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Old 2018-01-04, 03:29   #10
Romulan Interpreter
LaurV's Avatar
Jun 2011

2·5·859 Posts

Yet, this is still a divisibility sequence, the terms can be prime only if the number of 1 in the sequence is prime, that is, only if the n is prime. The proof is very simple, and similar to the "binary proof" for mersenne numbers: if the number of 1 in the sequence is not prime, then split them in equal strings and you just factored your number. For example, N=(10^15-1)/9=111 111 111 111 111=111*(1001001001001)=11111 11111 11111=11111*(10000100001), and you have already (at least) two different ways to factor N, without any calculus.

So, for such a number to be prime, n must be prime.

Last fiddled with by LaurV on 2018-01-04 at 03:30
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