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• ## Re: [PrimeNumbers] Re: Fw: A new conjecture on primes

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• ... Ignoring Sun s definition of P_i, if we set = , the corresponding value of w_1(3) would be equal to 4; not a prime. Thus, there must
Message 1 of 10 , Mar 21, 2012
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> > 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.
>
> Isn't it self evident that the first (least) integer which doesn't divide into another integer, is a prime number? Or am I misunderstanding something here ...

Ignoring Sun's definition of P_i, if we set <P1, P2, P3> = <1, 3, 4>, the
corresponding value of w_1(3) would be equal to 4; not a prime. Thus,
there must be something special about his definition of P_i, which makes
the resulting ones primes [or his conjecture is false :-) ].

Peter

[Non-text portions of this message have been removed]
• ... Hint: what is the least integer which does not divide 6? It s evident that the least integer in question is a prime power, but it s not necessarily a
Message 2 of 10 , Mar 21, 2012
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--- In primenumbers@yahoogroups.com, "Mark" <mark.underwood@...> wrote:
>
>
> Isn't it self evident that the first (least) integer which doesn't divide into another integer, is a prime number? Or am I misunderstanding something here ...
>
>
> Mark
>

Hint: what is the least integer which does not divide 6?

It's evident that the least integer in question is a prime power, but it's not necessarily a prime.
• True, true, a prime power, not simply a prime. Now I suppose the question is, why hasn t there been a counterexample to Sun s conjectures. I m guessing it is
Message 3 of 10 , Mar 21, 2012
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True, true, a prime power, not simply a prime. Now I suppose the question is, why hasn't there been a counterexample to Sun's conjectures. I'm guessing it is just because prime powers are relatively scarce compared to the primes.

Mark

--- In primenumbers@yahoogroups.com, "jbrennen" <jfb@...> wrote:
>
> --- In primenumbers@yahoogroups.com, "Mark" <mark.underwood@> wrote:
> >
> >
> > Isn't it self evident that the first (least) integer which doesn't divide into another integer, is a prime number? Or am I misunderstanding something here ...
> >
> >
> > Mark
> >
>
> Hint: what is the least integer which does not divide 6?
>
> It's evident that the least integer in question is a prime power, but it's not necessarily a prime.
>
• 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
Message 4 of 10 , Mar 21, 2012
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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

--- In primenumbers@yahoogroups.com, "Mark" <mark.underwood@...> wrote:
>
>
>
>
> True, true, a prime power, not simply a prime. Now I suppose the question is, why hasn't there been a counterexample to Sun's conjectures. I'm guessing it is just because prime powers are relatively scarce compared to the primes.
>
> Mark
>
>
>
> --- In primenumbers@yahoogroups.com, "jbrennen" <jfb@> wrote:
> >
> > --- In primenumbers@yahoogroups.com, "Mark" <mark.underwood@> wrote:
> > >
> > >
> > > Isn't it self evident that the first (least) integer which doesn't divide into another integer, is a prime number? Or am I misunderstanding something here ...
> > >
> > >
> > > Mark
> > >
> >
> > Hint: what is the least integer which does not divide 6?
> >
> > It's evident that the least integer in question is a prime power, but it's not necessarily a prime.
> >
>
• ... Nope -- Sun claims that for any N, the first number which does not divide ANY of the primorials A# - B# for A, B
Message 5 of 10 , Mar 21, 2012
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> 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
Message 6 of 10 , Mar 21, 2012
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--- 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.

> 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.
• ... 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 7 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 8 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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