## Re: Fw: A new conjecture on primes

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• ... 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
Message 1 of 10 , Mar 21, 2012
> 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 ...

Mark

--- In primenumbers@yahoogroups.com, Steven Harvey <harvey563@...> wrote:
>
> I thought the group might find this interesting.
>
> StevenHarvey
>
> harvey563@...
>
> ----- Forwarded Message -----
> From: Zhi-Wei Sun <zwsun@...>
> To: NMBRTHRY@...
> Sent: Monday, March 19, 2012 6:24 PM
> Subject: A new conjecture on primes
>
> Dear number theorists,
>
>   Prime numbers are very mysterious and there are many unsolved
> problems on primes.
>
>   Here I report that I have formulated a new conjecture on primes which
> allows us to produce primes via primes.
>
> 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.
>
>   (ii) For n=1,2,3,... define w_2(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_2(n) is always a prime.
>
>   (iii) We have w_1(n)<n^2 and w_2(n)<n^2 for all n=2,3,4,....
>
>   Clearly w_i(n) does not exceed w_i(n+1) for i=1,2. Since P_1,...,P_n
> are pairwise distinct modulo w_1(n), we see that w_1(n) is at least n
> and hence W_1={w_1(n): n=1,2,3,...} is an infinite set. For any integer
> m>1, there is an odd prime p_n=-1 (mod m) by Dirichlet's theorem and
> hence P_{n-1}+P_n=P_{n-1}(1+p_n) is a multiple of m. Thus W_2={w_2(n):
> n=1,2,3,...} is also infinite.
>
>   If w_i(n) is the k-th prime p_k, then k is at least n since p_k
> divides both P_{k+1}-P_k and P_{k+1}+P_k. Thus part (ii) of the
> conjecture implies the inequality w_2(n)>n, which I am even unable to prove.
>
>   A prime is said to be of the first kind (or the second kind) if it
> belongs to W_1={w_1(n): n=1,2,3,...} (or W_2={w_2(n): n=1,2,3,...},
> resp.). Here I list the first 20 primes of each kind.
>
> Primes of the first kind:  2, 3, 5, 11, 23, 29, 37, 41, 47, 73, 131,
> 151, 199, 223, 271, 281, 353, 457, 641, 643, ...
>
> Primes of the second kind: 2, 3, 5, 7, 11, 19, 23, 47, 59, 61, 71, 101,
> 113, 223, 487, 661, 719, 811, 947, 1327, ...
>
>   I have added the two sequences w_1(1),w_1(2),w_1(3),...  and
> w_2(1),w_2(2),w_2(3),... to OEIS, see
> http://oeis.org/A210144%c2%a0 and  http://oeis.org/A210186
> where you can find values of w_1(n) for n=1,...,1172 and values of
> w_2(n) for n=1,...,258.
>
>   In my opinion, the above conjecture might be very challenging and it
> contains certain mysterious information about primes. It seems that
> parts (i) and (ii) of the conjecture remain true if we replace P_k by
> (P_k)^d or (-1)^k*(P_k)^d where d is any positive integer.
>
>
>
>   Zhi-Wei Sun
>   http://math.nju.edu.cn/~zwsun
>
> [Non-text portions of this message have been removed]
>
• ... 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 2 of 10 , Mar 21, 2012
> > 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 3 of 10 , Mar 21, 2012
--- 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 4 of 10 , Mar 21, 2012
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 5 of 10 , Mar 21, 2012
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 6 of 10 , Mar 21, 2012
> 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 7 of 10 , Mar 21, 2012
--- 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 8 of 10 , Mar 21, 2012
--- 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 9 of 10 , Mar 21, 2012
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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