Resolution criteria

This market will resolve to "Yes" if, by the end of 2025, 78,557 is proven to be the smallest Sierpiński number. This proof would require demonstrating that all odd numbers less than 78,557 are not Sierpiński numbers, meaning there exists a positive integer n such that k × 2ⁿ + 1 is prime for each k. If no such proof is established by the end of 2025, the market will resolve to "No".

Background

A Sierpiński number is an odd natural number k for which k × 2ⁿ + 1 is composite for all natural numbers n. In 1962, mathematician John Selfridge proved that 78,557 is a Sierpiński number by showing that all numbers of the form 78,557 × 2ⁿ + 1 have a factor in the covering set {3, 5, 7, 13, 19, 37, 73}. (en.wikipedia.org)

The Sierpiński problem seeks to determine the smallest such number. As of now, 78,557 is the smallest known Sierpiński number, but it has not been definitively proven to be the smallest possible. To confirm this, it must be shown that for every odd k less than 78,557, there exists an n such that k × 2ⁿ + 1 is prime. (en.wikipedia.org)

Considerations

The distributed computing project PrimeGrid is actively working to eliminate remaining candidates by finding primes of the form k × 2ⁿ + 1 for k < 78,557. As of recent updates, five candidates remain: k = 21,181; 22,699; 24,737; 55,459; and 67,607. The discovery of a prime for each of these k values would confirm that 78,557 is the smallest Sierpiński number. (en.wikipedia.org)