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erdos #936: [math]2^n \pm 1[/math] and [math]n! \pm 1[/math] powerful for only finitely many [math]n[/math]?

conditional (abc): yes — cushing-pascoe '16, crowdmath '20
unconditional structural constraints:

[math]2^n - 1[/math] (mersenne)

[math]\forall n \ge 2, n \neq 6[/math]: zsigmondy [math]\exists[/math] primitive [math]p, \operatorname{ord}_p(2)=n[/math]
[math]2^n-1[/math] powerful [math]\implies[/math] LTE [math]\implies p^2 \mid 2^n-1 \implies p[/math] is wieferich
[math]n=6[/math] exception: [math]63=3^2 \cdot 7[/math], [math]3 \nmid \operatorname{ord}_3(2)[/math] (since [math]3 \mid 2^2-1[/math])
[math]\implies #{\text{powerful mersennes}} \le #{\text{wieferich primes}} = 2[/math] (known)
density sieve:

[math]p=3[/math] kills [math]n \equiv 2,4 \pmod{6}[/math] 33.3% gone
covering set {3,5,7,13,17,241,...} [math]\ge 63.3%[/math] excluded unconditionally

[math]n! \pm 1[/math]

[math]n!+1[/math] powerful [math]\land[/math] [math]n+1[/math] prime [math]\implies n+1[/math] wilson prime (known: 5,13,563)
[math]n!-1[/math] powerful [math]\land[/math] [math]n+2[/math] prime [math]\implies W_{n+2} \equiv 1 \pmod{n+2}[/math] (near-wilson)
both conditions filter out almost all prime candidates

[math]2^n + 1[/math]

[math]n[/math] odd: [math]3 \mid n[/math] strictly necessary
[math]n[/math] even: [math]2^n+1[/math] = sum of two squares

[math]\forall p \equiv 3 \pmod{4}[/math]: [math]v_p(2^n+1)[/math] even (fermat two-squares)
powerful condition auto-satisfied for dirichlet density 1/2 of prime factors

tl;dr: wieferich/wilson bottlenecks + explicit density sieves rule out most [math]n[/math]
full unconditional resolution open