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 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]

See also

• 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

1. ^ 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, MR0422125) attributes this bound to Landau.
2. ^ Shanks, D. (1975), "Non-hypotenuse numbers", Fibonacci Quarterly 13 (4): 319–321, MR0387219 .

This number theory-related article is a stub. You can help Wikipedia by expanding it.v · Categories: Integer sequences Number theory stubs Wikimedia Foundation. 2010. Игры ⚽ Нужно сделать НИР? Star Dancer Heleno Fragoso Look at other dictionaries: Pythagorean triple — A Pythagorean triple consists of three positive integers a , b , and c , such that a 2 + b 2 = c 2. Such a triple is commonly written ( a , b , c ), and a well known example is (3, 4, 5). If ( a , b , c ) is a Pythagorean triple, then so is ( ka …   Wikipedia List of mathematics articles (N) — NOTOC N N body problem N category N category number N connected space N dimensional sequential move puzzles N dimensional space N huge cardinal N jet N Mahlo cardinal N monoid N player game N set N skeleton N sphere N! conjecture Nabla symbol… …   Wikipedia Pythagorean theorem — See also: Pythagorean trigonometric identity The Pythagorean theorem: The sum of the areas of the two squares on the legs (a and b) equals the area of the square on the hypotenuse (c) …   Wikipedia Hypotenuse — A hypotenuse is the longest side of a right triangle, the side opposite of the right angle. The length of the hypotenuse of a right triangle can be found using the Pythagorean theorem, which states that the square of the length of the hypotenuse… …   Wikipedia Constante de Landau-Ramanujan — En théorie des nombres, la constante de Landau Ramanujan apparaît dans le résultat selon lequel le nombre d entiers naturels inférieurs à x qui sont la somme de deux carrés est asymptotiquement proportionnel à Plus précisément, le nombre d… …   Wikipédia en Français 18+ © Academic, 2000-2024 Contact us: Technical Support, Advertising Dictionaries export, created on PHP, Joomla, Drupal, WordPress, MODx. Mark and share Search through all dictionaries Translate… Search Internet Share the article and excerpts Direct link … Do a right-click on the link above and select “Copy Link”