In mathematics, a nonhypotenuse number is a natural number whose square cannot be written as the sum of two nonzero squares. The name stems from the fact that an edge of length equal to a nonhypotenuse number cannot form the hypotenuse of a right angle triangle with integer sides.
The numbers 1, 2, 3 and 4 are all nonhypotenuse numbers. The number 5, however, is not a nonhypotenuse number as 52 equals 32 + 42.
The first fifty nonhypotenuse numbers are:
Although nonhypotenuse numbers are common among small integers, they become more-and-more sparse for larger numbers. Yet, there are infinitely many nonhypotenuse numbers, and the number of nonhypotenuse numbers not exceeding a value x scales asymptotically with x/√.
The nonhypotenuse numbers are those numbers that have no prime factors of the form 4k+1. Equivalently, they are the number that cannot be expressed in the form where K, m, and n are all positive integers. A number whose prime factors are not all of the form 4k+1 cannot be the hypotenuse of a primitive integer right triangle (one for which the sides do not have a nontrivial common divisor), but may still be the hypotenuse of a non-primitive triangle.
The nonhypotenuse numbers have been applied to prove the existence of addition chains that compute the first square numbers using only additions.