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15543Extension of Cunningham chains

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  • Jack Brennen
    Nov 12, 2004
      First I give an equivalent definition of a Cunningham chain:

      - A Cunningham chain of the first kind (second kind) is a sequence
      of integers of the form k*2^n-1 (k*2^n+1) where k is a fixed
      positive odd integer, and n ranges over two or more consecutive
      nonnegative integers, and where each member of the sequence has
      exactly 2 positive divisors.

      You'll note this is a pretty standard definition of a Cunningham
      chain, except I've defined it in terms of number of divisors
      rather than primality. Why? Because I'm looking at an extension
      of Cunningham chains where all of the members of the sequence have
      an equal number of positive divisors, but that number is not 2.

      So far, I've found a "4-divisor Cunningham chain" (of the second kind)
      of length 18:

      80935905*2^13+1 == 254993 * 2600177
      80935905*2^14+1 == 1951 * 679679071
      80935905*2^15+1 == 71 * 37353630071
      80935905*2^16+1 == 11 * 482201406371
      80935905*2^17+1 == 15667 * 677119483
      80935905*2^18+1 == 389 * 54542061389
      80935905*2^19+1 == 19 * 2233353882139
      80935905*2^20+1 == 4133 * 20534102957
      80935905*2^21+1 == 2797 * 60684624613
      80935905*2^22+1 == 2181271 * 155629351
      80935905*2^23+1 == 13 * 52226121551557
      80935905*2^24+1 == 853 * 1591886471677
      80935905*2^25+1 == 1489 * 1823880672049
      80935905*2^26+1 == 11 * 493774240123811
      80935905*2^27+1 == 44287967 * 245281823
      80935905*2^28+1 == 689383 * 31515234007
      80935905*2^29+1 == 76446859 * 568396579
      80935905*2^30+1 == 28879 * 3009254692399

      It looks like chains with 4 divisors each seem
      to be easiest to find, but 8 divisors each might be worth looking
      into as well. The best "8-divisor Cunningham chain" I've found,
      without spending a lot of time on it:

      302457*2^13+1 == 5 * 97 * 5108717
      302457*2^14+1 == 59 * 83 * 1011937
      302457*2^15+1 == 11 * 12041 * 74827
      302457*2^16+1 == 19 * 1993 * 523459
      302457*2^17+1 == 5 * 1129 * 7022789
      302457*2^18+1 == 23 * 2521 * 1367423
      302457*2^19+1 == 37 * 71 * 60363371
      302457*2^20+1 == 73 * 373 * 11647477
      302457*2^21+1 == 5 * 329267 * 385279
      302457*2^22+1 == 337 * 11273 * 333929
      302457*2^23+1 == 67 * 119677 * 316423
      302457*2^24+1 == 13 * 251 * 1555129151

      Can anybody find any longer such chains each with an equal number
      of divisors, either with 4, 8, or some other number?


      Jack
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