lean_workbook_plus_18283

⚠️ Retired — specification defect

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

Actually, it is quite easy to prove that the minimal solution of the Pell Equation $x^2-py^2=1$ must have $x$ even, $y$ odd by a similar trick, but we can proceed directly, so that we don't need to double the argument: \nWe start with the minimal solution $(a,b)$ of $X^2-pY^2=1$. So $(a-1)(a+1)=pb^2$. So one of the following four cases must occur: \n1. $a-1=pm^2, a+1=n^2$. So $n^2-pm^2=2$. \n2. $a-1=n^2, a+1=pm^2$. So $n^2-pm^2=-2$. \n3. $a-1=2pm^2, a+1=2n^2$. So $n^2-pm^2=1$. \n4. $a-1=2n^2, a+1=2pm^2$. So $n^2-pm^2=-1$. \nNote that $-1$ and $-2$ are not quadratic residues $\mod p$, so cases 2. and 4. cannot occur. \nMoreover, case 3. would contradict the minimality of $(a,b)$. \nSo case 1. must occur and $(n,m)$ is a solution of the equation $X^2-pY^2=2$.

Why this node was retired

The posted statement is

theorem lean_workbook_plus_18283  (p : ℕ)
  (a b : ℕ)
  (h₀ : Nat.Prime p)
  (h₁ : a^2 - p * b^2 = 1)
  (h₂ : ∀ x y : ℕ, x^2 - p * y^2 = 1 → a ≤ x ∧ b ≤ y) :
  Even a ∧ Odd b   :=  by sorry

Minimality includes the trivial Pell solution(1,0), and natural subtraction can truncate; the source nonresidue assumptions on p are missing.

The recorded counterexample refutes the statement as encoded. It says nothing about the problem shown above, which is a different proposition.

Proposed corrected statement

Use integer Pell arithmetic, minimize among x,y>0 solving x²−py²=1, and recover the source restriction that−1 and−2 are nonresidues modulo p. A parity conclusion is not valid for arbitrary prime p.

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.