Re^3: Rotationally Prime Numbers Revisited

As we all know, (10**n-1)/9 is surely not a prime when n is composed, because (10**(k*l)-1)/9 = ((10**(k*l)-1)/(10**k-1)) * ((10**k - 1)/9). Thus, they do have special factoring properties, but this doesn't mean that they are either more often or less often primes then other numbers.

Comment on Re^3: Rotationally Prime Numbers Revisited

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Re^4: Rotationally Prime Numbers Revisited by hv (Prior) on Mar 25, 2005 at 12:54 UTC
Indeed, I'd argue they're more likely to be primes: numbers of the form `M(a, p) = (a^p - 1)/(a - 1)` form an extended class of Mersenne numbers, and in particular will not share a factor with any `M(a, q), q < p`, and draw their factors from the restricted set `{p, <2kp + 1>}`. I don't know if there's a way to adapt the Lucas-Lehmer test to check directly for divisors in this extended class. Hugo	[reply] [d/l] [select]
Re^5: Rotationally Prime Numbers Revisited by tilly (Archbishop) on Mar 25, 2005 at 19:33 UTC
While they may be more likely to be primes, I'm not convinced that the effect is that large. True, possible factors are restricted, but numbers in that set of possible factors are far, far more likely to be divisors than mere chance would predict. How do these balance out? It certainly isn't obvious to me. Let's do some heuristics. the 38'th Mersenne prime is 26972593-1. Naively from the Prime Number theorem you'd expect to find an average of 22.12 primes in that sequence to that point. We actually got 38. So we got about 70% more primes than we were expecting to have. The limited results that we have for (10p-1)/9 suggest a similar advantage. Which is significant, but not astoundingly different than the naive approximation.	[reply]