lean_workbook_plus_21538

⚠️ 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.

Let $a,b,c$ be three positive real numbers such that $\frac{1}{a}+\frac{1}{b}+\frac{1}{c}=a+b+c$ , prove that : $\frac{1}{(2a+3b+4c)^n}+\frac{1}{(2b+3c+4a)^n}+\frac{1}{(2c+3a+4b)^n}\le \frac{1}{3^{2n-1}}$ .

Why this node was retired

The posted statement is

theorem lean_workbook_plus_21538 (a b c : ℝ) (ha : 0 < a) (hb : 0 < b) (hc : 0 < c) (hab : a + b + c = 1 / a + 1 / b + 1 / c) : ∀ n : ℕ, (1 / (2 * a + 3 * b + 4 * c) ^ n + 1 / (2 * b + 3 * c + 4 * a) ^ n + 1 / (2 * c + 3 * a + 4 * b) ^ n) ≤ 1 / 3 ^ (2 * n - 1)   :=  by sorry

Natural subtraction changes n=0, but the source bound also fails at n=30,a=3/2,b=1,c=2/3: its first denominator is26/3 and27^30>3·26^30.

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 an integer exponent2n−1 or specify the intended index range, and recheck the bound for positive n.

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.