The Connection Between Fermat Numbers and Brocard-Ramanujan Type Diophantine Equations

V. Pandichelvi, B. Umamaheswari

Abstract: In this manuscript, the study focuses on the Brocard-Ramanujan type Diophantine equation n!+1=F_t² where F_t=2^k+1, k=2^t, n, t≥0 is a Fermat number, is examined and demonstrates (n,k,t)=(4,2,1) is a sole positive integer solution. Additionally, a Python program to validate this result together with the geometrical presentation is developed.

Keywords: Fermat number, Brocard-Ramanujan Diophantine equation, Integer solution

How to Cite?: V. Pandichelvi, B. Umamaheswari, "The Connection Between Fermat Numbers and Brocard-Ramanujan Type Diophantine Equations", Volume 14 Issue 9, September 2025, International Journal of Science and Research (IJSR), Pages: 802-805, https://www.ijsr.net/getabstract.php?paperid=SR25917104020, DOI: https://dx.doi.org/10.21275/SR25917104020