Generalized Version of the Characteristic Number of Two Simultaneous Pell's Equations

A nonextendable Diophantine quadruple arising from a triple of Lucas numbers

“…This modulus involves only two specific odd primes, namely 3 and 5. Because of the inadequacy of such a restricted modulus for handling several problems, a method involving a general modulus was established in [Ramasamy 2006]. The present problem involves computational complexities and a new method is devised to overcome the computational difficulty by employing a result in this same reference.…”

“…Conclusion of the argument for solutions of the form (24). As mentioned, the characteristic number (in the generalized version given in [Ramasamy 2006] and explained earlier in this section) places several polynomials at our disposal for solving the problem. Each polynomial can potentially exclude several values of t. Once all values of t are excluded, we need not examine the remaining polynomials.…”

Sequences in finite fields yielding divisors of Mersenne, Fermat and Lehmer numbers, I

“…To settle the question concerning the common solutions of two Pell's equations, the concept of the characteristic number of two simultaneous Pell's equations was introduced by Mohanty and the author in [7]. A generalized version of this method was presented by the author in [8]. Two functions viz.…”

“…Define a(t) = A 2 t−1 and b(t) = B 2 t−1 , where A r + B r √ D denotes a solution of the Pell's equation A 2 − DB 2 = 1, D being a square-free natural number. The properties possessed by these functions was the focus of attention in [8]. These functions satisfy the relations a(t + 1) = 2(a(t)) 2 − 1 and b(t + 1) = 2a(t)b(t).…”