That successors, N, N’, have no common factor is easier to learn than there are infinitely many primes. It is easier to learn in the sense that we can understand the reason why it is true more directly.
(N’-N)/q = N’/q – N/q = j-i
where j and i are naturals.
(N’-N)/q = 1/q
j-i = 1/q
this can’t be.
The statement of the lemma suggests how to go about the proof. Whereas with the infinitely many primes, there is no suggestion of how to start the proof. Thus it is mysterious. Thus it is harder to learn. Thus it is harder to remember.
Successors have no common factor is easier to learn, prove and remember.