I borrow a possible description of algorithmic number theory from the presentation flyer of the book by Eric Bach and Jeffrey Shallit:

Algorithmic number theory represents the marriage of number theory with the theory of computational complexity. It may be briefly defined as finding integer solutions to equations, or proving their non-existence, making efficient use of resources such as time and space. Implicit in this definition is the question of how to efficiently represent the objects in question on a computer.

The problems of algorithmic number theory are important both for their intrinsic mathematical interest and their application to random number generation, codes for reliable and secure information transmission, computer algebra, and other areas.