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Rubik's graph is the Cayley graph of Rubik's group. The graph diameter of this graph is sometimes known as God's number, and was shown in Aug. 2010 to be equal to 20 (Rokicki ...

Brown numbers are pairs (m,n) of integers satisfying the condition of Brocard's problem, i.e., such that n!+1=m^2 where n! is the factorial and m^2 is a square number. Only ...

For elliptic curves over the rationals Q, the group of rational points is always finitely generated (i.e., there always exists a finite set of group generators). This theorem ...

Every even number is the difference of two consecutive primes in infinitely many ways (Dickson 2005, p. 424). If true, taking the difference 2, this conjecture implies that ...

A triangle-replaced graph T(G) is a cubic graph in which each vertex is replaced by a triangle graph such that each vertex of the triangle is connected to one of the ...

The metabidiminished icosahedron is a convex equilateral solid that is Johnson solid J_(62). The unit metabidiminished icosahedron has volume V=1/6(5+2sqrt(5)) (1) and Dehn ...

The Sims graph is a distance-transitive graph on 352 vertices (Brouwer et al. 1989, p. 370).

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 ...

For F_n the nth Fibonacci number, F_(n-1)F_(n+1)-F_n^2=(-1)^n. This identity was also discovered by Simson (Coxeter and Greitzer 1967, p. 41; Coxeter 1969, pp. 165-168; Wells ...

A method for generating random (pseudorandom) numbers using the linear recurrence relation X_(n+1)=aX_n+c (mod m), where a and c must assume certain fixed values, m is some ...