Euler’s Theorem has 1<a<N in the definition of \varphi(N). It should be 1\leq a \leq N. The first inequality allows 1, which is necessary, the second has no effect (unless N=1 when it is also necessary).
Different authors will handle this slightly differently, but it's pretty standard to define two integers to be coprime iff the only positive integer that divides both is 1. So all is well.
Far more standardized is the definition of coprime ideals, and since 0=1 mod 1, I am in the clear. Units are also an alternative way to define this, as you point out.
Euler’s Theorem has 1<a<N in the definition of \varphi(N). It should be 1\leq a \leq N. The first inequality allows 1, which is necessary, the second has no effect (unless N=1 when it is also necessary).
Thank you! Fixed.
Hmm. 0\leq a< N is not what I was expecting ? I was expecting 1\leq a \leq N.
I think 0 can be safely omitted, but N cannot.
No, it works. If you take N=1, 0=1.
I’m not sure we count 0 as coprime to anything do we?
Jones and Jones define \varphi(n) as the number of integers a=1,2,…,n such that gcd(a,n)=1.
I think including 0 complicates matters, but gcd(0,n)=n for all positive integers n, so yes, your definition works.
Defining \varphi(n) as the number of units in \Z/n\Z more closely aligns with the first, but meh.
Loving these posts btw. Keep up the good work.
Different authors will handle this slightly differently, but it's pretty standard to define two integers to be coprime iff the only positive integer that divides both is 1. So all is well.
Far more standardized is the definition of coprime ideals, and since 0=1 mod 1, I am in the clear. Units are also an alternative way to define this, as you point out.