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When ac is divisible by a number b that is relatively prime to a, then c must be divisible by b.

Schur (1916) proved that no matter how the set of positive integers less than or equal to |_n!e_| (where |_x_| is the floor function) is partitioned into n classes, one class ...

A number is said to be simply normal to base b if its base-b expansion has each digit appearing with average frequency tending to b^(-1). A normal number is an irrational ...

A modification of Legendre's formula for the prime counting function pi(x). It starts with |_x_| = (1) where |_x_| is the floor function, P_2(x,a) is the number of integers ...

A number which is simultaneously a nonagonal number N_m and hexagonal number Hex_n and therefore satisfies the Diophantine equation 1/2m(7m-5)=n(2n-1). (1) Completing the ...

A friendly number is a number that is a member of a friendly pair or a higher-order friendly n-tuple. Numbers that are not friendly are said to be solitary. There are some ...

An even number of the form 4n+2 (i.e., an integer which is divisible by 2 but not by 4). The first few for n=0, 1, 2, ... are 2, 6, 10, 14, 18, ... (OEIS A016825)

A field K is said to be an extension field (or field extension, or extension), denoted K/F, of a field F if F is a subfield of K. For example, the complex numbers are an ...

Legendre's formula counts the number of positive integers less than or equal to a number x which are not divisible by any of the first a primes, (1) where |_x_| is the floor ...

An alternating permutation is an arrangement of the elements c_1, ..., c_n such that no element c_i has a magnitude between c_(i-1) and c_(i+1) is called an alternating (or ...