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Seeking formal proof: primes, powers, factors
Posted:
Jul 3, 2014 8:51 AM


Greetings, experts!
I hope this is an appropriate place to ask this question; if not, please point me to some place where it's more appropriate.
(Caveat: I'm not a tertiarytrained mathematician in any sense; my skills and terminology are basically high school algebra level.)
Given any prime number P and whole number B, either P is a factor of B, or P is a factor of B^(P1)1.
I can state with confidence that this is true, by looking to long division. With a base of 10, this is the standard conversion of vulgar fraction to repeating decimal fraction  for example, the expansion of 1/7 is calculated thus:
10/7 = 1, remainder 3 30/7 = 4, remainder 2 20/7 = 2, remainder 6 60/7 = 8, remainder 4 40/7 = 5, remainder 5 50/7 = 7, remainder 1 10/7 = ... and we're back to the top.
It repeats on period 6, which means that 7 * 142857 = 999999, which is 10^61.
The remainder must be a whole number smaller than the original prime, and it can be 0 only if the prime is a factor of the base, which means that there are only six possible remainders for the division. Since the "state" of the division is simply the previous remainder, there are only six possible states, which means the cycle MUST repeat after six digits. (Some expansions repeat sooner than that, eg 1/3 repeats on period 1. It's always a factor of the maximum  in this case, 2  so it's still true that it repeats on period 2.)
None of this depends on the division being done in base 10. It works in any other base, just as well. (With a base of 2, we're looking at Mersenne numbers; ergo for any prime number P, the P1th Mersenne number is a multiple of P.) And it works for any prime number. Hence the original postulation.
So, since it's provably unfalsifiable, it ought to be possible to prove mathematically. But how? What aspect of mathematics makes this true? Surely there must be something.
Hope you find this sort of thing as fascinating as I do!
Chris Angelico



