section_path: Analytic properties/Prime values of quadratic polynomials
section_name: Prime values of quadratic polynomials
paragraph_text: Prime values of quadratic polynomials
  
  Euler noted that the function
  
  yields prime numbers for 1 ≤ n ≤ 40 {\displaystyle 1\leq n\leq 40}, although composite numbers appear among its later values. The search for an explanation for this phenomenon led to the deep algebraic number theory of Heegner numbers and the class number problem. The Hardy–Littlewood conjecture F predicts the density of primes among the values of quadratic polynomials with integer coefficients in terms of the logarithmic integral and the polynomial coefficients. No quadratic polynomial has been proven to take infinitely many prime values.
  
  The Ulam spiral arranges the natural numbers in a two-dimensional grid, spiraling in concentric squares surrounding the origin with the prime numbers highlighted. Visually, the primes appear to cluster on certain diagonals and not others, suggesting that some quadratic polynomials take prime values more often than others.