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A subsequence of {a} is a sequence {b} defined by b_k=a_(n_k), where n_1<n_2<... is an increasing sequence of indices (D'Angelo and West 2000). For example, the prime numbers ...

Let p be a prime number, G a finite group, and |G| the order of G. 1. If p divides |G|, then G has a Sylow p-subgroup. 2. In a finite group, all the Sylow p-subgroups are ...

A subfield which is strictly smaller than the field in which it is contained. The field of rationals Q is a proper subfield of the field of real numbers R which, in turn, is ...

Let a and b be nonzero integers such that a^mb^n!=1 (except when m=n=0). Also let T(a,b) be the set of primes p for which p|(a^k-b) for some nonnegative integer k. Then ...

The abc conjecture is a conjecture due to Oesterlé and Masser in 1985. It states that, for any infinitesimal epsilon>0, there exists a constant C_epsilon such that for any ...

Define a Bouniakowsky polynomial as an irreducible polynomial f(x) with integer coefficients, degree >1, and GCD(f(1),f(2),...)=1. The Bouniakowsky conjecture states that ...

Legendre's constant is the number 1.08366 in Legendre's guess at the prime number theorem pi(n)=n/(lnn-A(n)) with lim_(n->infty)A(n) approx 1.08366. Legendre first published ...

3 is the only integer which is the sum of the preceding positive integers (1+2=3) and the only number which is the sum of the factorials of the preceding positive integers ...

A magic square that remains magic when its border is removed. A nested magic square remains magic after the border is successively removed one ring at a time. An example of a ...

A concise notation based on the concept of the tangle used by Conway (1967) to enumerate prime knots up to 11 crossings. An algebraic knot containing no negative signs in its ...