## Re: [PrimeNumbers] Prime diophantine equations

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• ... 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
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> 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
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