Nonhypotenuse number explained

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:

1, 2, 3, 4, 6, 7, 8, 9, 11, 12, 14, 16, 18, 19, 21, 22, 23, 24, 27, 28, 31, 32, 33, 36, 38, 42, 43, 44, 46, 47, 48, 49, 54, 56, 57, 59, 62, 63, 64, 66, 67, 69, 71, 72, 76, 77, 79, 81, 83, 84

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/.^[1]

The nonhypotenuse numbers are those numbers that have no prime factors of the form 4k+1.^[2] Equivalently, they are the number that cannot be expressed in the form

K(m ^{2+n} ^{2)} where K, m, and n are all positive integers. A number whose prime factors are not 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

n square numbers using only n + o(n) additions.

See also

• Pythagorean theorem
• Landau-Ramanujan constant
• Fermat's theorem on sums of two squares

Notes and References

1. . This review of a manuscript of Beiler's (which was later published in J. Rec. Math. 7 (1974) 120–133,) attributes this bound to Landau.
2. .