# Strong pseudoprimes to twelve prime bases

@article{Sorenson2017StrongPT,
title={Strong pseudoprimes to twelve prime bases},
author={J. Sorenson and Jonathan Webster},
journal={Math. Comput.},
year={2017},
volume={86},
pages={985-1003}
}
• Published 2 September 2015
• Computer Science, Mathematics
• Math. Comput.
Let $\psi_m$ be the smallest strong pseudoprime to the first $m$ prime bases. This value is known for $1 \leq m \leq 11$. We extend this by finding $\psi_{12}$ and $\psi_{13}$. We also present an algorithm to find all integers $n\le B$ that are strong pseudoprimes to the first $m$ prime bases; with a reasonable heuristic assumption we can show that it takes at most $B^{2/3+o(1)}$ time.
18 Citations

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#### References

SHOWING 1-10 OF 33 REFERENCES
On strong pseudoprimes to several bases
With Y'k denoting the smallest strong pseudoprime to all of the first k primes taken as bases we determine the exact values for 5, q6, q7, q8 and give upper bounds for V/9, / W t,' 1 . We discuss theExpand
Strong pseudoprimes to the first eight prime bases
• Computer Science, Mathematics
• Math. Comput.
• 2014
A 19decimal-digit number Q11 = 3825 12305 65464 13051 is found which is a strong pseudoprime to the first 11 prime bases and Z. Z. Zhang conjectured that ψ9 = ψ10 =ψ11 = Q11 and this conjecture is proved by algorithms. Expand
Two kinds of strong pseudoprimes up to 1036
Let n > 1 be an odd composite integer. Write n - 1 = 2 s d with d odd. If either b d ≡ 1 mod n or b 2r d ≡ -1 mod n for some r = 0,1,..., s - 1, then we say that n is a strong pseudoprime to base b,Expand
The Pseudosquares Prime Sieve
The pseudosquares prime sieve is presented, which finds all primes up to n in sublinear time using very little space and the primes generated by the algorithm are proven prime unconditionally. Expand
On the difficulty of finding reliable witnesses
• Mathematics, Computer Science
• ANTS
• 1994
It is shown that there are finite sets of odd composites which do not have a reliable witness, namely a common witness for all of the numbers in the set. Expand
A Wieferich Prime Search up to 6.7 × 10 15
• Mathematics
• 2011
A Wieferich prime is a prime p such that 2 p−1 ≡ 1 (mod p 2 ). Despite several intensive searches, only two Wieferich primes are known: p = 1093 and p = 3511. This paper describes a new searchExpand
A Space-Efficient Fast Prime Number Sieve
• Computer Science, Mathematics
• Inf. Process. Lett.
• 1996
A new algorithm is presented that matches the running time of the best previous prime number sieve, but uses less space by a factor of Θ ( log n ). Expand
Explicit bounds for primality testing and related problems
Many number-theoretic algorithms rely on a result of Ankeny, which states that if the Extended Riemann Hypothesis (ERH) is true, any nontrivial multiplicative subgroup of the integers modulo m omitsExpand
A Binary Recursive Gcd Algorithm
• Mathematics, Computer Science
• ANTS
• 2004
This work presents a quasi-linear time recursive algorithm that computes the greatest common divisor of two integers by simulating a slightly modified version of the binary algorithm. Expand
On the Order of Finitely Generated Subgroups of Q*(mod p) and Divisors ofp−1
Abstract LetΓbe a finitely generated subgroup of Q * with rankr. We study the size of the order |Γp| ofΓ mod pfor density-one sets of primes. Using a result on the scarcity of primesp⩽xfor whichp−1Expand