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Divisors of Mersenne numbers under doubling

Primarily AI-generated textHuman understanding: some partsmath.NT — Number Theory

Contributed by Sunay Venkat ↗

Version 1 / Nov 12, 2026 / CC BY 4.0

Abstract

This paper studies a question of Erdős about how the total number of divisors of Mersenne numbers grows when the range is doubled. We prove that the doubling ratio is always at least 3, sharp, and nearly 4 whenever the relevant Fermat numbers are composite, as all known ones are. We also reduce the open question of whether the ratio tends to infinity to a statement about indices with few divisors. This leads to a new conjecture on the prime factors of Mersenne numbers next to near-Wagstaff numbers, which we study through implications, barrier results and a proven polynomial analogue.

Provenance statement

This proof was originally meant to work towards solving an Erdos problem, but was later spun off into its own paper after producing several novel ideas. The proof was created by Anthropic's Opus 5.5.

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Version history

  1. v1 · 13 pagesSubmitted by Sunay VenkatInitial depositCurrentNov 12, 2026