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Infinitude of composite Fermat numbers

Determine whether infinitely many Fermat numbers 2^{2^n}+1 are composite.

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Background

The authors compare the difficulty of their main problem to questions about Fermat numbers. It is known that Fermat numbers are prime for 0≤n≤4 and composite for 5≤n≤32, but proving the infinitude of composite Fermat numbers has resisted progress and is widely recognized as a challenging open problem.

References

It is a long-standing problem to prove (or disprove) the infinitude of the composite Fermat numbers.

— Intervals without primes near an iterated linear recurrence sequence (2504.14968 - Saito, 21 Apr 2025) in Introduction (discussion of Fermat numbers)