Browse Groups

• ## Re: [PrimeNumbers] Prime diophantine equations

(2)
• NextPrevious
• ... There is no such pair. The only prime pentagonal number is P2 == 5; in that case, q is 4 and is thus not prime. It shouldn t be too hard to figure out why
Message 1 of 2 , Feb 24, 2001
View Source
> If Pn is the nth pentagonal number then
> Pn = (3*n 1)*n/2 = 1 + 4 + 7 +  + (3*n - 2)

> Can anyone find 2 primes p and q so that q = 3*n 2 and p = Pn ?

There is no such pair. The only prime pentagonal number is P2 == 5;
in that case, q is 4 and is thus not prime.

It shouldn't be too hard to figure out why 5 is the only prime
pentagonal number.

> Hexagonal sums are of the form Hn where
> Hn = 1 + 6 + 6 +  + 6 = 6*n  5 so as 6 is not prime we won't
> find a p, q pair for hexagonal sums. However as all odd primes > 3
> are of the form 6*n 1 or 6*n  5 then about half of the primes are
> also hexagonal numbers!

Not correct. Hexagonal numbers are numbers of the form:

(4*n-2)*n/2, or (2*n-1)*n.

There are no prime hexagonal numbers.

Jack
Your message has been successfully submitted and would be delivered to recipients shortly.
• Changes have not been saved
Press OK to abandon changes or Cancel to continue editing
• Your browser is not supported
Kindly note that Groups does not support 7.0 or earlier versions of Internet Explorer. We recommend upgrading to the latest Internet Explorer, Google Chrome, or Firefox. If you are using IE 9 or later, make sure you turn off Compatibility View.