Re: [PrimeNumbers] Is there a pattern?

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• ... 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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