Pell and Pell-Lucas Numbers Associated with Brocard-Ramanujan Equation

Abstract

In this paper, the diophantine equations of the form $A_{n_{1}}A_{n_{2}}\cdots A_{n_{k}}\pm 1=B_{m}^{2}$ where $(A_{n})_{n\geq 0}$ and $(B_{m})_{m\geq 0}$ are either the Pell sequence or Pell-Lucas sequence are solved by applying the Primitive Divisor Theorem. This is another version of Brocard-Ramanujan equation.

Keywords

• Pell number,Pell-Lucas number,Brocard-Ramanujan equation,Diophantine equation

References

1. Berndt, B. C., Galway, W. F., On the Brocard-Ramanujan Diophantine equation $n!+1=m^{2}$, Ramanujan J., 4(1)(2016), 41--42.
2. Carmichael, R. D., \On the numerical factors of the arithmetic forms $\alpha ^{n}\pm \beta ^{n}$, Ann. of Math. Second S., 15(1/4)(1913), 30--48.
3. Dabrowski, A., On the Brocard-Ramanujan problem and generalizations, Colloq. Math., 126(1)(2012), 105--110.
4. Luca, F., The Diophantine equation $P(x)=n!$ and a result of M. Overholt, Glas. Math. Ser. III, 37(2)(2002), 269--273.
5. Marques, D., The Fibonacci version of the Brocard-Ramanujan diophantine equation, Portug. Math., 68(2011), 185--189.
6. Pongsriiam, P., Fibonacci and Lucas numbers associated with Brocard-Ramanujan equation, Commun. Korean Math. Soc., 32(3)(2017), 511--522.
7. Szalay, L., Diophantine equations with binary recurrences associated to Brocard-Ramanujan problem, Port. Math., 69(3)(2012), 213--220.