Show that the product of three consecutive positive integers cannot be the n-th power of an integer, for any integer n>1.

Let  the sequence of consecutive integers be x, x-1, x+1. Then the product is x(x^2-1).<o:p></o:p>

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Assuming this is a kth power, then since the highest common factor of x and x^2-1 is 1, both x and x^2-1 must be kth powers.<o:p></o:p>

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If x is a kth power, x^2 is a kth power.<o:p></o:p>

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But two kth powers (x^2 and x^2-1) cannot differ by 1.  Contradiction.<o:p></o:p>

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Hence the product (of three consecutive integers) cannot be a kth power.<o:p></o:p>

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Posted by goFish on 2005-09-14 09:11:17