In this paper, we find all the solutions of the Diophantine equation P ℓ +P m +P n = 2 a , in nonnegative integer variables (n, m, ℓ, a) where P k is the k-th term of the Pell sequence {P n } n≥0 given by P 0 = 0, P 1 = 1 and P n+1 = 2P n +P n−1 for all n ≥ 1.MSC: 11D45, 11B39; 11A25

We show that the field Q(x, y), generated by two singular moduli x and y, is generated by their sum x + y, unless x and y are conjugate over Q, in which case x + y generates a subfield of degree at most 2. We obtain a similar result for the product of two singular moduli. CDC, the IRN GandA (CNRS) and the ALGANT Program. Our calculations were performed using the PARI/GP package [9]. The sources are available from the second author.

Abstract. In this paper, we prove that there is no number with the Lehmer property in the sequence of Pell numbers.

In this paper, we find all solutions of the exponential Diophantine equation +1 − = in positive integer variables ( , , ), where is the -th term of the Balancing sequence.