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:"It is impossible for a cube to be the sum of two cubes, a fourth power to be the sum of two fourth powers, or in general for any number that is a power greater than the second to be the sum of two like powers. I have discovered a truly marvelous demonstration of this proposition that this margin is too narrow to contain."

:"It is impossible for a cube to be the sum of two cubes, a fourth power to be the sum of two fourth powers, or in general for any number that is a power greater than the second to be the sum of two like powers. I have discovered a truly marvelous demonstration of this proposition that this margin is too narrow to contain."

This problem has confounded mathematicians for centuries, and there still is no proof for it using [[elementary techniques]].  Most doubt that Fermat was able to prove it himself.

This problem has confounded mathematicians for centuries, and there still is no proof for it using [[elementary techniques]].  [[Gauss]] and other mathematicians doubt that Fermat was able to prove it himself, but [[Cal Tech]] mathematics Professor E.T. Bell, who wrote the standard biography of all the great mathematicians, wryly observed that "the fox who could not get at the grapes declared they were sour." <ref>E.T. Bell, "Men of Mathematics" 72 (1937).</ref>  "And so for all of [Fermat's] positive assertions with the one exception of the seemingly simple one which he made in his Last Theorem and which mathematicians, struggling for nearly 300 years, have been unable to prove: whenever Fermat asserted that he had ''proved'' anything, the statement, with the one exception noted, has subsequently been proved.  Both his scrupulously honest character and his unrivalled penetration as an arithmetician substantiate the claim made for him by some, but not by all, that he knew what he was talking about when he asserted that he possessed a proof of his theorem."<ref>''Ibid.'' at 71.</ref>

The theorem is as follows:

The theorem is as follows: