The Real Conjecture

[approx]

okay, I haven't had long to look at it, but it seems to me that you're choosing some primes for n and then exponentiating and seeing if n+prime=m is zero mod each of the rest of the primes. Seems to make sense. I still have to look at it some more to see how you choose your partitions and exponents, but I see where you're going with it (I think). Thanks for posting it.

approx

 

[CRGreathouse]

The version of the code that I posted doesn't actually test all numbers of the appropriate size, and it has some sections that could be sped up significantly. I posted it to show you the general idea; if you want a fast version you might be better off starting afresh. I've already made modifications to my version.

 

[Xevarion]

I think the fact that this conjecture is false would follow from the ABC conjecture. I didn't work it out carefully but it seems reasonable. (At least the previous one, with 2 and 3 only, would follow from ABC. And this problem seems to be essentially the same.)

 

[approx]

related Q

This is related to my conjecture...
I'm a little rusty, but I'm guessing the product as i goes from 9 to infinity of
[p(i)-2]/p(i) is zero?

 

[CRGreathouse]

I'd have to imagine. The result might not be that hard to prove using bounds on the nth prime, since the limit doesn't really depend on the properties of the primeness at all.

 

[Xevarion]

I'm pretty sure it's 0. See http://en.wikipedia.org/wiki/Prime_number_theorem#Approximations_for_the_nth_prime_number