In number theory, a Wieferich prime is a prime number p such that p2 divides 2p − 1 − 1,[4] therefore connecting these primes with Fermat's little theorem, which states that every odd prime p divides 2p − 1 − 1. Wieferich primes were first described by Arthur Wieferich in 1909 in works pertaining to Fermat's Last Theorem, at which time both of Fermat's theorems were already well known to mathematicians.[5][6]
Since then, connections between Wieferich primes and various other topics in mathematics have been discovered, including other types of numbers and primes, such as Mersenne and Fermat numbers, specific types of pseudoprimes and some types of numbers generalized from the original definition of a Wieferich prime. Over time, those connections discovered have extended to cover more properties of certain prime numbers as well as more general subjects such as number fields and the abc conjecture.
As of April 2023[update], the only known Wieferich primes are 1093 and 3511 (sequence A001220 in the OEIS).
^Franco, Z.; Pomerance, C. (1995), "On a conjecture of Crandall concerning the qx + 1 problem" (PDF), Mathematics of Computation, 64 (211): 1333–36, Bibcode:1995MaCom..64.1333F, doi:10.2307/2153499, JSTOR 2153499.
^Banks, W.D.; Luca, F.; Shparlinski, I.E. (2007), "Estimates for Wieferich numbers" (PDF), The Ramanujan Journal, 14 (3): 361–378, doi:10.1007/s11139-007-9030-z, S2CID 39279379, archived from the original (PDF) on 2013-05-03, retrieved 2011-03-12.
^McIntosh, R.J.; Roettger, E.L. (2007), "A search for Fibonacci–Wieferich and Wolstenholme primes" (PDF), Mathematics of Computation, 76 (260): 2087–2094, Bibcode:2007MaCom..76.2087M, CiteSeerX 10.1.1.105.9393, doi:10.1090/S0025-5718-07-01955-2
^The Prime Glossary: Wieferich prime
^Israel Kleiner (2000), "From Fermat to Wiles: Fermat's Last Theorem Becomes a Theorem", Elemente der Mathematik, 55: 21, doi:10.1007/PL00000079, S2CID 53319514.
^Leonhard Euler (1736), "Theorematum quorundam ad numeros primos spectantium demonstratio" (PDF), Novi Comm. Acad. Sci. Petropol. (in Latin), 8: 33–37.
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