In this paper, it will be shown that it is possible to detect if a number p is prime [1] or not, using the following expression involving the q-polygama function [2] (represented as ψ) and being c whatever positive real number higher than zero: q_2=1/2π ∫_(-π)^π〖e^pjω ((ψ_q (1)+ln⁡(1-e^(-c-jω) ))/(ln⁡(e^(-c-jω) )) -(e^(-c-jω)+e^(-2c-2jω))/(1-e^(-c-jω) )) 〗 dω (1) If the result is zero, p is a prime number. If it is different from zero, the result will give information about the number of factors that the number p has. We will check that using this as a basis, we can obtain the following integral: q_3=1/2π ∫_(-π)^π〖e^pjω (∑_(k=2)^(k=∞)〖e^(-c(2+k)) e^(-2kjω) 1/(1-〖e^(-c) e〗^(-kjω) )〗) 〗 dω (2) And it is possible to obtain the sum of the factors of a semiprime number p [3] with it. Once we have the sum and the product of the factors, it is immediate to obtain the two factors of the semiprime number. The solution of that integral (solving it numerically) is obtained in polynomial time (quadratic). To do so, the second element of the product inside the integral has had to be calculated before and stored in a table (data base). Once this prework is done, the result is given in polynomial time (linear) independently of p or its size. You can check that this is possible because the second element of the product within the integral does not depend on p: ∑_(k=2)^(k=∞)〖e^(-c(2+k)) e^(-2kjω) 1/(1-〖e^(-c) e〗^(-kjω) )〗 (3)