Existence of n such that M - 2^n or M + 2^n has no Prime factors less than 331

Theorem

Let $M \in \Z$ be an integer.

Then there exists a positive integer $n \in \Z_{\ge 0}$ such that either $M - 2^n$ or $M + 2^n$ has no prime factors less than $331$.

Proof

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Sources

• 1975: Frederick R. Cohen and J.L. Selfridge: Not Every Number is the Sum or Difference of Two Prime Powers (Math. Comp. Vol. 29, no. 129: pp. 79 – 81)  www.jstor.org/stable/2005463

• 1986: David Wells: Curious and Interesting Numbers ... (previous) ... (next): $331$
• 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $331$