On 2017-09-25 17:13, Michael Greenwald wrote:
On 2017-09-25 16:43, James Davis wrote:
Most of these patterns are related to the fact that 10^n+1 is a divisor of (10^(2n)-1).
I've played with these, but I don't understand - why is 10^+1 being a divisor of 10^(2n)-1 related?
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Because it's a step in showing that if {x} is the set of prime divisors of 10^n + 1, they are also divisors of 10^((2k+1)n) + 1 for all k.
If X = product{x} divides 10^{(2k+1)n} + 1, it also divides 10^{(2k+3)n} + 1, because: ( 10^{(2k+1)n} + 1 ) * (10^{2n} - 1 ) is divisible by X, and = 10^{(2k+3)n} + 1 - (10^{2n} - 1) - ( 10^{(2k+1)n} + 1 ). ( X divides 10^{2n} - 1 and (by assumption) 10^{(2k+1)n} + 1, so it must also divide 10^{(2k+3)n} + 1 )
So the factors of 10^2 + 1 should show up at n=2, 6, 10, 14 etc. The factors of 10^3 should show up at n=3, 9, 15, 21, etc. etc.
Never mind: I misunderstood what you were asking, and I answered a much simpler, and much less interesting question than Tom Rokicki answered.