OP-00004 Open

Primes of the form n² + 1

Are there infinitely many primes $p$ such that $p - 1$ is a perfect square? In other words: Are there infinitely many primes of the form $n^2 + 1$?

Source: Landau, ICM Cambridge 1912, the first of his four problems; Euler tabulated prime values of n² + 1 in a letter to Goldbach (1752). See Guy, Unsolved Problems in Number Theory, A1.

Classification

Primary: NT — Number Theory
Keywords: Landau's problems, primes represented by polynomials

References

@inproceedings{landau1912,
  author = {Landau, Edmund},
  title = {Gel{\"o}ste und ungel{\"o}ste Probleme aus der Theorie der Primzahlverteilung und der Riemannschen Zetafunktion},
  booktitle = {Proceedings of the Fifth International Congress of Mathematicians (Cambridge, 1912)},
  volume = {1},
  pages = {93--108},
  year = {1913}
}

@book{guy2004,
  author = {Guy, Richard K.},
  title = {Unsolved Problems in Number Theory},
  edition = {3},
  publisher = {Springer},
  year = {2004}
}

@article{iwaniec1978,
  author = {Iwaniec, Henryk},
  title = {Almost-primes represented by quadratic polynomials},
  journal = {Inventiones Mathematicae},
  volume = {47},
  pages = {171--188},
  year = {1978}
}

@misc{wikipedia,
  title = {Landau's problems},
  howpublished = {Wikipedia},
  url = {https://en.wikipedia.org/wiki/Landau%27s_problems}
}

@misc{formalconjectures,
  author = {{The Formal Conjectures Authors}},
  title = {Formal Conjectures},
  url = {https://github.com/google-deepmind/formal-conjectures/blob/0771383387505c96d1b2f6a3d35088ad00892c5c/FormalConjectures/Wikipedia/PrimesAndPerfectSquares.lean#L31},
  note = {Lean statement under Apache-2.0; problem text under CC BY 4.0, or CC BY-SA 4.0 where it is based on Wikipedia}
}

Canonical Lean statement

A submission resolves this problem by proving the constant Problem_OP_00004.

/-- OP-00004: Primes of the form n² + 1 -/
def Problem_OP_00004 : Prop :=
  {n | Prime (n ^ 2 + 1)}.Infinite

Formalised by The Formal Conjectures Authors (Apache-2.0); reviewed by clemens-admin.

Available to match against in: Lean v4.33.1 with Mathlib v4.33.1.

Resolving this problem in Lean

Add the problems package as a dependency, import this problem's module, and prove the constant. Any published revision works.

In lakefile.toml:

[[require]]
name = "RegistryProblems"
git = "https://github.com/cthalhammer/registry-problems"
rev = "<published revision>"

In your proof:

import RegistryProblems.OP_00004

theorem resolution : RegistryProblems.Problem_OP_00004 := by
  sorry

In preprint.toml, mapping the claim to the declaration that establishes it:

[[statements]]
label = "Theorem 1"
lean = "resolution"
latex = "..."
problem = "OP-00004"
claim = "full"

To disprove it instead, prove the negation and claim "disproof", or "counterexample" if the proof exhibits one:

theorem disproof : ¬ RegistryProblems.Problem_OP_00004 := by
  sorry

You can also write the statement out in full; the verifier checks it matches up to definitional equality. Partial results and reductions are not matched against this statement.

Citing this problem

This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.

OP-00004, "Primes of the form n² + 1", clopen.solutions.