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, 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.[3]
The nonhypotenuse numbers have been applied to prove the existence of addition chains that compute the first square numbers using only additions.[4]
^D. S.; Beiler, Albert H. (1968), "Albert Beiler, Consecutive Hypotenuses of Pythagorean Triangles", Mathematics of Computation, 22 (103): 690–692, doi:10.2307/2004563, JSTOR2004563. This review of a manuscript of Beiler's (which was later published in J. Rec. Math.7 (1974) 120–133, MR0422125) attributes this bound to Landau.
