Nonhypotenuse number
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 (sequence A004144 in the OEIS)
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/√log x.[1]
The nonhypotenuse numbers are those numbers that have no
The nonhypotenuse numbers have been applied to prove the existence of addition chains that compute the first square numbers using only additions.[4]
See also
- Pythagorean theorem
- Landau-Ramanujan constant
- Fermat's theorem on sums of two squares
References
- MR0422125) attributes this bound to Landau.
- MR 0387219.
- ISBN 978-0-486-21096-4
- MR 0557832