Puzzle 1235 6^n+n^6

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Puzzle 1235 6^n+n^6

Sebastián Martín Ruiz sent the following nice puzzle:

Let F(n)=6^n+n^6 OEIS A001594

For n=1 F(1)=7 is prime

For n>1 I have searched up to n=50000 (around 38900 digits) and no prime result was found.

Q) Prove that F(n) is composite for all n>1 or find a prime counterexample.

From Aug 30 to Sep 6, 2025, contributions came from Michal Branicky

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Michael wrote:

I tested n = 2..150000, finding no other prime F(n).

For n > 1:
* n must be odd else 2 | F(n);
* n mod 14 == 7 else 2 | F(n) or 7 | F(n);
* n mod 210 in {35,49,91,119,161,175} else 2, 3, 5, or 7 divides F(n).

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