Sierpiński and Riesel numbers among repdigits, repunits, repintegers and repstrings: six improved bounds and two questions answered

Abstract

Bispels, Cohen, Harrington, Lowrance, Pontes, Schaumann and Wong asked, for each of eight repetitive shapes of integer, for the smallest base or the smallest repeating block that admits a Sierpiński or a Riesel number of that shape, and answered each of the eight questions only with an upper bound. We improve six of the eight bounds and settle the remaining two exactly. The improvements are β₁ ≤ 53, β₂ ≤ 40, β₂' ≤ 39 and β₁' ≤ 94 for the four base questions, against 147, 87, 180 and 16518444216571; and κ ≤ 16519, κ' ≤ 16519 for the two 2-repinteger block questions, against 18107. For the two 2-repstring questions we prove tildeκ=tildeκ'=1: the block k=1 already works, so those two questions are closed rather than improved, and the witnesses are Selfridge's and Riesel's own covering systems, reused without changing a single residue class. Three of the improvements are invisible to a search over covering systems whose period divides 144, where all the classical covering systems live; they need period 60, and we isolate the arithmetic reason. Every statement is proved from an explicit covering certificate and has been machine-checked in Lean 4.

Open review

This founding-collection manuscript received AI review before publication. Independent human review is open. Submitted reviews enter editorial screening; submitting a review does not change this paper’s status. Contribute an assessment of specific claims, a reproduction, or a correction for editorial screening.

Archived files

1. Version 1 · current (opens in a new tab)

   Source snapshot 2026-08-30 15:34 UTC

   File fingerprint

   1eb1ac3386a0a30d2f087123c474b79118f34b321ca4d93994473daec5d25951