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• ## Re: Fw: A new conjecture on primes

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• ... No, I m wrong again. The least number to not divide another number will indeed be a prime power, but I m wrong to assume this would be true when applied
Message 1 of 10 , Mar 21, 2012
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--- In primenumbers@yahoogroups.com, "Mark" <mark.underwood@...> wrote:
>
>
>
> --- In primenumbers@yahoogroups.com, Peter Kosinar <goober@> wrote:
> >
> > > If I'm not mistaken Sun is saying that the least number to not divide, say, 11# - 7#, is a prime. But the first number to not divide this is 8.
> > >
> > > Mark
> >
> > Nope -- Sun claims that for any N, the first number which does not divide
> > ANY of the primorials A# - B# for A, B <= N, is a prime.
> >
> > Peter
> >
>
> Ah, I see, thanks for the clarification. So as a counterexample we're looking for a prime power (with exponent greater than one) which is the least number that does not divide any combination of A# - B# for A,B up to a given n. I think I can see why that should be next to impossible.
>

No, I'm wrong again. The least number to not divide another number will indeed be a prime power, but I'm wrong to assume this would be true when applied to more than one number. The problem is more subtle and difficult than I supposed.

Mark

>
>
> > CONJECTURE ON PRIMES (Z. W. Sun, March 17-18, 2012). For k=1,2,3,... let
> > P_k denote the product of the first k primes p_1,...,p_k.
> >
> > (i) For n=1,2,3,... define w_1(n) as the least integer m>1 such that m
> > divides none of P_i-P_j with i,j distinct and not more than n. Then
> > w_1(n) is always a prime.
>
• ... As would be expected when coming from Zhi-Wei Sun: http://en.wikipedia.org/wiki/Sun_Zhiwei If he presents it as a conjecture, you can be sure of two
Message 1 of 10 , Mar 21, 2012
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On 3/21/2012 3:56 PM, Mark wrote:
>
> No, I'm wrong again. The least number to not divide another number
> will indeed be a prime power, but I'm wrong to assume this would be
> true when applied to more than one number. The problem is more
> subtle and difficult than I supposed.
>

As would be expected when coming from Zhi-Wei Sun:

http://en.wikipedia.org/wiki/Sun_Zhiwei

If he presents it as a conjecture, you can be sure of two things...
It's very likely true, and will be very hard to prove.
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