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Questions about modulo
Posted:
Nov 5, 2000 12:52 PM
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Hello,
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
but for mod 11 this doesn't work.
Again for mod 5 (5-1 is divisible by 4):
2+3=5
but not for mod 7.
Why is this? Can anyone help?
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