Loading ...
Sorry, an error occurred while loading the content.

[PrimeNumbers] RE: twin prime number

Expand Messages
  • mikeoakes2@aol.com
    ... 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
    • 0 Attachment
      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
    • mikeoakes2@aol.com
      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
      • 0 Attachment
        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
      • jbrennen
        ... (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
        • 0 Attachment
          --- In primenumbers@yahoogroups.com, mikeoakes2@... wrote:
          >
          >
          > 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
        • jbrennen
          ... 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
          • 0 Attachment
            --- 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
            > 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???
          Your message has been successfully submitted and would be delivered to recipients shortly.