Four distinct positive integers $a, b, c, d$ satisfy the equation $a^3+b^3 = c^3+d^3$ . An example for $(a,b,c,d)$ is $(9,10,1,12)$ as $9^3+10^3 = 1^3+12^3$. Is there an infinite number of such examples for $(a,b,c,d)$? If there is no an infinite number of examples for $(a,b,c,d)$ then how many?
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2Multiply any solution throughout with a positive number to get another solution. So there are an infinite solutions unless you restrict these. – Macavity Feb 23 '16 at 02:20
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$(a,a,a,a)$ works for all $a$ – ASKASK Feb 23 '16 at 02:21
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Request OP to first check for similar problems before posting a problem. Math.SE has a lot of such known problems already posted and with solutions. – Shailesh Feb 23 '16 at 02:21
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24153319581254312065344 can be expressed as the sum of two cubes in six different ways. – Ross Millikan Feb 23 '16 at 02:23
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See Leonhard Euler, Disquitiones Artithmeticae, Vol. I, Ch. $272$, Pag. $556-576$. – Lucian Feb 23 '16 at 08:21