E047: Fermat numbers: Pépin test + factor witnesses¶

Tags: number-theory, counterexample-search, visualization, fermat See: Valid Tags.

Highlights¶

Runs Pépin’s test for \(F_n = 2^{2^n} + 1\) (definitive for Fermat numbers).
Finds small, explicit factor witnesses by bounded trial division.
Visualizes how explosively \(F_n\) grows and where the first counterexample appears.

Demonstrate (computationally) why the statement “all Fermat numbers are prime” fails, and produce concrete witnesses.

How quickly do Fermat numbers become composite in practice, and how easy is it to exhibit a witness factor for the first failures?

Compute \(F_n\) for \(n\le n_{max}\).
Apply Pépin’s test (for \(n\ge 1\)) to classify prime vs. composite.
If composite, attempt to find a small factor by trial division up to a configurable bound.
Plot \(\log_{10}(F_n)\) and the smallest discovered factor (if any).

If Pépin says “composite”, the number is definitively composite (not a probabilistic claim).
The factor search is bounded: “no factor found” does not mean “prime”.
The classic factor for \(F_5\) (and often for \(F_6\)) appears quickly under the default bounds.

If this experiment is included in the docs gallery, include the published snapshot (report + params).

n	F_n	Pépin says prime?	smallest factor found
0	3	✅	—
1	5	✅	—
2	17	✅	—
3	257	✅	—
4	65537	✅	—
5	4294967297	❌	641
6	18446744073709551617	❌	274177

Pépin’s test is definitive for Fermat numbers (n ≥ 1).
The factor search here is bounded trial division, meant to produce small, explicit witnesses (e.g., for F_5 and F_6).

See Kř\'ıžek et al. [2013], Wikipedia contributors [2025], The OEIS Foundation Inc. [2025].