{"history":[],"article":{"id":"ade8dae9-3130-4fd5-88ae-bf4516691898","title":"Idoneal number","slug":"idoneal-number","content":":::summary\n:::\n\n# \nIn mathematics, [Euler's](https://bliptext.com/articles/leonhard-euler) **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](https://bliptext.com/articles/relatively-prime) to Dy2) is a prime power or twice a prime power.\nIn particular, a number that has two distinct representations as a sum of two squares is [composite](https://bliptext.com/articles/euler-s-factorization-method).\nEvery idoneal number generates a set containing infinitely many primes and missing infinitely many other primes.\n\n# Definition\nA 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.\nIt is sufficient to consider the set\n{ n + k2\np\n,\np2\n,\n2 · p\nor 2s for some integer s, where\np\nis a prime, then\nn\nis idoneal.\n\n# Conjecturally complete listing\nThe 65 idoneal numbers found by [Leonhard Euler](https://bliptext.com/articles/leonhard-euler) and [Carl Friedrich Gauss](https://bliptext.com/articles/carl-friedrich-gauss) and conjectured to be the only such numbers are\nResults of [Peter J. Weinberger](https://bliptext.com/articles/peter-j-weinberger) from 1973 imply that at most two other idoneal numbers exist, and that the list above is complete if the [generalized Riemann hypothesis](https://bliptext.com/articles/generalized-riemann-hypothesis) holds (some sources incorrectly claim that Weinberger's results imply that there is at most one other idoneal number).","current_revision":null,"created_by":"kACnuQmH5k9Nr5VNVEpVeg7XpvUmMctD","created_at":"2025-01-26T10:18:32.215Z","updated_at":"2025-01-26T10:18:32.215Z","search_vector":"'idon':1 'number':2","search_content":" \n \n\n# \nIn mathematics, [Euler's](https://bliptext.com/articles/leonhard-euler) **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](https://bliptext.com/articles/relatively-prime) to Dy2) is a prime power or twice a prime power.\nIn particular, a number that has two distinct representations as a sum of two squares is [composite](https://bliptext.com/articles/euler-s-factorization-method).\nEvery idoneal number generates a set containing infinitely many primes and missing infinitely many other primes.\n\n# Definition\nA 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.\nIt is sufficient to consider the set\n{ n + k 2\n p \n,\n p 2\n,\n2 · p \nor 2s for some integer s , where\n p \nis a prime, then\n n \nis idoneal.\n\n# Conjecturally complete listing\nThe 65 idoneal numbers found by [Leonhard Euler](https://bliptext.com/articles/leonhard-euler) and [Carl Friedrich Gauss](https://bliptext.com/articles/carl-friedrich-gauss) and conjectured to be the only such numbers are\nResults of [Peter J. Weinberger](https://bliptext.com/articles/peter-j-weinberger) from 1973 imply that at most two other idoneal numbers exist, and that the list above is complete if the [generalized Riemann hypothesis](https://bliptext.com/articles/generalized-riemann-hypothesis) holds (some sources incorrectly claim that Weinberger's results imply that there is at most one other idoneal number).","revisionCount":0},"hasMore":false,"nextCursor":null}

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