Sriman Dutta Posted October 18, 2016 Share Posted October 18, 2016 Hi, Can anybody find two numbers such that the difference between their squares is a cube and the difference between their cubes is a square ? :-) Link to comment Share on other sites More sharing options...
DrKrettin Posted October 18, 2016 Share Posted October 18, 2016 Am I on the right track if I take the two numbers to be 2^n plus and minus 1? If n is a multiple of 3, then at least I get the first condition. Link to comment Share on other sites More sharing options...
Sriman Dutta Posted October 18, 2016 Author Share Posted October 18, 2016 There must be two numbers a and b such that a^2 - b^2 = n^3 and a^3 - b^3 = m^2 , m and n are two different numbers. Link to comment Share on other sites More sharing options...
DrKrettin Posted October 18, 2016 Share Posted October 18, 2016 Yes, I got that. I meant whether a good starting point might be taking a = 2^n - 1 and b = 2^n + 1 where n is a positive integer. Then a^2 - b^2 = (a + b)(a - b) = 2^(n+2) which is a cube for n= 1,4,7.. Link to comment Share on other sites More sharing options...
Sriman Dutta Posted October 19, 2016 Author Share Posted October 19, 2016 OK And the other condition ? Link to comment Share on other sites More sharing options...
uncool Posted October 21, 2016 Share Posted October 21, 2016 (edited) DrKrettin: that wouldn't work, as then a^3 - b^3 = (a - b)(a^2 + ab + b^2) = 2*something odd, which can't be a square (except in the case that n = 0, a = 2, b = 0, and it still isn't a square). Edited October 21, 2016 by uncool Link to comment Share on other sites More sharing options...
DrKrettin Posted October 21, 2016 Share Posted October 21, 2016 DrKrettin: that wouldn't work, as then a^3 - b^3 = (a - b)(a^2 + ab + b^2) = 2*something odd, which can't be a square (except in the case that n = 0, a = 2, b = 0, and it still isn't a square). Yes - I'd decided that approach was getting nowhere. There doesn't seem to be an analytical way of doing this. Link to comment Share on other sites More sharing options...
imatfaal Posted October 21, 2016 Share Posted October 21, 2016 10 and 6 1 Link to comment Share on other sites More sharing options...
DrKrettin Posted October 21, 2016 Share Posted October 21, 2016 Nice. Do you know whether that is a unique answer, or are there other larger ones? Link to comment Share on other sites More sharing options...
imatfaal Posted October 21, 2016 Share Posted October 21, 2016 Nice. Do you know whether that is a unique answer, or are there other larger ones? No idea - ran through a few numbers in my head on my ride to work today. I cannot think of an analytical method - or any way other than sieve tbh Link to comment Share on other sites More sharing options...
Sriman Dutta Posted October 21, 2016 Author Share Posted October 21, 2016 Perfect imatfaal and +1 Link to comment Share on other sites More sharing options...
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