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A circle having a given number of lattice points on its circumference. The Schinzel circle having n lattice points is given by the equation {(x-1/2)^2+y^2=1/45^(k-1) for n=2k ...

For every positive integer n, there exists a circle in the plane having exactly n lattice points on its circumference. The theorem is based on the number r(n) of integral ...

A Fourier series-like expansion of a twice continuously differentiable function f(x)=1/2a_0+sum_(n=1)^inftya_nJ_0(nx) (1) for 0<x<pi, where J_0(x) is a zeroth order Bessel ...

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 ...

In the above illustration, black dots appear to form and vanish at the intersections of the gray horizontal and vertical lines. When focusing attention on a single white dot, ...

A subset E of a topological space S is said to be of second category in S if E cannot be written as the countable union of subsets which are nowhere dense in S, i.e., if ...

Every smooth nonzero vector field on the 3-sphere has at least one closed orbit. The conjecture was proposed in 1950 and proved true for Hopf maps. The conjecture was ...

A sequent is an expression Gamma|-Lambda, where Gamma and Lambda are (possibly empty) sequences of formulas. Here, Gamma is called the antecedent and Lambda is called the ...

A problem also known as the points problem or unfinished game. Consider a tournament involving k players playing the same game repetitively. Each game has a single winner, ...

The randomization of a deck of cards by repeated interleaving. More generally, a shuffle is a rearrangement of the elements in an ordered list. Shuffling by exactly ...