Additive Number Theory pp 303-317 | Cite as

# Support Bases of Solutions of a Functional Equation Arising From Multiplication of Quantum Integers and the Twin Primes Conjecture

## Summary

Let *P* be the support base of a solution *Γ*, with the field of coefficients of characteristic zero, of the functional equations arising from the multiplication of quantum integers discussed in [A. Borisov, M. Nathanson, Y. Wang, Quantum integers and cyclotomy, J. Number Theory (to appear); M. Nathanson, A functional equation arising from multiplication of quantum integers, J. Number Theory, 103(2), 214–233 (2003)]. It is known from the work of Nathanson as well as our work that there is a close relationship between *P* and the constructibility of *Γ* from quantum integers. In this paper, we prove that if the Twin Primes conjecture holds, then *Γ* is constructible from quantum integers if *P* contains infinitely many pairs of twin primes. This shows, in particular, that if the Twin Primes conjecture holds, then *Γ* is constructible from quantum integers whenever *P* has finite complement.

## Keywords

Polynomial functional equations Q-series Quantum integers Quantum algebra## References

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