## The square terms in Lucas sequences.(English)Zbl 0851.11011

J. Number Theory 58, No. 1, 104-123 (1996); addendum ibid. 61, 420 (1996).
Let $$P$$ and $$Q$$ be relatively prime odd integers and define the sequences $$\{U_n\}$$ and $$\{V_n\}$$ by $$U_n = PU_{n - 1} - QU_{n - 2}$$ with $$U_0 = 0$$, $$U_1 = 1$$ and $$V_n = PV_{n - 1} - QV_{n - 2}$$ with $$V_0 = 2$$, $$V_1 = P$$. The main results of the paper are the following. (i) If $$V_n$$ is a square, then $$n = 1,3$$ or 5. (ii) If $$2V_n$$ is a square, then $$n = 0,3$$ or 6. (iii) If $$U_n$$ is a square, then $$n = 0,1,2,3,6$$ or 12. (iv) If $$2U_n$$ is a square, then $$n = 0,3$$ or 6. These results are nice extensions of some earlier ones concerning special (e.g. Fibonacci and Lucas) sequences.
Reviewer: Péter Kiss (Eger)

### MSC:

 11B39 Fibonacci and Lucas numbers and polynomials and generalizations 11D09 Quadratic and bilinear Diophantine equations
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