In mathematics, Euler'sidoneal numbers (also called suitable numbers or convenient numbers) are the positive integers D such that any integer expressible in only one way as x2 ± Dy2 (where x2 is relatively prime to Dy2) is a prime power or twice a prime power. In particular, a number that has two distinct representations as a sum of two squares is composite. Every idoneal number generates a set containing infinitely many primes and missing infinitely many other primes.
A positive integer n is idoneal if and only if it cannot be written as ab + bc + ac for distinct positive integers a, b, and c.[1]
It is sufficient to consider the set { n + k2 | 3 . k2 ≤ n ∧ gcd (n, k) = 1 }; if all these numbers are of the form p, p2, 2 · p or 2s for some integer s, where p is a prime, then n is idoneal.[2]
Results of Peter J. Weinberger from 1973[3] imply that at most two other idoneal numbers exist, and that the list above is complete if the generalized Riemann hypothesis holds (some sources incorrectly claim that Weinberger's results imply that there is at most one other idoneal number).[4]
idoneal, number, mathematics, euler, idoneal, numbers, also, called, suitable, numbers, convenient, numbers, positive, integers, such, that, integer, expressible, only, where, relatively, prime, prime, power, twice, prime, power, particular, number, that, dist. In mathematics Euler s idoneal numbers also called suitable numbers or convenient numbers are the positive integers D such that any integer expressible in only one way as x2 Dy2 where x2 is relatively prime to Dy2 is a prime power or twice a prime power In particular a number that has two distinct representations as a sum of two squares is composite Every idoneal number generates a set containing infinitely many primes and missing infinitely many other primes Contents 1 Definition 2 Conjecturally complete listing 3 See also 4 Notes 5 References 6 External linksDefinition editA positive integer n is idoneal if and only if it cannot be written as ab bc ac for distinct positive integers a b and c 1 It is sufficient to consider the set n k2 3 k2 n gcd n k 1 if all these numbers are of the form p p2 2 p or 2s for some integer s where p is a prime then n is idoneal 2 Conjecturally complete listing editUnsolved problem in mathematics Are there 65 66 or 67 idoneal numbers more unsolved problems in mathematics The 65 idoneal numbers found by Leonhard Euler and Carl Friedrich Gauss and conjectured to be the only such numbers are 1 2 3 4 5 6 7 8 9 10 12 13 15 16 18 21 22 24 25 28 30 33 37 40 42 45 48 57 58 60 70 72 78 85 88 93 102 105 112 120 130 133 165 168 177 190 210 232 240 253 273 280 312 330 345 357 385 408 462 520 760 840 1320 1365 and 1848 sequence A000926 in the OEIS Results of Peter J Weinberger from 1973 3 imply that at most two other idoneal numbers exist and that the list above is complete if the generalized Riemann hypothesis holds some sources incorrectly claim that Weinberger s results imply that there is at most one other idoneal number 4 See also editList of unsolved problems in mathematicsNotes edit Eric Rains OEIS A000926 Comments on A000926 December 2007 Roberts Joe The Lure of the Integers The Mathematical Association of America 1992 Acta Arith 22 1973 p 117 124 Kani Ernst 2011 Idoneal numbers and some generalizations PDF Annales des Sciences Mathematiques du Quebec 35 2 Corollary 23 Remark 24 References editZ I Borevich and I R Shafarevich Number Theory Academic Press NY 1966 pp 425 430 D A Cox 1989 Primes of the Form x2 ny2 Wiley Interscience p 61 ISBN 0 471 50654 0 L Euler An illustration of a paradox about the idoneal or suitable numbers 1806 G Frei Euler s convenient numbers Math Intell Vol 7 No 3 1985 55 58 and 64 O H Keller Ueber die Numeri idonei von Euler Beitraege Algebra Geom 16 1983 79 91 Math Rev 85m 11019 G B Mathews Theory of Numbers Chelsea no date p 263 P Ribenboim Galimatias Arithmeticae in Mathematics Magazine 71 5 339 1998 MAA or My Numbers My Friends Chap 11 Springer Verlag 2000 NY J Steinig On Euler s ideoneal numbers Elemente Math 21 1966 73 88 A Weil Number theory an approach through history from Hammurapi to Legendre Birkhaeuser Boston 1984 see p 188 P Weinberger Exponents of the class groups of complex quadratic fields Acta Arith 22 1973 117 124 Ernst Kani Idoneal Numbers And Some Generalizations Ann Sci Math Quebec 35 No 2 2011 197 227 External links editK S Brown Mathpages Numeri Idonei M Waldschmidt Open Diophantine problems Weisstein Eric W Idoneal Number MathWorld Retrieved from https en wikipedia org w index php title Idoneal number amp oldid 1155326189, wikipedia, wiki, book, books, library,
