2006/01/01 by Andrew Granville, Greg Martin
Mathematics · #Advanced Mathematical Identities #Analytic Number Theory Research #Arithmetic #Combinatorics #Figuring #History and Theory of Mathematics #Mathematics #Physics #Prime (order theory) #Pure mathematics #Statistics #Value (mathematics)
paper · doi:10.1080/00029890.2006.11920275
crossref issued 2006/01/01 · crossref published 2006/01/01 · crossref published-print 2006/01/01 · openalex publication_date 2006/01/01 · crossref published-online 2018/01/31 · crossref created 2018/01/31 · crossref deposited 2018/03/01 · openalex created_date 2019/07/30 · crossref indexed 2026/03/25 · openalex updated_date 2026/08/03
This talk is a survey of “prime number races”. Chebyshev noticed in the first half of the nineteenth century that for any given value of x, there always seem to be more primes of the form 4n+3 less than x than there are of the form 4n+1. Similar observations have been made with primes of the form 3n+2 and 3n+1, primes of the form 10n+3, 10n+7 and 10n+1, 10n+9, and many others besides. More generally, one can consider primes of the form qn + a, qn + b, qn + c, . . . for our favorite constants q, a, b, c, . . . and try to figure out which forms are “preferred” over the others – not to mention figuring out what, precisely, being “preferred” means. We describe these phenomena in greater detail and explain the efforts that have been made at understanding them. This talk should be accessible to graduate students. 1272547773