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Fermat's Last Theorem

The assertion that the equation x^n + y^n = z^n has no solutions in non-zero integers x,y,z when n > 2.

The equation was originally proposed by Fermat in the margin of his copy of the works of Diophantus in 1605, who claimed to have a proof which, famously, 'the margin was too small to contain'.

It was proved by Andrew Wiles in a proof announced in 1993 and completed with Richard Taylor in 1994.

The proof deduces the result as a consequence of the proof of the deeper Shimura-Taniyama-Weil conjecture on elliptic curves over the rational numbers.

Attempts to prove Fermat's Last Theorem in the 19th century led to the development of much of modern algebraic number theory.

Pell's Equation

The equation x^2 - d y^2 = 1 for given d. It is soluble in integers x,y for all positive non-square d.

The fundamental solution is that with the smallest non-zero value of y. All other solution can be obtained from the fundamental.

If (x,y) is a solution then x + y.sqrt(d) is a unit in the ring of integers of the field containing the square root of d. Solution of the equation is thus connected with the problem of finding the fundamental unit of the ring of integers in this field.

The solutions to Pell's equation can be obtained from the continued fraction expansion of sqrt(d)

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Fermat's Last Theorem

Pell's Equation