Thread: Coset conundrum
View Single Post
Old 2021-08-28, 21:28   #2
Jan 2017

23×3×5 Posts

Originally Posted by Dr Sardonicus View Post
A) Prove that \sum_{i=1}^{h}r_{i}\;=\;m\times n for a positive integer m.
If you multiply the elements of the set {b^i for all i} by b, you get the same set with permuted elements. Thus multiplying the sum by b does not change it mod n. Thus S*b = S mod n, S*(b-1)=0 mod n, and S must be 0 mod n.
B) Prove that m = 1 if and only if n is a repunit to the base b, and also that one of the ri is equal to 1.
The "and also that one of the ri is equal to 1" part seems ambiguous or wrong. For k != 1, m may or may not equal 1?

n = 1111, b = 10, k = 2: m = 1
n = 1111, b = 10, k = 21: m = 2
uau is offline   Reply With Quote