The Almost-Proof of Fermat’s Last Theorem


Fermat’s Theorem continues to remain one of the biggest open problems in Mathematics. The needull discusses one big flaw in all the proposed solutions to the theorem.

Except there was a catch. As my story “New Number Systems Seek Their Lost Primes” describes, by expanding the number system to include new values, mathematicians lost something essential: unique prime factorization. Primes are the atoms of a number system — its fundamental building blocks — and unique prime factorization ensures that any number, such as 12, can be expressed uniquely as a product of primes: 2 x 2 x 3. The expanded number systems used to solve Fermat’s Last Theorem yielded competing prime factorizations, making these systems an ultimately shaky basis on which to construct a proof.

“Even today, in many false proofs of Fermat’s Last Theorem found by amateurs, somewhere or other this is the mistake — they’re assuming in some of these bigger number systems that numbers can be uniquely decomposed into primes,” said Manjul Bhargava, a mathematician at Princeton University. “It’s so counterintuitive to think that could fail for a bigger number system, but it sometimes does.”

The complete article

Kevin Hartnett – Quanta

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