Wikipedia:Reference desk/Archives/Mathematics/2015 November 30
Mathematics desk | ||
---|---|---|
< November 29 | << Oct | November | Dec >> | Current desk > |
Welcome to the Wikipedia Mathematics Reference Desk Archives |
---|
The page you are currently viewing is an archive page. While you can leave answers for any questions shown below, please ask new questions on one of the current reference desk pages. |
November 30
editEuler's totient function
editFor two positive integers m and n, when is true? GeoffreyT2000 (talk) 03:54, 30 November 2015 (UTC)
- When the primes dividing m all divide n. For example ϕ(100⋅40)=1600=100ϕ(40). Note, it's always the case that ϕ(mn)≤mϕ(n). --RDBury (talk) 07:00, 30 November 2015 (UTC)
- Coprimality depends only on the distinct prime factors. If every prime dividing m also divides n, then mn has the same distinct prime factors as n and hence an integer is coprime to mn if and only if it is coprime to n. GeoffreyT2000 (talk) 14:48, 30 November 2015 (UTC)
- That's not what I meant. For example every prime dividing 4 (namely 2) also divides 6. In this case ϕ(4⋅6)=4ϕ(6) but you can't say an integer is coprime to 4 iff it's coprime to 6 (3 being a counterexample). --RDBury (talk) 15:13, 30 November 2015 (UTC)
- But it is true that an integer is coprime to 24 if and only if it is coprime to 6. I said "coprime to mn", not "coprime to m". GeoffreyT2000 (talk) 15:29, 30 November 2015 (UTC)
- My bad, I misread what you wrote. --RDBury (talk) 16:55, 30 November 2015 (UTC)
- But it is true that an integer is coprime to 24 if and only if it is coprime to 6. I said "coprime to mn", not "coprime to m". GeoffreyT2000 (talk) 15:29, 30 November 2015 (UTC)
- That's not what I meant. For example every prime dividing 4 (namely 2) also divides 6. In this case ϕ(4⋅6)=4ϕ(6) but you can't say an integer is coprime to 4 iff it's coprime to 6 (3 being a counterexample). --RDBury (talk) 15:13, 30 November 2015 (UTC)
- Coprimality depends only on the distinct prime factors. If every prime dividing m also divides n, then mn has the same distinct prime factors as n and hence an integer is coprime to mn if and only if it is coprime to n. GeoffreyT2000 (talk) 14:48, 30 November 2015 (UTC)