Diophantine equations involving powers of factorials

2025/09/23 by Novaković, Saša
#11A25 #11A41 #11D99 #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2509.18860

Abstract

We are motivated by a result of Alzer and Luca who presented all the integer solutions to the relations (k!)ⁿ-kⁿ=(n!)^k-n^k and (k!)ⁿ+kⁿ=(n!)^k+n^k. We consider the equations (k!)^n!± kⁿ=(n!)^k!± n^k and (k!)ⁿ± k^n!=(n!)^k± n^k! and prove a similar statement.

Citations

• Some Diophantine equations involving arithmetic functions and Bhargava factorials
• A note on some polynomial-factorial diophantine equations
• Explicit bounds for primes in arithmetic progressions

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