On Y-coordinates of Pell equations which are Fibonacci numbers

Abstract Let $$d \ge 2$$ d ≥ 2 be an integer which is not a square. We show that if $$(F_n)_{n\ge 0}$$ ( F n ) 0 the Fibonacci sequence and $$(X_m, Y_m)_{m\ge 1}$$ X m , Y 1 m th solution of Pell equation $$X^2 -dY^2 = \pm 1$$ - = ± , then $$Y_m F_n$$ has at most two positive solutions ( n ) except for $$d=2$$ when it three $$(m,n)=(1,2),(2,3),(3,5)$$ 3 5 .