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In mathematics, a Mersenne prime is a prime number that is one less than a power of two. That is, it is a prime number of the form Mn = 2n − 1 for some integer n. They are named after Marin Mersenne, a French Minim friar, who studied them in the early 17th century. If n is a composite number then so is 2n − 1. Therefore, an equivalent definition of the Mersenne primes is that they are the prime numbers of the form Mp = 2p − 1 for some prime p.
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A prime gap is the difference between two successive prime numbers. The n-th prime gap, denoted gn or g(pn) is the difference between the (n + 1)-th and the n-th prime numbers, i.e.
In mathematics, a prime triplet is a set of three prime numbers in which the smallest and largest of the three differ by 6. In particular, the sets must have the form or. With the exceptions of and, this is the closest possible grouping of three prime numbers, since one of every three sequential odd numbers is a multiple of three, and hence not prime.
In number theory, a prime k-tuple is a finite collection of values representing a repeatable pattern of differences between prime numbers. For a k-tuple, the positions where the k-tuple matches a pattern in the prime numbers are given by the set of integers n such that all of the values are prime. Typically the first value in the k-tuple is 0 and the rest are distinct positive even numbers.
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References
- ↑ Preda Mihăilescu (2004). "Primary Cyclotomic Units and a Proof of Catalan's Conjecture". J. Reine Angew. Math. 2004 (572): 167–195. doi:10.1515/crll.2004.048. MR 2076124.
- ↑ Jeanine Daems A Cyclotomic Proof of Catalan's Conjecture.
- ↑ Weisstein, Eric W. "Double Wieferich Prime Pair". MathWorld .
- ↑ OEIS: A124121 , For example, currently there are two known double Wieferich prime pairs (p, q) with q = 5: (1645333507, 5) and (188748146801, 5).
- ↑ List of all known Barker sequence