Consecutive binomial coefficients in Pythagorean triples and squares in the Fibonacci sequence (Q2785033)

All triples consisting of consecutive binomial coefficients \(C(n,k)\), \(C(n,k + 1)\), \(C(n,k + 2)\) forming Pythagorean triples are found. The theorem is the following: If the three numbers listed previously form a Pythagorean triple, then \(n = 62\) and \(k = 26\) or \(34\). Results by \textit{P. Ribenboim} and \textit{W. L. McDaniel} [J. Number Theory 58, 104-123 (1996); Addendum ibid. 61, 420 (1996; Zbl 0851.11011)] and \textit{J. H. E. Cohn} [J. Lond. Math. Soc. 39, 537-540 (1964; Zbl 0127.26705); Pac. J. Math. 41, 631-646 (1972; Zbl 0248.10016)] on square Fibonacci numbers are utilized.