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If n=1,2 (mod 4), and the squarefree part of n is divisible by a prime p=3 (mod 4), then no difference set of order n exists. Equivalently, if a projective plane of order n ...

A number which is simultaneously a heptagonal number Hep_n and hexagonal number Hex_m. Such numbers exist when 1/2n(5n-3)=m(2m-1). (1) Completing the square and rearranging ...

A number which is simultaneously a heptagonal number H_n and pentagonal number P_m. Such numbers exist when 1/2n(5n-3)=1/2m(3m-1). (1) Completing the square and rearranging ...

For any two nonzero p-adic numbers a and b, the Hilbert symbol is defined as (a,b)={1 if z^2=ax^2+by^2 has a nonzero solution; -1 otherwise. (1) If the p-adic field is not ...

A number which is simultaneously a pentagonal number P_n and triangular number T_m. Such numbers exist when 1/2n(3n-1)=1/2m(m+1). (1) Completing the square gives ...

For algebraic alpha |alpha-p/q|<1/(q^(2+epsilon)), with epsilon>0, has finitely many solutions. Klaus Roth received a Fields medal for this result.

A Heronian triangle is a triangle having rational side lengths and rational area. The triangles are so named because such triangles are related to Heron's formula ...

Fermat's last theorem is a theorem first proposed by Fermat in the form of a note scribbled in the margin of his copy of the ancient Greek text Arithmetica by Diophantus. The ...

An equation is said to be a closed-form solution if it solves a given problem in terms of functions and mathematical operations from a given generally-accepted set. For ...

The number of cells in a generalized Chinese checkers board (or "centered" hexagram). Unlike the polygonal numbers, there is ambiguity in the case of the star numbers as to ...