Prove that the quadratic congruence 6x^2+5x+1=0(mod p) has a solution for every prime p, even though the equation 6x^2+5x+1=0 has no solution in the integers.
Prove that the quadratic congruence 6x^2+5x+1=0(mod p) has a solution for every prime p, even though the equation 6x^2+5x+1=0 has no solution in the integers.
There is a solution x=1 if p=2 or p=3.
The solution in the reals is . If p is not equal to 2 or 3 then 12 is coprime to p, and so 12 will have an inverse (mod p). Then will be solutions to (mod p).