Near misses for disproving Fermat’s conjecture

Photo by ThisisEngineering RAEng on Unsplash

According to Fermat’s so-called “last theorem,” the equation x³ + y³ = z³ has no solution in integers. Neither does x⁴ + y⁴ = z⁴, nor x⁵ + y⁵ = z⁵, and so on and so forth. If the exponent is an integer greater than 2, then at least one of x, y or z must not be an integer.

Pierre de Fermat claimed in 1637 to have found a proof of this “theorem,” but his proof was never found, even though he lived for a couple of decades after making that claim.

Andrew Wiles proved Fermat’s conjecture in 1994, using prior results that Fermat could not possibly have known about. And it took Wiles years of work and some time in a dead end to actually prove the conjecture.

It would have been a lot easier if the conjecture had been wrong and the smallest counterexample was a reasonably small number. And Fermat did have his share of conjectures that turned out to be wrong.

For example, Fermat asserted that the numbers 3, 5, 17, 257, 65537, 4294967297, 18446744073709551617, etc., are always prime. The smallest prime factor of that last number I listed is 274177, which would seem forbiddingly out of reach to someone without a calculator.

But, with just a little diligence, Fermat himself could have found that in fact 4294967297 = 641 × 6700417, or at least that 4294967297 is divisible by 641, and therefore not prime. Indeed that particular conjecture was disproved a long time before calculators became commonplace.

In the time before Wiles’s proof of Fermat’s most famous conjecture, there were people who claimed to have disproved it. For example, it appears that 13⁵ + 16⁵ = 17⁵.

We see that 13⁵ + 16⁵ = 1419869. But the fifth root of that is actually roughly 17.0000287, not 17.0. The correct equation is 13⁵ + 16⁵ = 17⁵ + 12.

Here’s another false counterexample that is much easier to show as not genuine: (−5)³ + 7³ = 6³. Remember that the cube of a negative number is also negative. So we have −125 + 343 = 218 = 6³ + 2.

Indeed there are several equations like x⁵ + y⁵ = z⁵ + c that have solutions in integers, provided of course that c not be 0. Prof. Noam Elkies from Harvard has found quite a few of these “near misses.”

Today’s April Fool’s Day. Maybe you’ll fall for some prank or other. But if anyone tells you that they have disproved Fermat’s “last theorem”, you won’t be falling for that one.

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is a composer and photographer from Detroit, Michigan. He has been working on a Java program to display certain mathematical diagrams.

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Alonso Del Arte

Alonso Del Arte

is a composer and photographer from Detroit, Michigan. He has been working on a Java program to display certain mathematical diagrams.

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