A Binary Inequality with Prime and Square-Free Number

This paper is devoted to the study of a binary inequality that links prime numbers with square-free integers. The main goal is to show that, for any exponent lying in a certain interval greater than one and smaller than a critical constant, and for every sufficiently large positive number $$N$$ , the inequality under investigation admits a solution. In such a solution one of the variables is a prime number, while the other is restricted to be square-free. The approach relies on analytic number theory, in particular on estimates for exponential sums involving prime numbers together with sieve methods to control the contribution of square-free integers. The obtained result demonstrates that even under the arithmetic restriction of being square-free, the inequality can still be solved for infinitely many sufficiently large values of $$N$$ . This extends previous research, where either primes or unrestricted integers were considered, and provides a new contribution to the understanding of additive problems involving primes and special classes of integers.