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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: Try some n-values, such as n = 3, n=4. Notice anything?


Solution Commentary: The number 1324 raised to any power must end in either the digit 6 or the digit 4. The other two numbers, 731 and 1961, raised to an integral power will always end in the digit 1. Thus, the summed numbers on the fet side of the expression will end in either a 7 or a 5, while the right side will end in a 1. Thus, Krieger’s numbers cannot be a solution for any n-value.

When the NYT reporter pointed out this fact, Krieger responded: "You mean that you doubt me? Well, when the time comes, I will explain everything.”

Unfortunately, or fortunately, no one has ever heard any more from Krieger....