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.
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.
In this paper, we find all the solutions of the title Diophantine equation in nonnegative integer variables (m, n, x) , where P k is the k th term of the Pell sequence.
We consider the family of Lucas sequences uniquely determined by Un+2(k) = (4k + 2)Un+1(k) − Un(k), with initial values U0(k) = 0 and U1(k) = 1 and k ≥ 1 an arbitrary integer. For any integer n ≥ 1 the discriminator function D k (n) of Un(k) is defined as the smallest integer m such that U0(k), U1(k), . . . , Un−1(k) are pairwise incongruent modulo m. Numerical work of Shallit on D k (n) suggests that it has a relatively simple characterization. In this paper we will prove that this is indeed the case by showing that for every k ≥ 1 there is a constant n k such that D k (n) has a simple characterization for every n ≥ n k . The case k = 1 turns out to be fundamentally different from the case k > 1.
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