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Fw: [PrimeNumbers] Seeking cause of coincidence--Update

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  • James Merickel
    Nobody expressed any interest in this, but here is one last entry before I turned this program off.  This is a tie result.  The stem value of 126 is repeated
    Message 1 of 1 , Oct 4, 2012
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      Nobody expressed any interest in this, but here is one last entry before I turned this program off.  This is a tie result.  The stem value of 126 is repeated as the first generating a reverse concatenation prime at base 287217406.  So, at any rate, a final digit of 6 was again the result.
      JimMe 

      --- On Tue, 8/14/12, James Merickel <moralforce120@...> wrote:


      From: James Merickel <moralforce120@...>
      Subject: [PrimeNumbers] Seeking cause of coincidence
      To: primenumbers@yahoogroups.com
      Date: Tuesday, August 14, 2012, 8:25 AM



       



      Dear group members,
      I could be accused of either laziness or hastiness for posting before myself finding an adequate answer, but it is what I think an interesting question that I have yet to resolve other than intuitively.  The following program--explained at bottom--with output in PARI/GP is the issue (Please pardon poor programming style):
       
      {
      b=2;rec=3;while(b,n=2;g=1;e=b;while(g,N=n;k=n-1;if(k==e,e*=b);E=e;while(k,N*=E;N+=k;
      if(ispseudoprime(N),g=0;break());if(k==E/b,E/=b);k--;next());if(g,n++);next());if(n>=rec,
      rec=n;print1(n":"b"\n"));b++;next())
      }
       
      4:4
      4:10
      4:16
      4:22
      4:25
      4:28
      4:31
      4:34
      4:40
      5:42
      6:46
      6:58
      8:60
      10:88
      10:100
      10:148
      10:166
      12:250
      13:252
      22:400
      26:1666
      28:5104
      31:8011
      34:19246
      37:23791
      40:36664
      40:55612
      40:67006
      40:79420
      43:84226
      44:87802
      48:89482
      49:102886
      52:116386
      52:144766
      54:201520
      54:354106
      54:505606
      54:652891
      61:684712
      64:759946
      64:1184626
      64:1432426
      72:1744366
      76:3043756
      90:4800436
      90:9602776
      94:24374026
      96:25069276
      96:26586886
      126:28041136
      ----------------------------------
      Okay, now what this program does (or is supposed to) is generate the values of bases in which the stem value for reverse concatenation primes--like 43 and 10987 in base 10, where the 'stem value' is 4 in the first case and 10 in the second--is necessarily at least a certain record value.  For example, the above results say that for bases under 42 at least one of the values represented by 21, 32, 321, 43, 432 and 4321 is prime (with an adjustment for bases <5), but that none of these are for base 42 itself, with that base's 54321 being the prime with the smallest stem value.  The surprise here is how it appears that beyond a certain point it is exceedingly unlikely the units digit of a base that ties or sets a new record is not 6.  My guess is this is because a certain amount of coincidence is required for smaller values and that the values next in line can be guaranteed composite with this unit digit.  Does anybody want to formalize this or correct
      it with a better explanation?  I would wager that if I ever get another element in the sequence (something I somewhat doubt because of the jump from 96 to 126) it will also have base terminating with digit 6, but I do not have a very good formal reason for feeling this is a good wager.
       
      Jim Merickel
       
       

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