BROCARD-RAMANUJAN BENZERİ ÇEŞİTLİ DENKLEMLER ÜZERİNE

In this study, we investigate the positive integer solutions of the Diophantine equation𝑚3=(𝑚+1)!+𝑚m 3 =(m+1)!+mwhich resembles the well-known Brocard–Ramanujan type equations. Equations involving factorial and power terms, such as 𝑛!+1=𝑚2n!+1=m 2 , are of particular interest due to the rarity of their integer solutions and the implications for number theory. Initially, small positive integers were tested directly, revealing that the equation holds only for 𝑚=2m=2 and 𝑚=3m=3. Considering that the factorial function grows significantly faster than the cubic term, it was conjectured that no further solutions exist. This conjecture was then rigorously confirmed using mathematical induction. All symbolic computations were verified with the aid of SageMath and Wolfram Mathematica software. As a result, it was proven that the only positive integer solutions to the given equation are 𝑚=2m=2 and 𝑚=3m=3. This finding aligns with general results in the literature, which suggest that exponential Diophantine equations involving factorial expressions typically have only finitely many solutions.