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Re: [PrimeNumbers] Prime diophantine equations

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  • Jack Brennen
    ... 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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