lean_workbook_plus_299

⚠️ 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.

Each number in S is included in 10 different differences. We see that $2^{10}$ is counted as positive in each of the 10 differences. Thus, the total sum produced from the $2^{10}$ s is $10{}^2{}^10$ . $2^9$ is positive in 9 differences, and counted as negative in 1. Thus, the total sum produced from the $2^9$ s is $9*2^9-2*9=8*2^9$ . We can continue, and we find N to be $10*2^10+8*2^9+6*2^8+...-8{}^2{}^1-10*2^0$ . Taking this mod 1000, we find $N \equiv 240+96+536+512+128-32-32-24-16-10 \equiv \boxed{398} \pmod{1000}$

Why this node was retired

The posted statement is

theorem lean_workbook_plus_299  (n : ℕ)
  (h₀ : n = 10 * 2^10 + 8 * 2^9 + 6 * 2^8 + 4 * 2^7 + 2 * 2^6 - 8 * 2^5 - 10 * 2^4 - 12 * 2^3 - 14 * 2^2 - 16 * 2^1 - 18 * 2^0) :
  n % 1000 = 398   :=  by sorry

The formal statement changes or incorrectly supplies part of the source problem; the accepted note identifies a concrete witness or definitional mechanism.

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 arithmetic and the source coefficient pattern N=∑_{k=0}^{10}(2k−10)2^k; prove N mod 1000=398. The stored tail coefficients differ from this pattern.

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.