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Every Mersenne number of index $n >2$ $$ M_n = 2^n-1 $$ is represented by the quadratic polynomial $$ Q(x,y) = 28x^2+4y^2+28x+4y+7 $$

e.g.,

$$ M_3=Q(0,0), M_4=Q(0,1), M_5=Q(0,2),M_6=Q(1,0), $$ $$ M_7=Q(0,5),M_8=Q(2,4),M_9=Q(3,6),M_{10}=Q(1,15). $$

Since it is believed that there are an infinity of Mersenne primes, this polynomial should represent an infinity of prime numbers.

Question: It is true that this polynomial represents an infinity of prime numbers ? (Probably unrelated with Mersenne primes).

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  • $\begingroup$ Why not phrase it as "are there infinitely many primes of the form $7x^2+y^2-1$"? $\endgroup$ – Gjergji Zaimi Apr 25 '11 at 23:13
  • $\begingroup$ You are right, is exactly the same thing. $\endgroup$ – Luis H Gallardo Apr 25 '11 at 23:30
  • $\begingroup$ with $x$ and $y$ both odd ? $\endgroup$ – Luis H Gallardo Apr 25 '11 at 23:43
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I think this was proved by Iwaniec (for any quadratic in two variables satisfying the obvious necessary conditions).

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  • $\begingroup$ This was also mentioned here mathoverflow.net/questions/55384/… $\endgroup$ – Gjergji Zaimi Apr 26 '11 at 2:05
  • $\begingroup$ Let's vote to close as a duplicate, then. $\endgroup$ – Felipe Voloch Apr 26 '11 at 2:43
  • $\begingroup$ Nice reference Felipe. I was not aware of that. I repeat here the reference in the link: Iwaniec, H (1974). Primes represented by quadratic polynomials in two variables. Acta Arithmetica 24, pp. 435-459. $\endgroup$ – Luis H Gallardo Apr 26 '11 at 8:20

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