My number theory jargon is not really up to scratch, but I'll try and explain my question and perhaps someone can give me a plain english answer?
I'm interested in multiplication modulo a prime number. I've found various interesting things out and have been able to get to grips with why they happen. But one thing has elluded me.
I know that I can generate a sequence of numbers modulo prime, by repeated multiplication and taking the modulo. One such sequence is :
1 3 2 6 4 5 1 modulo 7
I notice that all the numbers less than 7 appear once in the sequence, in what looks like no particular order. Some multipliers produce a whole sequence others just produce a subset of the largest sequence. For example :
1 2 4 1 modulo 7
If I just focus on those numbers that produce a whole sequence, a pattern arises that I can't figure out. Here's the first few primes and the associated numbers which can produce a whole sequence :
mod 3: 2 mod 5: 2, 3 mod 7: 3, 5 mod 11: 2, 6, 7, 8 mod 13: 2, 6, 7, 11
By inspection on each of these numbers on a per prime basis, it looks like these numbers are symetrically distributed about prime/2 only if the prime-1 is divisible by 4.
For example for mod 13 (13-1 is divisible by 4): 2+11 = 13 6+7 = 13