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Re: [PrimeNumbers] Is there a pattern?

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  • Jens Kruse Andersen
    ... Welcome to the list. The Internet has many good resources for people interested in primes. Most would agree the best site is The Prime Pages at
    Message 1 of 4 , Nov 28, 2005
      Amit Sharma wrote:

      > 3^3-2^2 is prime; 4^4-3^3 is prime; 7^7-6^6 is prime; 11^11-10^10 is
      > prime; 17^17-16^16 is prime.... Is there a pattern???? Can someone
      > explain?

      Welcome to the list.
      The Internet has many good resources for people interested in primes.
      Most would agree the best site is The Prime Pages at http://primes.utm.edu/

      A great program to test when arbitrary formulas give large primes is
      PrimeForm/GW at http://groups.yahoo.com/group/primeform/

      Setting up a search for n^n-(n-1)^(n-1) took 1 minute.
      PrimeForm quickly found it is prime (or probably prime) for
      n = 2, 3, 4, 7, 11, 17, 106, 120, 1907, with no more below 2000.
      1907 missed in the former post.
      As one would expect, primes tend to get rarer among larger numbers.
      Apart from that, I cannot see a pattern and would be surprised to find one.

      I searched the n sequence at http://www.research.att.com/~njas/sequences/
      Your 3, 4, 7, 11, 17 is enough to find
      http://www.research.att.com/projects/OEIS?Anum=A072164
      It confirms my results and has no more primes. It also links to
      http://www.primepuzzles.net/puzzles/puzz_185.htm which has the same.

      Numbers on this form are too hard to prove prime when they are very large,
      but a program like PrimeForm can say whether they are a prp (probable prime).

      http://www.primenumbers.net/prptop/prptop.php stores prp's above 10000 digits.
      It shows 7918^7918-7917^7917 (30870 digits) found by Henri Lifchitz in 2001.
      I don't know whether there has been an exhaustive search
      for n = 2000 to 7918 or further.
      I guess prp's above 10000 digits (n>2889) would have been listed there.
      PrimeForm could easily search n from 2000 to 2889.

      --
      Jens Kruse Andersen
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