Editing Prime number (section)

=== Prime values of quadratic polynomials ===
[[File:Ulam 2.png|thumb|upright=1.1|The [[Ulam spiral]]. Prime numbers (orange) cluster on some diagonals and not others. Prime values of <math>4n^2 - 2n + 41</math> are shown in blue.|alt=The Ulam spiral]]
Euler noted that the function
: <math>n^2 - n + 41</math>
yields prime numbers for {{tmath| 1\le n\le 40 }}, although composite numbers appear among its later values.<ref>{{cite book |last1=Hua |first1=L. K. |title=Additive Theory of Prime Numbers |publisher=American Mathematical Society |year=2009 |isbn=978-0-8218-4942-2 |series=Translations of Mathematical Monographs |volume=13 |location=Providence, RI |pages=176–177 |mr=0194404 |oclc=824812353 |orig-year=1965}}</ref><ref>The sequence of these primes, starting at <math>n=1</math> rather than {{tmath|1= n=0 }}, is listed by {{cite book|title=103 curiosità matematiche: Teoria dei numeri, delle cifre e delle relazioni nella matematica contemporanea|language=it|first1=Paolo Pietro|last1=Lava|first2=Giorgio|last2=Balzarotti|publisher=Ulrico Hoepli Editore S.p.A.|year=2010|isbn=978-88-203-5804-4|page=133|contribution-url=https://books.google.com/books?id=YfsSAAAAQBAJ&pg=PA133|contribution=Chapter 33. Formule fortunate}}</ref> The search for an explanation for this phenomenon led to the deep [[algebraic number theory]] of [[Heegner number]]s and the [[class number problem]].<ref>{{cite book|title=Single Digits: In Praise of Small Numbers|first=Marc|last=Chamberland|publisher=Princeton University Press|year=2015|isbn=978-1-4008-6569-7|contribution=The Heegner numbers|pages=213–215|contribution-url=https://books.google.com/books?id=n9iqBwAAQBAJ&pg=PA213}}</ref> The [[Hardy–Littlewood conjecture F]] predicts the density of primes among the values of [[quadratic polynomial]]s with integer [[coefficient]]s in terms of the logarithmic integral and the polynomial coefficients. No quadratic polynomial has been proven to take infinitely many prime values.<ref name="guy-a1">{{cite book|title=Unsolved Problems in Number Theory|series=Problem Books in Mathematics|edition=3rd|first=Richard|last=Guy|author-link=Richard K. Guy|publisher=Springer|year=2013|isbn=978-0-387-26677-0|pages=7–10|contribution-url=https://books.google.com/books?id=1BnoBwAAQBAJ&pg=PA7|contribution=A1 Prime values of quadratic functions}}</ref>

The [[Ulam spiral]]<ref>{{Cite journal |last1=Stein |first1=M.L. |last2=Ulam |first2=S.M. |last3=Wells |first3=M.B. |date=1964 |title=A Visual Display of Some Properties of the Distribution of Primes |url=https://www.jstor.org/stable/2312588 |journal=The American Mathematical Monthly |volume=71 |issue=5 |pages=516–520 |doi=10.2307/2312588|jstor=2312588 }}</ref> 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.<ref name="guy-a1"/>