- In an email dated Fri, 3 2 2006 2:34:06 pm GMT, "leavemsg1" <leavemsg1@...> writes:

>I found an interesting POTM(possibly only to me) prime pattern.

Using PFGW,I find the following:-

>

>When using Dario A.'s factorization program, I noticed a simple fact.

>

>If z= p1^2 * p2^2 * p3^2 * ... * pn^2 +1 (where p1, p2, p3, etc. form

>the ordered prime number sequence 2, 3, 5, etc.), then either z or z-2

>must contain one and only one twin prime factor > pn.

>

>Can anyone find a counter-example? I check it up to pn= 89.

(2#)^2+(-1) trivially factors prime!: 3

(2#)^2+(1) trivially factors prime!: 5

(3#)^2+(-1) trivially factors as: 5*7

(3#)^2+(1) trivially factors prime!: 37

(5#)^2+(-1) trivially factors as: 29*31

(5#)^2+(1) trivially factors as: 17*53

(7#)^2+(-1) trivially factors as: 11*19*211

(7#)^2+(1) trivially factors prime!: 44101

(11#)^2+(-1) trivially factors as: 2309*2311

(11#)^2+(1) trivially factors prime!: 5336101

(13#)^2+(-1) trivially factors as: 59*509*30029

(13#)^2+(1) trivially factors as: 73*12353437

(17#)^2+(-1) has factors: 19*61*97*277*8369

(17#)^2+(1) has factors: 1409*184968389

I am unable to correlate this set of results with your statement

"either z or z-2 must contain one and only one twin prime factor > pn"

It would be enlightening if you could illustrate what that statement - and in particular the term "twin prime factor" - means, in terms of the first few examples above.

-Mike Oakes - Observe that (by PFGW):-

(43#)^2+(-1) has factors: 167

((43#)^2+(-1))/(167) is composite

(43#)^2+(1) has factors: 1117*11261

((43#)^2+(1))/(1117*11261) is composite

and refer to the useful web page:-

http://www.mathematical.com/twin2to10m.htm

Assuming that by "twin prime factor" you mean a factor which is one of a twin-prime pair, then, since none of 167, 1117, 11261 is such a member, the above shows your statement to be false - unless (which I haven't checked - have you?)

either

((43#)^2+(-1))/(167)

or

((43#)^2+(1))/(1117*11261)

is a member of a twin-prime pair.

So how is it that you claim to have checked it up to pn=89?

-Mike Oakes - --- In primenumbers@yahoogroups.com, mikeoakes2@... wrote:
>

(43#)^2-1 has a factor which is a twin prime: 92608862041

>

> Observe that (by PFGW):-

> (43#)^2+(-1) has factors: 167

> ((43#)^2+(-1))/(167) is composite

> (43#)^2+(1) has factors: 1117*11261

> ((43#)^2+(1))/(1117*11261) is composite

>

(43#)^2+1 has a factor which is a twin prime: 410621 - --- In primenumbers@yahoogroups.com, "leavemsg1" <leavemsg1@...>

wrote:> If z= p1^2 * p2^2 * p3^2 * ... * pn^2 +1 (where p1, p2, p3, etc.

form

> the ordered prime number sequence 2, 3, 5, etc.), then either

Well, if pn = 61,

> z or z-2 must contain one and only one twin prime factor > pn.

>

> Can anyone find a counter-example? I check it up to pn= 89.

>

then z = 13756564401909684622656803563109083750619892901

And neither z nor z-2 contains any twin prime factor.

z = 96973 * 141859738297357868918738242223186698881337

z - 2 = 223 * 1193 * 85738903 * 1146665184811 * 525956867082542470777

Unless of course, your definition of "twin prime factor" is

different than mine???