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Prove A Variation of Fermat's Theorem

There always exists a real number n such that a^n = b^n + c^n , where a, b and c are any integers.

The problem is not Fermat's Last Theorem, but a variation of it with real exponents.

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I would be grateful to anyone able to show that there always exists a *real* number n such that

a^n = b^n + c^n , where a, b and c are any integers.

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As long as you don't require n to be an integer, this is fairly straightforward.

We assume that a, b, and c are *positive* or at minimum *non-negative* integers -- if they are negative you run into serious problems with real exponents because the even roots are not defined, so the decimal real roots are sometimes defined and sometimes, not, NOT a function you want to play with.

For b and c fixed positive integers, b^n + c^n is a *continuous* function of n, with range
0< b^n + c^n < infinity
(A continuous ...

Solution Summary

It is shown that there always exists a real number n such that a^n = b^n + c^n , where a, b and c are any integers.

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