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import FormalConjecturesUtilGreen's Open Problem 23
References:
Green, Ben. "100 open problems." (2024).
[FrKlMo25] Frantzikinakis, N., O. Klurman, and J. Moreira. "Partition regularity of Pythagorean pairs." Forum of Mathematics, Pi 13. Cambridge University Press (2025).
namespace Green23
Suppose that $\mathbb{N}$ is finitely coloured. Are there $x,y$ of the same colour such that $x^2 + y^2$ is a square?
Solved in [FrKlMo25].
@[category research solved, AMS 5 11]
theorem green_23 :
answer(True) ↔
-- For every finite colouring of the natural numbers
∀ (k : ℕ) (c : ℕ → Fin k),
-- there exist two numbers of the same colour whose squares sum to a square
∃ x y : ℕ,
0 < x ∧ 0 < y ∧ -- Exclude trivial x^2 + 0^2 = x^2 solution
c x = c y ∧ -- Same colour
IsSquare (x^2 + y^2)
:= ⊢ True ↔ ∀ (k : ℕ) (c : ℕ → Fin k), ∃ x y, 0 < x ∧ 0 < y ∧ c x = c y ∧ IsSquare (x ^ 2 + y ^ 2)
All goals completed! 🐙
end Green23