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July 26

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Diophantine Equation

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Let a be some natural number. What are the solutions in integers of the equation ? עברית (talk) 16:32, 26 July 2017 (UTC)[reply]

Any even difference of 2 squares is divisible by 4, so there are many solutions to that equation! Georgia guy (talk) 16:38, 26 July 2017 (UTC)[reply]
Hint: Try rewriting as , and then factor and solve in terms of the factors of 4a. Considering whether the factors are even or odd should let you simplify further. --Deacon Vorbis (talk) 16:47, 26 July 2017 (UTC)[reply]
Consider a prime number p≠2 that divides both x and y, then p² must divide a since it divides 4a and not 4, and then (x/p, y/p) is solution for a/p². Hence, the general solution should be easily generated from the solutions where by stripping a from its square divisors.
For the divisibility by 2: Obviously x and y are both odd or both even. If they are even, the equation reduces to where . If they are odd, which reduces to . Both equations look quite attackable by looking at the pairs m,n such that a=m*n, and retrieving x,y as a function of m,n. TigraanClick here to contact me 17:24, 26 July 2017 (UTC)[reply]
A curious fact is that the equation always has at least two solutions , and , . These are the only solutions if is simple. Ruslik_Zero 17:32, 26 July 2017 (UTC)[reply]
Thank you! עברית (talk) 19:40, 26 July 2017 (UTC)[reply]