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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.

Two lengths are called incommensurate or incommensurable if their ratio cannot be expressed as a ratio of whole numbers. Irrational numbers and transcendental numbers are ...

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 continuous homomorphism of a group into the nonzero complex numbers. A multiplicative character omega gives a group representation on the one-dimensional space C of complex ...

The odd part Od(n) of a positive integer n is defined by Od(n)=n/(2^(b(n))), where b(n) is the exponent of the exact power of 2 dividing n. Od(n) is therefore the product of ...

Find two numbers such that x^2=y^2 (mod n). If you know the greatest common divisor of n and x-y, there exists a high probability of determining a prime factor. Taking small ...

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 sum of the aliquot divisors of n, given by s(n)=sigma(n)-n, where sigma(n) is the divisor function. The first few values are 0, 1, 1, 3, 1, 6, 1, 7, 4, 8, 1, 16, ... ...

For a discrete function f(n), the summatory function is defined by F(n)=sum_(k in D)^nf(k), where D is the domain of the function.

If n>1, (a,n)=1 (i.e., a and n are relatively prime), and m is the least integer >sqrt(n), then there exist an x and y such that ay=+/-x (mod n) where 0<x<m and 0<y<m (Nagell ...