Number theory Diophantine equations Theorem

Fermat's Last Theorem for n = 4

Builds on Fermat's right triangle theorem Proof by infinite descent

Statement

There are no positive integers $a$ , $b$ and $c$ with $a^4 + b^4 = c^4$ .

Remarks

A stronger statement holds: $x^4 + y^4 = z^2$ has no solution in positive integers. The theorem also follows from Fermat's right triangle theorem, in the form that $x^4 - y^4 = z^2$ has no solution in positive integers. The proof uses infinite descent and the classification of Pythagorean triples. Since $4$ divides every power of $2$ above $2$ , the case $n = 4$ settles Fermat's Last Theorem for all such exponents.

History

Fermat's proof of his right triangle theorem, by infinite descent, gives this case.

Reading path

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Divisibility definition
Proof by infinite descent concept
Greatest common divisor definition
Pythagorean triple definition
Fermat's right triangle theorem statement
Fermat's Last Theorem for n = 4 statement

Tag	`0E741HSMJ1`
URI	`m2://0E741HSMJ1`
Kind	statement, theorem
Writing	W0 stub
MSC	11D25
Status	This copy of the site has no build data. Run `m2 status 0E741HSMJ1`.
Uses	Fermat's right triangle theorem, Proof by infinite descent

Fermat's Last Theorem for n = 4

Statement

Remarks

History

Reading path

Further reading

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