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.”
Needull in a haystack 
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