lean_workbook_plus_36456

⚠️ 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.

Find all pairs $(m,n) \in \mathbb{Z}$ such that\n\n $\huge{\displaystyle\color{blue} \begin{cases} 2m \equiv -1 \pmod n \n^2 \equiv -2 \pmod m \end{cases}}$\n\nWe'll assume that operations $\pmod 0$ are undefined, so $m, n \ne 0$ . Note that $n$ must be odd from the first equation. Then $n^2+2$ is odd, so $m$ must be odd as well. It follows that $m \equiv \frac{n-1}{2} \pmod n$ and $n^2+2 \equiv 0 \pmod m$ . Let $m=\frac{a(n-1)}{2}$ and $n^2+2=mb$ for some $a,b \in \mathbb{Z}$ . Then by combining equations and letting $ab=c$ , we get $2n^2-cn+(4+c)=0$ . The discriminant must be a perfect square, so $c^2-8c-32=d^2$ . Letting $d=(c+x)$ , we get $-8c-32=2cx+x^2$ . $x$ must be even, so we set $x=2y$ and simplify the equation to $-2c-8=cy+y^2$ , or $c=-\frac{12}{y+2}-(2-y)$ after rearranging. For $c$ to be an integer, $2-y$ must be a factor of $12$ . This gives possible $y$ values of $-10,-4,-2,-1,0,1,3,4,5,6,8,14$ , which correspond to probable $c$ values of $-17,-12,-11,3,4,9$ after removing duplicates. Plugging these into the discriminant and solving the quadratic equation gives us $=-3,-4,-5,2,4,10$ as theoretical values of $n$ . We know $n$ is odd, so that narrows it down to $-3, -5$ . Plugging these in and analyzing possible values of $m$ gives us four solutions: $(1,-3), (-11, -3), (-3, -5), (27, -5)$ , where a solution pair is represented as $(m,n)$ .

Why this node was retired

The posted statement is

theorem lean_workbook_plus_36456  (m n : ℤ)
  (h₀ : 2 * m ≡ -1 [ZMOD n])
  (h₁ : n^2 ≡ -2 [ZMOD m])
  (h₂ : 0 < m ∧ 0 < n) :
  (m = 1 ∧ n = -3) ∨ (m = -11 ∧ n = -3) ∨ (m = -3 ∧ n = -5) ∨ (m = 27 ∧ n = -5)   :=  by sorry

The source’s incomplete four-pair list is also wrong; the formalization additionally adds positivity incompatible with its negative-n conclusions.

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

Proposed corrected statement

Restore nonzero integer variables and give all22 pairs: (1,1),(3,1),(1,3),(27,5),(3,7),(17,7),(19,13),(−1,1),(−3,1),(−11,3),(−3,5), together with each pair’s second-coordinate sign reversal. If restricting to positive variables, use the first seven only.

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.