## [PrimeNumbers] RE: twin prime number

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• ... Using PFGW,I find the following:- (2#)^2+(-1) trivially factors prime!: 3 (2#)^2+(1) trivially factors prime!: 5 (3#)^2+(-1) trivially factors as: 5*7
Message 1 of 5 , Feb 3, 2006
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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.
>
>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.

Using PFGW,I find the following:-
(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
Message 2 of 5 , Feb 3, 2006
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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
• ... (43#)^2-1 has a factor which is a twin prime: 92608862041 (43#)^2+1 has a factor which is a twin prime: 410621
Message 3 of 5 , Feb 3, 2006
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>
>
> 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: 92608862041

(43#)^2+1 has a factor which is a twin prime: 410621
• ... form ... Well, if pn = 61, then z = 13756564401909684622656803563109083750619892901 And neither z nor z-2 contains any twin prime factor. z = 96973 *
Message 4 of 5 , Feb 3, 2006
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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
> 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.
>

Well, if pn = 61,

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???
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