Fermat Numbers: The Conjecture That Turned Out to Be Wrong
Pierre de Fermat was one of the most prolific mathematical minds of the seventeenth century, and few ideas consumed him as persistently as the question of prime numbers of a specific form. His conjecture about what we now call Fermat numbers is a remarkable story — not because he was right, but precisely because he wasn’t.
Fermat’s Lifelong Belief About Fermat Numbers
Fermat was passionately convinced — and, as he himself wrote, “almost certain” — that all numbers of the form 2^(2ⁿ) + 1 are prime.
He wrote about this in 1640 to Bernard Frénicle de Bessy, his primary correspondent on matters of number theory.
He wrote about it to Marin Mersenne, the central figure in the scientific circles of the time, through whom most scholarly correspondence passed.
He wrote about it to Pascal in 1654, when Pascal was loading him with problems about probability — and Fermat, having quickly worked through the probability questions, kept returning to his favorite problem.
Finally, in 1659, he wrote about it once more to Pierre de Carcavi, in a letter surveying all of his mathematical results.
In each case, Fermat pointed to the first five numbers of this form as evidence that the pattern held.
Euler Disproves the Conjecture
Unlike Fermat’s famous Last Theorem, this conjecture was refuted rather quickly. In 1732, Euler found the sixth Fermat number — and it turned out to be composite:
F = 4,294,967,297 = 641 × 6,700,417
Moreover, every Fermat number discovered after the first five has also turned out to be composite. To be fair, only 33 such numbers have been computed in total — they grow astronomically large.
What Remains Unknown About Fermat Numbers
Here is the twist: it is still unknown whether any prime Fermat numbers exist beyond the first five.
The conjecture has effectively flipped to its opposite. What Fermat believed to always be true, mathematicians now suspect may almost never be true — yet no one can prove it.
Another unsolved puzzle left behind by Fermat, still waiting for its answer.
