slug: fibonacci-pythagorean-perimeter-refutation bibkey: huber2023a134492 doi: null url: https://oeis.org/A134492 triage: theorem motivation_gids:

• D5/S3/Arith/FibonacciPythagoreanPerimeterRefutation.result

Fibonacci Numbers That Are Pythagorean Perimeters

Problem

OEIS A134492 is the sequence a(n) = Fibonacci(6n). Its COMMENTS section carries, at revision #62 of Sep 22 2025:

Conjecture: For n >= 2, the terms of this sequence are exactly those Fibonacci numbers which are the sum of the three numbers of a Pythagorean triple (checked up to F(80)). - Felix Huber, Nov 03 2023

With quantifiers written out, and P(N) for ∃ a b c > 0, a² + b² = c² ∧ a + b + c = N:

∀ N, (∃ k, F(k) = N) → (P(N) ↔ ∃ n ≥ 2, F(6n) = N).

The comment says “a Pythagorean triple” with no qualifier, so the triples are all triples of positive integers, primitive or not.

Motivation

The frozen theorem D5/S3/Arith/FibonacciPythagoreanPerimeterRefutation.result refutes it.

Gap

Issue 9430 records the screen carried out before the probe. The entry was read in full at revision #62: the conjecture stands unqualified, with no counterexample and no proof recorded. A web pass over the sequence number together with the terms of the conjecture returned nothing. Citation indices were not exhaustively reachable, so this is a bounded negative finding.

Route

Every Pythagorean triple is d times a primitive one with parameters m > n ≥ 1, gcd(m, n) = 1, m - n odd, so its perimeter is 2 d m (m + n). Writing s = m and t = m + n, the perimeters are exactly the numbers 2 s t d with s < t < 2s, gcd(s, t) = 1 and t odd. So N is a perimeter precisely when N is even and N/2 has a factorisation s · t · d of that shape.

Two consequences fix where the conjecture can fail. First, perimeters are even, so a Fibonacci number at an index not divisible by three, being odd, is never one; those indices agree with the conjecture for trivial reasons. Second, at an index divisible by three the question becomes whether F(k)/2 admits a pair of coprime divisors s < t < 2s with t odd — a question about how the divisors of F(k) are spaced, with no reason to answer no once F(k) has enough prime factors.

Running that test over the indices divisible by three finds the first failure at index forty-five. F(45) = 1134903170, F(45)/2 = 61 · 85 · 109441, and the parameters s = 61, t = 85, that is m = 61, n = 24, give the primitive triple (3145, 2928, 4297). Scaling by 109441:

344191945² + 320443248² = 470267977² ,
344191945 + 320443248 + 470267977 = 1134903170 = F(45) .

Forty-five is not a multiple of six, and F(45) is not a term of the sequence because F(42) = 267914296 < 1134903170 < 4807526976 = F(48) and Nat.fib is monotone. The recorded proof carries only the triple and that monotonicity step; the divisor analysis above is how the triple was found, not part of the proof.

Falsifier

A different value for F(45), or an arithmetic slip in the triple, would invalidate the witness; both are checked by the kernel. The witness does not depend on the parametrisation of Pythagorean triples: it is three explicit positive integers whose squares and whose sum are computed directly.

Evidence

Forty-five is the smallest counterexample. At indices not divisible by three the Fibonacci numbers are odd and hence not perimeters, matching the conjecture; at the indices 3, 6, ..., 42 divisible by three the conjecture holds. Further failures of the same kind occur at indices 57, 63, 69, 75, 81.

The counterexample lies inside the range the comment reports having checked, F(80). Under the alternative reading that restricts the triples to primitive ones the conjecture also fails inside that range: F(18) and F(54) are terms of the sequence but are not perimeters of primitive triangles, and F(81) is one without 81 being a multiple of six. So no reading of the comment survives its own stated verification range.

Triage

theorem; Tier 1 named external conjecture on an OEIS comment line, preregistered in issue 9430 before the probe. The admission basis is open-problem-resolution; the conservative classification is proof_shape: bind-only with escape_witness: none, since the proof is one certified witness together with monotonicity of Nat.fib. The computational use is a certified-instance with a typed refutes edge from result to claim.

ASSUMED-UNVERIFIED

The literature screen is bounded: the OEIS entry at revision #62 and a web pass over the sequence number and the conjecture’s terms were opened; citation-index result pages were not exhaustively reachable, so no worldwide priority claim is made.

The further failing indices 57, 63, 69, 75, 81 and the primitive-reading failures at 18, 54, 81 were found by the divisor test described above and are not part of the recorded proof; they are reported as computation, not as kernel-checked facts.