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A number which is simultaneously octagonal and triangular. Let O_n denote the nth octagonal number and T_m the mth triangular number, then a number which is both octagonal ...

A nexus number is a figurate number built up of the nexus of cells less than n steps away from a given cell. The nth d-dimensional nexus number is given by N_d(n) = ...

Color each segment of a knot diagram using one of three colors. If 1. At any crossing, either the colors are all different or all the same, and 2. At least two colors are ...

A number which is simultaneously square and triangular. Let T_n denote the nth triangular number and S_m the mth square number, then a number which is both triangular and ...

A finite group G has a finite number of conjugacy classes and a finite number of distinct irreducible representations. The group character of a group representation is ...

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 ...

A number which is simultaneously octagonal and square. Let O_n denote the nth octagonal number and S_m the mth square number, then a number which is both octagonal and square ...

If alpha is any number and m and n are integers, then there is a rational number m/n for which |alpha-m/n|<=1/n. (1) If alpha is irrational and k is any whole number, there ...

The Frobenius number is the largest value b for which the Frobenius equation a_1x_1+a_2x_2+...+a_nx_n=b, (1) has no solution, where the a_i are positive integers, b is an ...

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