Claims

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1918_06_01_polya: Pólya's 1918 theorem that the largest prime factor of f(n) tends to infinity when f is a product of two distinct linear factors, applied to n(n+1); refereed in Mathematische Zeitschrift.

1935_01_01_mahler: Mahler's 1935 theorem on the largest prime factor of D x^2 - A, for A among plus or minus 1 and plus or minus 2, applied to (2n+1)^2 - 1 = 4n(n+1), gives F(n) > (log log n)/(1 + eps) for all large n; a journal paper.

1967_01_01_schinzel: Schinzel's theorem on the greatest prime factor of quadratic values along a sequence gives infinitely many n with the largest prime factor of n(n+1) at most n to the power O(1/log log log n); refereed in Acta Arithmetica.

2023_12_06_pasten: Pasten's 2024 theorem on the largest prime factor of xy(x+y), at x = 1 and y = n, gives the lower bound F(n) >> (log log n)^2 / log log log n, the best known; refereed in Inventiones Mathematicae.

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