The q-integers and the Mersenne numbers

The q-integers and the Mersenne numbers: Here we will show that the q-integers, the q-analogue of the integers that we can find in the q-calculus, are forming an additive group having a generalized sum similar to the sum of the Tsallis q-entropies of independent systems. The symmetric form of q-integers will be studied too. We will see that these numbers are linked to the Kaniadakis \kappa-calculus. In the article, a final discussion will be devoted to the link of the q-integers to the Mersenne numbers. Besides the discussion of the previously mentioned numbers, the general aim of the paper is that of popularizing the existence of the q-calculus.