lean_workbook_plus_8035

⚠️ Retired — specification defect

The Lean statement below does not encode the problem shown on this page, so its Proved status carries no information about that problem. Do not import this node or use it as a dependency.

The integer N is positive. There are exactly $2005$ ordered pairs $(x,y)$ \n\n of positive integers satisfying \n\n $1/x+1/y=1/N$ \n\n. \n\n Prove that $N$ is a perfect square. \n\n

Why this node was retired

The posted statement is

theorem lean_workbook_plus_8035 (N : ℕ) (hN : 0 < N) (hxy : { (x,y) : ℕ × ℕ | 1/x + 1/y = 1/N } = {2005}) : ∃ k : ℕ, k^2 = N   :=  by sorry

The target singleton and natural division make its hypotheses inconsistent.

A proof of a malformed proposition can be a correct proof of that proposition, so this is not a judgment on the accepted submission — but the Proved status must not be read as settling the problem shown above.

Proposed corrected statement

Count exactly 2005 ordered pairs of positive integers satisfying the equation using exact rational division; formalize cardinality, not equality of the solution set with a singleton numeral.

Diagnosis and correction from the public Prove2Me statement audit (wamlat/prove2me-errors). The correction is natural-language mathematics and is not Lean-verified — it is a specification for a corrected node, not a drop-in replacement. No corrected replacement node exists yet.