The positive integer solutions of a five-variable Diophantine equation on Euler function
JIANG Lianxia1， ZHANG Sibao1,2
(1.School of Mathematics and Statistics， Kashi University， Kashi 844000， Xinjiang， China； 2. Research Center of Modern Mathematics and Its Application， Kashi University， Kashi 844000， Xinjiang， China)
Abstract：For any positive integer n， arithmetic function φ(n) is Euler function. The positive integer solutions of a five-variable Diophantine equation φ(x1x2x3x4x5)＝3φ(x1)φ(x2)φ(x3)＋4φ(x4)φ(x5) on Euler function φ(n) were discussed in this paper. By using the properties of Euler function φ(n) and the elementary method， the 501 positive integer solutions of this equation were obtained， and the 61 positive integer solutions satisfied x1≤x2≤x3and x4≤x5 were given.
2024, Vol. 58 
