 Although conjectures have been formulated about the proportions of primes in higher-degree polynomials, they remain unproven, and it is unknown whether there exists a quadratic polynomial that (for integer arguments) is prime infinitely often.Euler's proof that there are infinitely many primes considers the sums of reciprocals of primes,

Euler showed that, for any arbitrary real number formula_22, there exists a prime formula_20 for which this sum is bigger than formula_22.This shows that there are infinitely many primes, because if there were finitely many primes the sum would reach its maximum value at the biggest prime rather than growing past every formula_22.