Math Is Fun Forum

namealreadychosen

Member

Registered: 2011-07-23

Posts: 16

screws - modular arithmetic

Let m()~n be the number of values which an nth power can take modulo m. e.g. 4()~2=2 because squares can be 0 or 1 modulo 2. I arbitrary invented the symbol ()~ and called it a screw (did I screw up on the name?).

Here is some of what I have worked out:
1()~x=1
2()~x=2
3()~2x-1=3
3()~2x=2
4()~1=4
4()~2x=2
4()~2x+1=3
5()~4x=2
5()~4x-2=3
5()~2x-1=5
6()~2x=4
6()~2x-1=6
7()~1=7
7()~3=4
7()~2x=4
7()~2x+3=7

What is the quickest way to calculate m()~n?
Is n()~n always the smallest factor >1 of n for integers n>1?
When does m()~2=m?
What is the maximum number of values that m()~x can have in terms of m?
Obviously m()~1=m, but are there any simple formulae for m()~n for other positive integers n?

Alg Num Theory

Member

Registered: 2017-11-24

Posts: 693

Website

Re: screws - modular arithmetic

https://en.wikipedia.org/wiki/Power_residue_symbol.

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Me, or the ugly man, whatever (3,3,6)