# The exponential Diophantine equation $x2+(3n2+1)y=(4n2+1)z$

In: Mathematica Slovaca, vol. 64, no. 5
Wang Jianping - Wang Tingting - Zhang Wenpeng

## Details:

Year, pages: 2014, 1145 - 1152
Keywords:
exponential Diophantine equation, Pell equation, Lucas number
Let $n$ be a positive integer. In this paper, using the results on the existence of primitive divisors of Lucas numbers and some properties of quadratic and exponential diophantine equations, we prove that if $n \equiv 3 \pmod 6$, then the equation $x2+(3n2+1)y=(4n2+1)z$ has only the positive integer solutions $(x,y,z)=(n,1,1)$ and $(8n3+3n,1,3)$.
Jianping, W., Tingting, W., Wenpeng, Z. 2014. The exponential Diophantine equation $x2+(3n2+1)y=(4n2+1)z$. In Mathematica Slovaca, vol. 64, no.5, pp. 1145-1152. 0139-9918. DOI: https://doi.org/10.2478/s12175-014-0265-z
Jianping, W., Tingting, W., Wenpeng, Z. (2014). The exponential Diophantine equation $x2+(3n2+1)y=(4n2+1)z$. Mathematica Slovaca, 64(5), 1145-1152. 0139-9918. DOI: https://doi.org/10.2478/s12175-014-0265-z