I was able to download the paper, but it took me a bit of clicking, as it was not entirely obvious what was the “button” to only download that paper. Nothing to do with your work, just the curious style of “Academy”. Anyway, you might be interested in the Euler conjecture tutorial, as there a similar number-theoretical approach is taken - Euler Sum of Powers Conjecture: Tutorial.
I have not yet read the paper, but I can imagine that large integers are required. There are various libraries to deal with them in Fortran. One is fmzm, see Arbitrary/multiple precision libraries.