Factorial Reduction

In the course of searching for continued fraction identities, Raayoni (2021) and Elimelech et al. (2023) noticed that while the numerator and denominator of continued fraction convergents p_n/q_n in general grow factorially (p_n,q_n∼(n!)^d for some positive integer d), the reduced numerator and denominator p_n/g_n and q_n/g_n for g_n=GCD(p_n,q_n) grew at most exponentially (p_n,q_n∼s^n).


This phenomenon has been termed "factorial reduction" and, while it is extremely rare in general (Elimelech et al. 2023), it holds for all identities originally found by the Ramanujan Machine (Raayoni et al. 2021, Elimelech et al. 2023). It is illustrated above for the Apéry constant continued fraction


See also

Continued Fraction, Convergent

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Elimelech, R.; David, O.; De la Cruz Mengual, C.; Kalisch, R.; Berndt, W.; Shalyt, M.; Silberstein, M.; Hadad, Y.; and Kaminer, I. "Algorithm-Assisted Discovery of an Intrinsic Order Among Mathematical Constants." 22 Aug 2023., G; Gottlieb, S.; Manor, Y.; Pisha, G.; Harris, Y.; Mendlovic, U.; Haviv, D.; Hadad, Y.; and Kaminer, I. "Generating Conjectures on Fundamental Constants With the Ramanujan Machine." Nature 590, 67-73, 2021.

Cite this as:

Weisstein, Eric W. "Factorial Reduction." From MathWorld--A Wolfram Web Resource.

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