Re: [PrimeNumbers] Re: p^p +(p+1)^(p+1)
- In an email dated 10/5/2005 5:44:29 pm GMT Daylight time, "jbrennen" <jb@...> writes:
>--- In firstname.lastname@example.org, Mark Underwood wrote:[snip]
>> PS on the positive side, no wonder I couldn't prove that 'result'! :)
>But you could prove that p^p+(p+1)^(p+1) is never divisible by
>2, 3, or 7... In general, numbers of this form seem to be
>overall less likely to be divisible by any particular prime.
>Here's a table showing the actual proportion of such numbers
>which are divisible by each prime q, along with the ratio of
>that proportion to 1/q:
Everything becomes rather clearer if we don't restrict "p" to be prime.
I studied the divisibiity properties of those forms extensively 3 years ago but seem to have lost my notes :(
I think it's relatively easy to prove by Fermat's little theorem that the values of n divisible by (odd) primes q repeat with a cycle of length (q-1)*q.
Using pfgw -d I have just found (must go to work now...) that n^n+(n+1)^(n+1) is divisible by primes q as follows:-
As your tables exemplify, there is no such pattern if "p" is constrained to be prime.