This paper is the first of several papers in which we prove, for the case where the fields of coefficients are of characteristic zero, four open problems posed in the work of Melvyn Nathanson (2003) [1] concerning the solutions of a functional equation arising from multiplication of quantum integers [n] q = q n−1 + q n−2 + · · · + q + 1.In this paper, we prove one of the problems. The next papers, namely [2-4] by Lan Nguyen, contain the solutions to the other 3 problems.

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Given a set of primes P , we determine the necessary and sufficient criterions for the existence of a sequence of polynomials Γ , with support base P , which is a solution of the functional equations arising from multiplication of quantum integers discussed in Melvyn B. Nathanson (2003) [1] and which cannot be generated by quantum integers.

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