> CONJECTURE ON PRIMES (Z. W. Sun, March 17-18, 2012). For k=1,2,3,... let

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 ...

> 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.

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.

>

> Any comments are welcome!

>

>

> Zhi-Wei Sun

> http://math.nju.edu.cn/~zwsun

>

> [Non-text portions of this message have been removed]

>- On 3/21/2012 3:56 PM, Mark wrote:
>

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

> 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.

>

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.