I thought the following story would make a good question to add flavour to some first-year undergraduate Number Theory courses. It’s relevant, interesting, well-documented, and shows how a certain amount of scientific literacy can be extremely useful when evaluating bogus claims. Here’s an extract from a 1938 TIME Magazine article (££)
In 1937 A.D., a German-Jewish mathematician named Samuel Isaac Krieger, who was taking a mineral bath near Buffalo, N. Y., suddenly leaped out, rushed naked into the adjoining room, began to scribble figures. He thought he had discovered something too: a solution to the equation given in Fermat’s last theorem.
Krieger claimed to have found some integer such that the equation held, but refused to reveal the value of . A journalist quickly proved this to be false. Can you see how?
Obviously, Andrew Wiles only proved Fermat’s Last Theorem (that there are no integers such that for ) to be true in full generality after 1994, so he couldn’t use that. While it was known at the time that the theorem was true for exponents including , the journalist only used elementary mathematics and did not need to use any specific knowledge about the theorem.
Solution after the break. Also, how not to solve FLT, and more on Krieger.