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High Quality Content by WIKIPEDIA articles! In mathematics, a theorem is a statement which has been proved on the basis of previously established statements, such as other theorems, and previously accepted statements, such as axioms. The derivation of a theorem is often interpreted as a proof of the truth of the resulting expression, but different deductive systems can yield other interpretations, depending on the meanings of the derivation rules. The proofs of theorems have two components, called the hypotheses and the conclusions. The proof of a mathematical theorem is a logical argument demonstrating that the conclusions are a necessary consequence of the hypotheses, in the sense that if the hypotheses are true then the conclusions must also be true, without any further assumptions. The concept of a theorem is therefore fundamentally deductive, in contrast to the notion of a scientific theory, which is empirical. ... Read more

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High Quality Content by WIKIPEDIA articles! In number theory, a Wieferich prime is a prime number p such that p2 divides 2p ? 1 ? 1; compare this 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. The only known Wieferich primes are 1093 and 3511 (sequence A001220 in OEIS), found by W. Meissner in 1913 and N. G. W. H. Beeger in 1922, respectively. If any other Wieferich primes exist, they must be greater than 6.7 × 1015. It has been conjectured that only finitely many Wieferich primes exist. It has also been conjectured (as for Wilson primes) that infinitely many Wieferich primes exist, and that the number of Wieferich primes below x is approximately log log x, which is the heuristic result followed from a plausible assumption that for a prime p, the (p-1)-th degree roots of unity modulo p2 are uniformly distributed in the multiplicative group of integers modulo p2. ... Read more

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High Quality Content by WIKIPEDIA articles! High Quality Content by WIKIPEDIA articles! Zhi-Wei Sun is a Chinese mathematician, working primarily on number theory, combinatorics, and group theory. Currently he works as a professor in Nanjing University.Born in Huai'an, Jiangsu, Sun and his twin brother Sun Zhihong proved a theorem about what are now known as the Wall-Sun-Sun primes that guided the search for counterexamples to Fermat's last theorem. In 2003, he presented a unified approach to three famous topics of Paul Erd?s in combinatorial number theory: covering systems, restricted sumsets, and zero-sum problems or EGZ Theorem. ... Read more

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High Quality Content by WIKIPEDIA articles! A pseudoprime is a probable prime (an integer which shares a property common to all prime numbers) which is not actually prime. Pseudoprimes can be classified according to which property they satisfy. The most important class of pseudoprimes come from Fermat's little theorem and hence are called Fermat pseudoprimes. This theorem states that if p is prime and a is coprime to p, then ap?1 ? 1 is divisible by p. If a number x is not prime, a > 1 is coprime to x and x divides ax?1 ? 1, then x is called a pseudoprime to base a. Some sources use variations of this definition, for example to only allow odd numbers to be pseudoprimes. A number x that is a pseudoprime for all values of a that are coprime to x is called a Carmichael number. ... Read more