## 2 replies on “Math From Scratch 21: Totients”

1. StarDrifter says:

Some nitpicking:

In section 1 you constrain phi(n) to the range [1, n) but I think you instead want to constrain the “integers that are coprime to n” to that range.

In section 2.3 did you switch the prime factorization exponents from using “e” to “n” in the middle or did I miss something? Sticking with “e” would seem more clear since you are already using n for something else.

In section 2.4 you say “(i.e. elements of the set which can be used as x)” but you don’t mention x anywhere else. I guess this a reference back to lesson 9 but it seems confusing to me.

In section 3 you say “we must prove that, if gcd (a, n) = 1, then the numbers ranging from 1 to n – 1 which are coprime to n form a cyclic subgroup” but it is not clear to me what the “if gcd (a, n) = 1” part gains us. You may also want to explicitly mention the operation that the subgroup is using instead of leaving it implied.

In section 3.5 you say “before completing the proof” but the proof has already been completed at this point.

• Thanks for the feedback. I’ve made the adjustments on my copy, and will be uploading the corrected and clarified version shortly.