Sum of two numbers with prime conditions

The conjecture for sequence A231201: for any n > 1, there exist x, y > 0 such that n = x + y and 2^x + y is prime.

From the catalogue. Imported from The Formal Conjectures Authors (Google DeepMind and contributors) (Apache-2.0) — original. Nobody has started on it here yet: tasks are created as soon as someone asks for one or submits a claim. A Lean proof is checked against the statement below by the Lean kernel; a curator confirms before the problem counts as resolved.

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Cite

@misc{cairn-oeis-231201,
  title        = {Sum of two numbers with prime conditions},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/oeis-231201}},
  year         = {2026},
  note         = {Open problem on Cairn Commons, CC BY 4.0. Accessed 2026-09-29}
}

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Claims

Verified

Disputed

Refuted

On the literature board

The problem

The question

The conjecture for sequence A231201: for any $n > 1$ , there exist $x, y > 0$ such that $n = x + y$ and $2^{x} + y$ is prime.

Number of ways to write $n = x + y$ , for $x, y > 0$ such that $2^{x} + y$ is prime.

Zhi-Wei Sun has offered a \$1000 prize for the first proof.

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.OEIS.«231201».

theorem conjecture (n : ℕ) (hn : 1 < n) : A n

What counts as progress

A Lean proof of the pinned statement (or of its negation, for a yes/no question) — checked by the Lean kernel against the upstream statement; a curator confirms before the problem is marked resolved.
Partial results: special cases, weaker bounds, reductions — as verified claims.
Computations and numerical evidence with published code (reproducible).
Literature: the problem may have been solved or partly solved already. Report it as a literature claim.
A precise flaw in the formal statement (a misformalisation) — report it upstream too.

References

A231201
Zhi-Wei Sun, "Table of n, a(n) for n = 1..10000", "Write n = k + m with 2^k + m prime", a message to Number Theory List, Nov. 16, 2013, "On a^n+ bn modulo m", arXiv:1312.1166 [math.NT], 2013-2014, "Problems on combinatorial properties of primes", arXiv:1402.6641 [math.NT], 2014-2015.

Source and licence

Imported from Formal Conjectures (OEIS), commit e6d1743831c2. Statements and descriptions © The Formal Conjectures Authors, Apache License 2.0; reformatted for this page.