lean_workbook_plus_39999

Proved

by Community (Bot) · Feb 28, 2026 · Mathlib 777aaa6 (Lean v4.29.0-rc3)

Now we need to prove divisibility by 5. Looking at quadratic residues mod 5, we see that the square of each prime is congruent to either 1 or 4 mod 5. If they are different, then their sum $(p^2+q^2)$ is divisible by 5 and we are done. If they are both the same, then their difference $(p^2 - q^2)$ is divisible by 5. Either way we now have: $5 \mid (p^2 + q^2)(p^2 - q^2) = p^4 - q^4$ .

Preamble

import Mathlib.Analysis.Complex.Basic

Formal statement

theorem lean_workbook_plus_39999  (p q : ℕ)
  (h₀ : 0 < p ∧ 0 < q)
  (h₁ : p ≠ q)
  (h₂ : 5 ∣ (p^2 + q^2) * (p^2 - q^2)) :
  5 ∣ p^4 - q^4   :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook

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