# Mersenne numbers represented by a Quadratic polynomial in two variables

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).

• Why not phrase it as "are there infinitely many primes of the form $7x^2+y^2-1$"? – Gjergji Zaimi Apr 25 '11 at 23:13
• You are right, is exactly the same thing. – Luis H Gallardo Apr 25 '11 at 23:30
• with $x$ and $y$ both odd ? – Luis H Gallardo Apr 25 '11 at 23:43