The property of having infinitely many prime numbers award these numbers to have many applications in various fields of sciences. One of the most important applications is their use in the creation of many public key cryptosystems' private keys. Therefore, the main aim of this paper is considering a well known form of primes generated by the Fermat-Euler equation p=x²+dy² and studying whether or not this form keeps the property of generating infinitely many primes if the unknowns x, y and p are terms in certain binary recurrence sequences called the Lucas sequences of the first kind \uₙ (a, b) \ or the second kind \vₙ (a, b) \. In other words, in this paper we present a technique for investigating the integer solutions (x, y, p) of the equation p=x²+dy², where the unknowns are terms in \uₙ (a, b) \ or \vₙ (a, b) \. We also apply this technique for determining the solutions (x, y, p) = (tᵢ (a, b), tⱼ (a, b), tₖ (a, b) ) with 1 i j k, where tₙ (a, b) represents the general term uₙ (a, b) or vₙ (a, b) under certain conditions on the integers a and b.
Hayder Hashim (2026) studied this question.