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Algebra

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4 Apr 2008 18:20:53 IST

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Algebra

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$\mbox{If} \ a+b+c = 0, a,b,c \in \mathbb{Z} \ \mbox{prove that} \ 2(a^4+b^4+c^4) \ \mbox{is a perfect square};$ .

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sandeep ramesh

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4 Apr 2008 18:33:59 IST

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EDIT

Anand Hegde

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4 Apr 2008 18:36:53 IST

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why rohith?

when a=1, b=1, c=-2, we get

2(1+1+16) =36

sandeep ramesh

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4 Apr 2008 18:37:23 IST

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the sum i believe was edited :)

sreeraman nagasubramaniyan

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4 Apr 2008 18:42:35 IST

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$2(a^4+b^4+c^4) \\ \\ = 2((a+b)^4-4a^3b-6a^2b^2-4ab^3+c^4) \\ \\ = 2(2c^4-4ab(a^2+b^2)-6a^2b^2) \\ \\ 2(2c^4-4ab(c^2-2ab)-6a^2b^2) \\ \\ = 2(2c^4-4abc^2+2a^2b^2) \\ \\ = (2(c^2-ab))^2$

Hari Shankar

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4 Apr 2008 19:05:37 IST

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anyone else?

Hari Shankar

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4 Apr 2008 19:12:32 IST

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Ok I filched from another forum. You can check out the other solutions here: http://www.mathlinks.ro/viewtopic.php?p=1089628#1089628

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