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.
The journalist exploded Krieger’s supposed counterexample by looking at the equation modulo 10: powers of numbers ending in 4 must always end in 4 or 6. Similarly powers of numbers ending with 1, always end with a 1. But the sum would then end in a 5 or 7, which is a contradiction.
Slightly more formally:
The example also always fails modulo 5, and hence must do so modulo 10 too, as 5 is a factor of 10. But it does hold in modulo 2: an even number plus an odd number is always odd.
From Futility Closet (where I first saw this posed as a problem):
“You mean that you doubt me?” Krieger asked the reporter. “Well, when the time comes, I will explain everything.”
A short mathematical diversion before we return to the anti-hero of our story: here we showed that one particular choice of could not be a solution to the FLT equation. To prove FLT you’d have to show it for all possible choices. How far might you be able to use similar approaches to prove Fermat’s Last Theorem?
Let’s concentrate on one specific exponent, say, . To prove Fermat’s Last Theorem in this case, we might hope to find one number such that the FLT equation for has no solutions . Avoiding any technicalities, it is sufficient to consider only prime , and to account for trivial solutions we shall look for infinitely many such primes. However, the following theorem due to Schur shows that this can’t happen: there can only be finitely many primes where the case has no solutions whatsoever in that modulus.
Theorem. Given some , there exists a prime such that for all larger primes , the congruence has a solution for some integers with .
So this result says that the above approach won’t work, and suggests more advanced mathematical machinery than congruences alone are required for a successful proof. You can find a more detailed explanation on Qiaochu Yuan’s blog Annoying Precision: the proof of the theorem is a nice application of Ramsey theory. You can read a free extract about Fermat’s Last Theorem and Wiles’s proof, from the indispensable Princeton Companion to Mathematics.
Now back to Krieger.
Before you start to bemoan the loss of a golden age of journalism, Krieger seems to have often successfully courted the press, claiming (if not always incorrectly, then at least frequently so) to have found large primes and the like. These other claims are generally much harder to disprove. In a 1935 article, The Milwaukee Journal reported:
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.
In an earlier piece in The Pittsburgh Post-Gazette, Krieger also references a number theoretic problem, he supposedly solved, that ‘stumped Einstein‘. He seems to have at least a pair of papers and perhaps a patent filed in 1942, which is the last reference I can find to him (aged about 38).
Krieger may have overlapped during his time in Göttingen (if he was actually there!) with Professor Edmund Landau who, from 1909 to 1934, was in charge of refereeing the Wolskehl Prize which to anyone who proved Fermat’s Last Theorem to be true (but offered no money for a counterexample) would pay out a 100,000 marks prize ($50,000 when Wiles eventually picked it up). According to Simon Singh’s Fermat’s Last Theorem (which I’ve heard is the most mentioned book by university maths applicants on their UCAS personal statements), Landau had so many entries that he created a standard response card with blanks where the entrant’s name, and the page and line number of their first mistake were to be written: “This invalidates the proof.” was the end to this terse form. His students, of course, were to fill in the blanks.