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April 8

Constructing Wieferich numbers according to Agoh, Dilcher and Skula

I am trying to understand how to use the method in https://doi.org/10.1006/jnth.1997.2162 to construct composite base-a Wieferich numbers from a given integer a > 1 and a base-a Wieferich prime p. In section 6. EXAMPLES the authors use Theorem 5.5 to construct the list of base-2 Wieferich numbers based on the two known Wieferich primes 1093 and 3511. They compute the prime factorizations of 1092 and 3510 and apply Theorem 5.5 to obtain the 104 known base-2 Wieferich numbers. How does this work for arbitrary Wieferich primes p and bases a? For example, suppose I have the prime number p = 5209 which is a Wieferich prime to base a = 1359624 (see https://oeis.org/A143548 comment no. 3 by T. D. Noe). 5208 = 2³ * 3 * 7 * 31. How does one obtain the base-1359624 Wieferich numbers generated by 5209 with Theorem 5.5? I also find the notation in the paper a bit confusing: q(a, m) is an Euler quotient per definition 1.2, but I don't really understand the definition of σ(a, p) at the top of p. 46. Is this simply the multiplicative order of a (mod p)? Toshio Yamaguchi (talk) 11:40, 8 April 2024 (UTC)[reply]

Is your question related to [1]? Which is the largest known Wieferich numbers to bases b<=25 2402:7500:916:7991:1CB3:D7F2:BD28:3F3A (talk) 08:11, 9 April 2024 (UTC)[reply]

Mhm, I guess it is related. If I understand correctly, Theorem 2 in that paper could be used to generate a Wieferich number w based on a given Wieferich prime p, although I have yet to understand how exactly that would work. Toshio Yamaguchi (talk) 09:22, 9 April 2024 (UTC)[reply]

A further question: On p. 2 of the Akbari, Siavashi paper there is a following statement I do not understand: "For a prime p and positive integer n, we denote the largest power of p in n by v_p(n)". What does "power of p in n" mean? Toshio Yamaguchi (talk) 09:38, 9 April 2024 (UTC)[reply]