Inmathematics, anonhypotenuse number is anatural number whose squarecannot be written as the sum of two nonzero squares. The name stems from the fact that an edge of length equal to a nonhypotenuse numbercannot form thehypotenuse of aright angle triangle with integer sides.

The numbers 1, 2, 3, and 4 are all nonhypotenuse numbers. The number 5, however, isnot a nonhypotenuse number as${\displaystyle 5^2=3^2+4^2}$.

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 (sequenceA004144 in theOEIS)

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 valuex scales asymptotically withx/√logx.^[1]

The nonhypotenuse numbers are those numbers that have noprime factors ofthe form 4k+1.^[2] Equivalently, they are the number that cannot be expressed in the form${\displaystyle K(m^2+n^2)}$ whereK,m, andn are all positive integers. A number whose prime factors are notall of the form 4k+1 cannot be the hypotenuse of aprimitive 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.^[3]

The nonhypotenuse numbers have been applied to prove the existence ofaddition chains that compute the first${\displaystyle n}$ square numbers using only${\displaystyle n+o(n)}$ additions.^[4]

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

• OEISsequence A004144 (Nonhypotenuse numbers)
• OEISsequence A125667 (Eta numbers)

• Integer sequences
• Pythagorean theorem

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