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Show That Krieger Is wrong!

Nineteen years have passed. It is time to celebrate Professor Andrew Wiles' proof of Fermat's Last Conjecture: There are no positive inetgers a, b, c such that an + bn = cn for integer n > 2.

In 1907, the University of Göttingen offered the Wolfskehl Prize of 100,000 German marks for the first person to prove/disprove the Conjecture. Many people tried, submitting their works of genius, but all failed until Wiles appeared with a solution in 1995 (and the prize was worth about $50,000).

One person who tired to disprove the Conjecture was mathematician Dr. Samuel Isaac Krieger. As reported in TIME (1937), Krieger, who was taking a mineral bath near Buffalo, N. Y., suddenly leaped out of the bath, rushed naked into the adjoining room, and began to scribble figures. He thought he had discovered something: a solution to Fermat's equation.

Kreiger later announced that the positive integers 1324, 731, and 1961 solved Fermat's equation, but he refused to reveal the necessary n-value (except stating it was less than 20).

Your Task: Show that for all n>2, Kriger's numbers are NOT a solution.

Note: One approach is by exhaustion...that is try all possible n-values 3, 4, 5, ...,18, 19. BUT, try to use another apporoach that involves a little more problem-solving and thinking...as was done by a New York Times reporter in 1938.

Unfortunately, or fortunately, no one heard much about Krieger any more....and he already had a checkered reputation.

Krieger often contacted the press to report his discovery (usually incorrect) of large primes.

In a 1935 issue of The Milwaukee Journal, an article states: Dr. Samuel Isaac Krieger, who arrived here Monday, flunked three times in arithmetic, back in preparatory school in Hamburg, Germany, but he didn’t let a little thing like that stop him from becoming a great mathematician.

He has been described as “the greatest genius in mathematics and the greatest mathematical mind I have ever seen” by no less a personage than famed Dr. Albert Einstein, relativity expert who is said to be somewhat of a mathematician himself and well aware that the answer to two plus two is four.

Earlier, The Pittsburgh Post-Gazette had reported that the same Kreiger had solved a number theoretic problem that had even "stumped Einstein." In addition to a pair of published papers and a patent filed in 1942, Krieger was not heard of again.....especially after the Fermat error on his part.

Source: http://outofthenormmaths.wordpress.com/2011/11/09/blitz-krieger/

Hint

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