Wieferich past and Future
نویسنده
چکیده
Fermat’s Last Theorem (FLT) is the assertion that for n ≥ 3, the equation X + y = Z has no solutions in integers X,Y, Z with XY Z 6= 0. It was proven by Fermat for n = 4 and by Euler for n = 3, cf. [Weil, page 104]. To prove it in general, then, it suffices to prove it when n is any prime p ≥ 5. For fixed p, the ”first case” of FLT is the assertion that there are no integer solutions with XY Z prime to p. In 1909, Arthur Wieferich, then a 25 year old student at Munster, astounded the mathematical world with the following theorem.
منابع مشابه
Estimates for Wieferich Numbers
We define Wieferich numbers to be those odd integers w ≥ 3 that satisfy the congruence 2φ(w) ≡ 1 (mod w2). It is clear that the distribution of Wieferich numbers is closely related to the distribution of Wieferich primes, and we give some quantitative forms of this statement. We establish several unconditional asymptotic results about Wieferich numbers; analogous results for the set of Wieferic...
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A Wieferich prime is a prime p such that 2p−1 ≡ 1 (mod p2). Despite several intensive searches, only two Wieferich primes are known: p = 1093 and p = 3511. This paper describes a new search algorithm for Wieferich primes using double-precision Montgomery arithmetic and a memoryless sieve, which runs significantly faster than previously published algorithms, allowing us to report that there are ...
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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...
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