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The Frobenius equation is the Diophantine equation a_1x_1+a_2x_2+...+a_nx_n=b, where the a_i are positive integers, b is an integer, and the solutions x_i are nonnegative ...

The sequence defined by H(0)=0 and H(n)=n-H(H(H(n-1))). The first few terms are 1, 1, 2, 3, 4, 4, 5, 5, 6, 7, 7, 8, 9, 10, 10, 11, 12, 13, 13, 14, ... (OEIS A005374).

The Diophantine equation x_1^2+x_2^2+...+x_n^2=ax_1x_2...x_n which has no integer solutions for a>n.

The recurrence relation (n-1)A_(n+1)=(n^2-1)A_n+(n+1)A_(n-1)+4(-1)^n valid for n=4, 5, ... with A(2)=0 and A(3)=1 and which solves the married couples problem (Dörrie 1965, ...

An integer sequence given by the recurrence relation a(n)=a(a(n-2))+a(n-a(n-2)) with a(1)=a(2)=1. The first few values are 1, 1, 2, 3, 3, 4, 5, 6, 6, 7, 7, 8, 9, 10, 10, 11, ...

The Diophantine equation x^2+y^2+z^2=3xyz. The Markov numbers m are the union of the solutions (x,y,z) to this equation and are related to Lagrange numbers.

A method used by Gauss to solve the quadratic Diophantine equation of the form mx^2+ny^2=A (Dickson 2005, pp. 391 and 407).

For every k>1, there exist only finite many pairs of powers (p,p^') with p and p^' natural numbers and k=p^'-p.

Given a Pythagorean triple (a,b,c), the fractions a/b and b/a are called Pythagorean fractions. Diophantus showed that the Pythagorean fractions consist precisely of ...

The formula giving the roots of a quadratic equation ax^2+bx+c=0 (1) as x=(-b+/-sqrt(b^2-4ac))/(2a). (2) An alternate form is given by x=(2c)/(-b+/-sqrt(b^2-4ac)). (3)