2
$\begingroup$

Does have at most 2 solutions for primes $p\geq 5$?

Mathematica confirms this for the first 10K primes:

Max[   Table[    Length[     Reduce[{      Mod[2*x^2 + 2*x + 3, Prime[i]] == 0, 0 <= x <= Prime[i]}, x,Integers]],    {i,3,10000}]  ]   Output: 2  

but I wasn't sure if it was true in general, and if it could be proved.

Inspired by: Elementary proof that $2x^2+xy+3y^2$ represents infinitely many primes?

  • 6
    you're looking for roots of a quadratic polynomial in the field $\mathbb{F}_p$ - so surely there are at most 22011-03-29
  • 3
    Don't call me Shirley2011-03-29
  • 0
    Of course, you are talking about "at most 2 solutions" **modulo** $\mathbf{p}$. By the way, you don't need to check $x=Prime[i]$, since $0\equiv p\pmod{p}$; and you don't need to check $0$ except for $p=3$, where it is a solution.2011-03-29

2 Answers 2