Sun · Write n=k+m with 2ᵏ+m prime

For every integer n>1, prove that there is an integer k with 1≤k<n for which 2ᵏ+n−k is prime.

prime numberspowers of tworepresentations

The problem

For every integer n>1, prove that there is an integer k with 1≤k<n for which 2ᵏ+n−k is prime.

A 2026 preprint establishes a positive-density result but does not settle the universal conjecture.

Open since2013exact public formulation year

Last checked2026.07.27Catalog verification

Offer 01$1,000Zhi-Wei Sun

Personal offer

The author’s 2017 paper version announced US$1,000 for the first solution.