Fermat’s last theorem is one of the most popular theorem of the history of mathematics. It was first conjectured, without proof, by Pierre de Fermat, a French Mathematician, in 1637. For 356 years the proof of Fermat’s statement remained a challenge to all mathematician. If you have never heard of this before, I strongly advice you to watch the BBC documentary. Andrew Wiles, a british mathematician, published a complete proof of Fermat’s Last Theorem in 1995.

Fermat’s Last Theorem states that there exist no three integers satisfying the equation for .

In this post we will consider a variation of Fermat’s last theorem. We wish to show that there is no three integers satisfying the equation for any integers , and where is the imaginary number ().

Let us start by assuming that such integers exists. It implies that . And since are supposed to be integers it follows that and are integers as well. Let us define , and so that one can rewrite .

From now, all is left to do is to show that we can not find three integers satisfying . To proceed, we observer that . Therefore .

Which leads to ,

and finally .

Rewriting our last equation as , we can now use . We get

.

Rewriting and , all we need to show is that it exist no two integers satisfying

.

Let us now write under its exponential form , where is an arbitrary integer. Note that and does not modify the previous equality. Identically, we have , where again is an integer. It follows that

. Taking the we get

, which can be rewritten as

, which finally leads to

.

Finally we should point out that and are transcendental numbers. It implies that to any power can not be written as a rational number. In other words it is impossible to write for any and integers. This finally ends the proof. It exists no three integers satisfying

for .

Note that it is known that for any integer is a transcendental number. See more on

- Weisstein, Eric W., “Irrational Number”,
*MathWorld*.
**Jump up^** Modular functions and transcendence questions, Yu. V. Nesterenko, Sbornik: Mathematics(1996), 187(9):1319

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