We aim to solve the equation 2(n) = `n2 + An + B, where `, A, and B are given integers. We find that this equation has infinitely many solutions only if ` = 1. Then we characterize the solutions to the equation 2(n) = n2 + An + B. We prove that, except for finitely many computable solutions, all the solutions to this equation with (A, B) = (L2m, F2 2m 1) are n = F2k+1F2k+2m+1, where both F2k+1 and F2k+2m+1 are Fibonacci primes. Meanwhile, we show that the twin prime conjecture holds if and only if the equation 2(n) n2 = 2n + 5 has infinitely many solutions.
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