منابع مشابه
A search for Wieferich and Wilson primes
An odd prime p is called a Wieferich prime if 2p−1 ≡ 1 (mod p); alternatively, a Wilson prime if (p− 1)! ≡ −1 (mod p). To date, the only known Wieferich primes are p = 1093 and 3511, while the only known Wilson primes are p = 5, 13, and 563. We report that there exist no new Wieferich primes p < 4×1012 , and no new Wilson primes p < 5×108. It is elementary that both defining congruences above h...
متن کاملOn Wieferich primes and p-adic logarithms∗†
We shall make a slight improvement to a result of p-adic logarithms which gives a nontrivial upper bound for the exponent of p dividing the Fermat quotient x − 1.
متن کاملThe continuing search for Wieferich primes
A prime p satisfying the congruence 2p−1 ≡ 1 (mod p) is called a Wieferich prime. Although the number of Wieferich primes is believed to be infinite, the only ones that have been discovered so far are 1093 and 3511. This paper describes a search for further solutions. The search was conducted via a large scale Internet based computation. The result that there are no new Wieferich primes less th...
متن کاملOverpseudoprimes, Mersenne Numbers and Wieferich Primes
We introduce a new class of pseudoprimes-so called ”overpseudoprimes” which is a special subclass of super-Poulet pseudoprimes. Denoting via h(n) the multiplicative order of 2 modulo n,we show that odd number n is overpseudoprime if and only if the value of h(n) is invariant of all divisors d > 1 of n. In particular, we prove that all composite Mersenne numbers 2 − 1, where p is prime, and squa...
متن کاملThe abc conjecture and non-Wieferich primes in arithmetic progressions
Article history: Received 18 June 2012 Revised 12 September 2012 Accepted 6 October 2012 Available online xxxx Communicated by Greg Martin MSC: 11A41 11B25
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ژورنال
عنوان ژورنال: Journal of Number Theory
سال: 2000
ISSN: 0022-314X
DOI: 10.1006/jnth.1999.2471