Claims

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2026_01_11_somani: For every a at least 2, the central binomial coefficients with indices a, 2a+2 and 8a^2+8a+1 have the same product as those with indices a+1, 2a and 8a^2+8a+2, so the equation has infinitely many distinct-index solutions.

2026_01_29_feng: Theorem 4 of the Gemini case study: for every k at least 3 the central binomial coefficients with indices k, 2k-2 and 8k^2-8k+2 have the same product as those with indices k-1, 2k and 8k^2-8k+1; Somani's family again.

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