Claims

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1974_12_01_good: Good's 1974 note evaluates the reciprocal Fibonacci sum over the indices 2^k as (7 - sqrt 5)/2, a quadratic irrational, so the instance n_k = 2^k of the question has answer yes.

1976_12_01_hoggatt_bicknell: Hoggatt and Bicknell evaluate the reciprocal Fibonacci sum over the indices 2^n k for every fixed k in closed form with the coefficient -1/2 on sqrt 5, so every instance n_j = 2^j k of the question has answer yes.

1987_07_01_badea: Badea's 1987 Corollary 4 proves the reciprocal Fibonacci sum over the indices 2^n + 1 irrational by his criterion for series of positive rationals, so the instance n_k = 2^k + 1 of the question has answer yes.

1993_01_01_badea: Badea's Corollary 3.2 of 1993 proves the reciprocal Fibonacci sum irrational whenever n(k+1) is at least 2n(k)-1 for all large k, which answers the problem for every ratio constant c at least 2; the range 1 < c < 2 is not covered.

2026_04_24_deepmind: A Lean proof found by AlphaProof that the reciprocal Fibonacci sum over the indices 2^k is irrational, the instance n_k = 2^k of the question.

2026_07_15_snyder: A 2026 Lean 4 development, posted through the Star Fleet Math site and the problem's proof-claims thread, claims that the reciprocal Fibonacci sum along any index sequence with ratios at least some c > 1 is irrational.

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