The first perfect number is 6, because 1, 2, and 3 are its proper positive divisors, and 1 + 2 + 3 = 6. Equivalently, the number 6 is equal to half the sum of all its positive divisors: ( 1 + 2 + 3 + ...
Many fundamental questions about Mersenne primes remain unresolved. It is not even known whether the set of Mersenne primes is finite or infinite. The Lenstra–Pomerance–Wagstaff conjecture asserts t ...
In number theory, the Gaussian moat problem asks whether it is possible to find an infinite sequence of distinct Gaussian prime numbers such that the difference between consecutive numbers in the sequ ...
The question of whether there exist infinitely many twin primes has been one of the great open questions in number theory for many years. This is the content of the twin prime conjecture, which states ...
Write M(p) instead of M_p. A special case of the double Mersenne numbers, namely the recursively defined sequence2, M(2), M(M(2)), M(M(M(2))), M(M(M(M(2)))), ... (sequence A007013 in OEIS)is called th ...
There are a number of conditions which are equivalent to a and b being coprime:No prime number divides both a and b.There exist integers x and y such that ax + by = 1 (see Bézout's identity).The inte ...
Amicable numbers were known to the Pythagoreans, who credited them with many mystical properties. A general formula by which some of these numbers could be derived was invented circa 850 by the Iraqi ...
In mathematics, Lehmer's totient problem, named for D. H. Lehmer, asks whether there is any composite number n such that Euler's totient function φ(n) divides n ? 1. This is true of every prime numb ...
In number theory, Tunnell's theorem gives a partial resolution to the congruent number problem, and under the Birch and Swinnerton-Dyer conjecture, a full resolution. The congruent number problem asks ...
The question of determining whether a given rational number is a congruent number is called the congruent number problem. This problem has not (as of 2012) been brought to a successful resolution. Tun ...
This means the following: take a point (α,β) in the plane, and then consider the sequence of points(2α,2β), (3α,3β), ... .For each of these consider the closest lattice point, as determined by m ...
Pairs of the numbers (n, m) that solve Brocard's problem are called Brown numbers. There are only three known pairs of Brown numbers:(4,5), (5,11), and (7,71).Paul Erd?s conjectured that no other sol ...
In mathematics, the generalized taxicab number Taxicab(k, j, n) is the smallest number which can be expressed as the sum of j kth positive powers in n different ways. For k = 3 and j = 2, they coincid ...
A number that belongs to a singleton club, because no other number is friendly with it, is a solitary number. All prime numbers are known to be solitary, as are powers of prime numbers. More generally ...
