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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.
A twin prime is a prime number that is either 2 less or 2 more than another prime number—for example, either member of the twin prime pair. In other words, a twin prime is a prime that has a prime gap of two. Sometimes the term twin prime is used for a pair of twin primes; an alternative name for this is prime twin or prime pair.
In number theory, a Sierpiński number is an odd natural number k such that is composite for all natural numbers n. In 1960, Wacław Sierpiński proved that there are infinitely many odd integers k which have this property.
In number theory, a lucky number is a natural number in a set which is generated by a certain "sieve". This sieve is similar to the Sieve of Eratosthenes that generates the primes, but it eliminates numbers based on their position in the remaining set, instead of their value.
In number theory, a weird number is a natural number that is abundant but not semiperfect.
In mathematics, a Riesel number is an odd natural number k for which is composite for all natural numbers n. In other words, when k is a Riesel number, all members of the following set are composite:
In number theory, a regular prime is a special kind of prime number, defined by Ernst Kummer in 1850 to prove certain cases of Fermat's Last Theorem. Regular primes may be defined via the divisibility of either class numbers or of Bernoulli numbers.
61 (sixty-one) is the natural number following 60 and preceding 62.
In mathematics, and more particularly in number theory, primorial, denoted by "#", is a function from natural numbers to natural numbers similar to the factorial function, but rather than successively multiplying positive integers, the function only multiplies prime numbers.
In mathematics, an aliquot sequence is a sequence of positive integers in which each term is the sum of the proper divisors of the previous term. If the sequence reaches the number 1, it ends, since the sum of the proper divisors of 1 is 0.
An untouchable number is a positive integer that cannot be expressed as the sum of all the proper divisors of any positive integer. That is, these numbers are not in the image of the aliquot sum function. Their study goes back at least to Abu Mansur al-Baghdadi, who observed that both 2 and 5 are untouchable.
In mathematics, Euclid numbers are integers of the form En = pn# + 1, where pn# is the nth primorial, i.e. the product of the first n prime numbers. They are named after the ancient Greek mathematician Euclid, in connection with Euclid's theorem that there are infinitely many prime numbers.
In mathematics, a colossally abundant number is a natural number that, in a particular, rigorous sense, has many divisors. Formally, a number n is said to be colossally abundant if there is an ε > 0 such that for all k > 1,
A Pierpont prime is a prime number of the form
In mathematics, the Mersenne conjectures concern the characterization of prime numbers of a form called Mersenne primes, meaning prime numbers that are a power of two minus one.
In recreational number theory, a minimal prime is a prime number for which there is no shorter subsequence of its digits in a given base that form a prime. In base 10 there are exactly 26 minimal primes:
The Euclid–Mullin sequence is an infinite sequence of distinct prime numbers, in which each element is the least prime factor of one plus the product of all earlier elements. They are named after the ancient Greek mathematician Euclid, because their definition relies on an idea in Euclid's proof that there are infinitely many primes, and after Albert A. Mullin, who asked about the sequence in 1963.
In number theory, primes in arithmetic progression are any sequence of at least three prime numbers that are consecutive terms in an arithmetic progression. An example is the sequence of primes, which is given by for .
In mathematics, Euler's idoneal numbers are the positive integers D such that any integer expressible in only one way as x2 ± Dy2 is a prime power or twice a prime power. In particular, a number that has two distinct representations as a sum of two squares is composite. Every idoneal number generates a set containing infinitely many primes and missing infinitely many other primes.
References
- ↑ Guy, Richard K. (1994). Unsolved problems in number theory (2nd ed.). Springer. pp. 7–8. ISBN 0-387-94289-0.
- Chris Caldwell, "The Prime Glossary: Fortunate number" at the Prime Pages.
- Weisstein, Eric W. "Fortunate Prime". MathWorld .
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