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For a subgroup H of a group G and an element x of G, define xH to be the set {xh:h in H} and Hx to be the set {hx:h in H}. A subset of G of the form xH for some x in G is ...

The Feit-Thompson conjecture asserts that there are no primes p and q for which (p^q-1)/(p-1) and (q^p-1)/(q-1) have a common factor. Parker noticed that if this were true, ...

There are two different statements, each separately known as the greatest common divisor theorem. 1. Given positive integers m and n, it is possible to choose integers x and ...

The continued fraction for K is [2; 1, 2, 5, 1, 1, 2, 1, 1, ...] (OEIS A002211). A plot of the first 256 terms of the continued fraction represented as a sequence of binary ...

For every k>=1, let C_k be the set of composite numbers n>k such that if 1<a<n, GCD(a,n)=1 (where GCD is the greatest common divisor), then a^(n-k)=1 (mod n). Special cases ...

Mann's theorem is a theorem widely circulated as the "alpha-beta conjecture" that was subsequently proven by Mann (1942). It states that if A and B are sets of integers each ...

Mills' constant can be defined as the least theta such that f_n=|_theta^(3^n)_| is prime for all positive integers n (Caldwell and Cheng 2005). The first few f_n for n=1, 2, ...

Murata's constant is defined as C_(Murata) = product_(p)[1+1/((p-1)^2)] (1) = 2.82641999... (2) (OEIS A065485), where the product is over the primes p. It can also be written ...

Three types of n×n matrices can be obtained by writing Pascal's triangle as a lower triangular matrix and truncating appropriately: a symmetric matrix S_n with (S)_(ij)=(i+j; ...

Sarnak's constant is the constant C_(Sarnak) = product_(p>=3)(1-(p+2)/(p^3)) (1) = 0.7236484022... (2) (OEIS A065476), where the product is over the odd primes.