudosquares prime test with strong pseudoprime tests [21] using results from [29] so that only a few bases were needed, due to the spotty trial-division information. 4.1.…”

Section: Computationsmentioning

confidence: 99%

Two algorithms to find primes in patterns

Sorenson

Webster

2020

Math. Comp.

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“…For the Fermat case we state known results for completeness, while for the Lucas case we state and prove the required results. We follow the notation in [SW17] when possible.…”

Section: Algorithmic Theorymentioning

confidence: 99%

Fast tabulation of challenge pseudoprimes

Shallue

Webster

2019

Open Book Series

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We provide a new algorithm for tabulating composite numbers which are pseudoprimes to both a Fermat test and a Lucas test. Our algorithm is optimized for parameter choices that minimize the occurrence of pseudoprimes, and for pseudoprimes with a fixed number of prime factors. Using this, we have confirmed that there are no PSW challenge pseudoprimes with two or three prime factors up to 2 80 . In the case where one is tabulating challenge pseudoprimes with a fixed number of prime factors, we prove our algorithm gives an unconditional asymptotic improvement over previous methods.

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“…New Results. We adapted the sieving techniques from [16,12] to use the space-saving wheel sieve, which was described in [17], and was used previously to find pseudosquares [18], pseudoprimes [19], and primes in patterns [20]. Our resulting algorithm has, so far, verified all previous computations for g(k), and extended them for all k ≤ 323.…”

Section: Introductionmentioning

confidence: 99%

An Algorithm and Estimates for the Erdős-Selfridge Function (work in progress)

Sorenson,

Webster

2019

Preprint

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Let p(n) denote the smallest prime divisor of the integer n. Define the function g(k) to be the smallest integer > k + 1 such that p( g(k) k ) > k. So we have g(2) = 6 and g(3) = g(4) = 7. In this paper we present the following new results on the Erdős-Selfridge function g(k):(1) We present a new algorithm to compute the value of g(k), and use it to both verify previous work [1,16,12] and compute new values of g(k), with our current limit being g(323) = 1 69829 77104 46041 21145 63251 22499.(2) We define a new function ĝ(k), and under the assumption of our uniform distribution heuristic we show that

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Strong pseudoprimes to twelve prime bases

Cited by 10 publications

References 21 publications

Two algorithms to find primes in patterns

Two algorithms to find primes in patterns

Fast tabulation of challenge pseudoprimes

An Algorithm and Estimates for the Erdős-Selfridge Function (work in progress)

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