WON plate198 | World!OfNumbers [ March 21, 2016 ] The sum of two numbers, their squares and their cubes Postings from Pieter Post The pair (184, 345) possesses the following property: The sum of the numbers is a square, the sum of their squares is a square and the sum of their cubes is also a square. 184 + 345 = 232 1842 + 3452 = 3912 1843 + 3453 = 68772 This pair is the only one I could find, at least with a greatest common divisor of 1. (➜ °) Naturally (736, 1380) and all quadratic multiples of (184, 345) fulfil the conditions, but I could find only one primitive solution. Maybe someone among my readers has enough computing capacity to discover a second pair ? [ April 28, 2016 ] Delighted to see the presentation made by Mr Pieter Post regarding the admiring properties of certain pair of numbers such as 184,345; 736,1380. My attempts to find second set of such pair of numbers have not fully materialised so far. Selected pairs from dozens of my humble findings are enclosed for your kind perusal. These were found by manual manipulations on Excel sheet. 889 + 4440 = 732 8892 + 44402 = not square 8893 + 44403 = 2970372 2044 + 3285 = 732 20442 + 32852 = 38692 20443 + 32853 = not square 2409 + 2920 = 732 24092 + 29202 = not square 24093 + 29203 = 1971732 3042 + 3042 = 782 {identical !} 30422 + 30422 = not square 30423 + 30423 = 2372762 1729 + 5160 = 832 {powers of Ramanujan's 1729} 17292 + 51602 = not square 17293 + 51603 = 3775672 I can submit to you these set of pairs on reaching 100^2. B.S.Rangaswamy [ May 1, 2016 ] Patrick, I only just saw WON Plate 198 today. Some initial things I notice are (184 * 2) - 345 = 23 391 / 23 = 17 6877 / 23 = 299 I don't know if these things mean anything or not but they are curious. Jeff Heleen [ June 6, 2016 ] I wish to share the following general inference from my studies of the postings by Mr Pieter Post “ When sum of two identical numbers is a square, sum of their respective cubes always is a square ” as in: 50 + 50 = 10^2 50^3 + 50^3 = 500^2 6272 + 6272 = 112^2 6272^3 + 6272^3 = 702464^2 2238728 + 2238728 = 2116^2 2238728^3 + 2238728^3 = 4737148448^2 Generalized: A^2/2 + A^2/2 = A^2 (A^2/2)^3 + (A^2/2)^3 = (A^6/8) + (A^6/8) = A^6/4 ➜ Square of (A^3/2) B.S.Rangaswamy [ December 14, 2021 ] (➜ °) The relations are true, but it's a mistake, gcd(184,345) = 23 and not 1 ! 184 = 8 * 23; 345 = 15 * 23 The general solutions are: k2 * 23 * 8; k2 * 23 * 15, with k = 1, 2, 3... The proof: k2 * 23 * 8 + k2 * 23 * 15 = (23 * k)2 k4 * 232 * 82 + k4 * 232 * 152 = (23 * k2)2 * (82 + 152) = (17 * 23 * k2)2 k6 * 233 * 83 + k6 * 233 * 153 = (23 * k2)3 * (83 + 153) = (23 * k2)3 * (8 + 15) * (82 + 152 – 8 * 15) = (13 * 232 * k3)2 Alexandru Petrescu A000198 Prime Curios! Prime Puzzle Wikipedia 198 Le nombre 198
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