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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 $$5^2$$ equals $$3^2 + 4^2$$.

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 (sequence A004144 in 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 prime factors of the form 4k+1.[2] Equivalently, any number that cannot be put into the form $$K(m^2+n^2)$$ where K, m, and n are all positive integers, is never a nonhypotenuse number. A number whose prime factors are not all of the form 4k+1 cannot be the hypotenuse of a primitive triangle, but may still be the hypotenuse of a non-primitive triangle.[3]

Nonhypotenuse Numbers (sequence A004144 in OEIS)
Eta Numbers (sequence A125667 in OEIS)
Pythagorean theorem
Landau-Ramanujan constant
Fermat's theorem on sums of two squares

References

D. S.; Beiler, Albert H. (1968), "Albert Beiler, Consecutive Hypotenuses of Pythagorean Triangles", Mathematics of Computation 22 (103): 690–692, doi:10.2307/2004563, JSTOR 2004563. This review of a manuscript of Beiler's (which was later published in J. Rec. Math. 7 (1974) 120–133, MR 0422125) attributes this bound to Landau.
Shanks, D. (1975), "Non-hypotenuse numbers", Fibonacci Quarterly 13 (4): 319–321, MR 0387219.
Beiler, Albert (1966). Recreations in the Theory of Numbers: The Queen of Mathematics Entertains (2 ed.). New York: Dover Publications. p. 116-117. ISBN 978-0-486-21096-4.

Mathematics Encyclopedia