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Contradiction. A Diophantine equation of the form x 2 + k = y3 (where k is a given nonzero integer) is called a Rachet equation - after Claude-Gaspar Bachet (1581-1638) who wrote poetry and philosophy as well as mathematics. In 1967 there were many integers k for which mathematicians could not solve the equation. However, in the years following 1967, Alan Baker and others developed a method for solving this equation for any given k. (See Ray. P. 3 also bears fruit in the following result which we shall use in our proof that x 3 + y3 = Z3 has no solution in nonzero integers.

Since (x,y) = 1, it follows that (u,w) = 1, and u and ware not both odd. Since X 3+y3 = Z3, it follows that 2u 3+6uw 2 = Z3, or 2u(u 2 +3w 2 ) = Z3. Case 1. u is not divisible by 3. Since u and w have different parity, u 2 +3w 2 is odd. Since (u, w) = 1, it follows that (2u, u2 + 3w 2 ) = 1 and hence there are integers t and s such that 2u = t 3 and u 2 + 3w 2 = 8 3 . 6, there are integers a and b such that u = a3- 9ab2 and w = 3a 2 b- 3b3. Since (u,w) = 1, it follows that (a,3b) = 1, and a and 3b have different parity.

Lagrange based his work on the following two theorems, which had been proved by Leonhard Euler (1707-1783). B. FOUR SQUARE THEOREM Hence iJ every prime is a sum oj Jour squares, then every natural number is a sum oj Jour squares. Proof: The equation can be verified by straight calculation. The 'hence' follows from the fact that every natural number has a prime factorisation. 2 For every odd prime p there is an integer m such that o < m < p and mp is a sum oj Jour squares. Proof: The squares 02, 12, 22, ...

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