Fix some number N. What fraction of the integers less than or equal to N are prime?<\/p>\n\n\n\n
Thinking about it, we know that primes occur less and less often as N grows. Can we quantify this somehow?<\/p>\n\n\n\n
Let Pi(N) denote the number of primes less than or equal to N that are prime. Then we expect that the fraction Pi(N)\/N must change (decrease?) with N. In fact there is an amazing theorem called the Prime Number Theorem which says that<\/p>\n\n\n\n
Pi(N)\/N is asymptotic to 1\/ln(N)<\/p>\n\n\n\n
which means that the ratio of those two quantities approaches 1 as N goes to infinity! Thus Pi(N) is closely approximated by N\/ln(N). In fact, a better estimate for Pi(N) is that it is very closely approximated by this\u00a0integral:<\/p>\n\n\n\n
INTEGRAL2 to x<\/sub>\u00a0dt\/ln(t) .<\/p>\n\n\n\n Presentation Suggestions:<\/strong> The Math Behind the Fact:<\/strong> How to Cite this Page:<\/strong> References:<\/strong> Fun Fact suggested by: <\/strong> Fix some number N. What fraction of the integers less than or equal to N are prime? Thinking about it,...<\/p>\n","protected":false},"author":7,"featured_media":0,"parent":0,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"","meta":{"footnotes":""},"tags":[7,100,6,151,4,10],"class_list":["post-401","page","type-page","status-publish","hentry","tag-calculus","tag-distribution-of-primes","tag-hard","tag-how-many-primes","tag-medium","tag-numtheory"],"jetpack_sharing_enabled":true,"_links":{"self":[{"href":"https:\/\/math.hmc.edu\/funfacts\/wp-json\/wp\/v2\/pages\/401","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/math.hmc.edu\/funfacts\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/math.hmc.edu\/funfacts\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/math.hmc.edu\/funfacts\/wp-json\/wp\/v2\/users\/7"}],"replies":[{"embeddable":true,"href":"https:\/\/math.hmc.edu\/funfacts\/wp-json\/wp\/v2\/comments?post=401"}],"version-history":[{"count":3,"href":"https:\/\/math.hmc.edu\/funfacts\/wp-json\/wp\/v2\/pages\/401\/revisions"}],"predecessor-version":[{"id":1580,"href":"https:\/\/math.hmc.edu\/funfacts\/wp-json\/wp\/v2\/pages\/401\/revisions\/1580"}],"wp:attachment":[{"href":"https:\/\/math.hmc.edu\/funfacts\/wp-json\/wp\/v2\/media?parent=401"}],"wp:term":[{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/math.hmc.edu\/funfacts\/wp-json\/wp\/v2\/tags?post=401"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}
Write out a table of Pi(N) for the first few values of N (or flash a transparency) just to give students a concrete feel for this function before telling them the answer.<\/p>\n\n\n\n
The proof of the Prime Number Theorem requires some hard asymptotic analysis. Several people have proved various versions of the Prime Number Theorem; among them Chebyshev, Hadamard, de la Vallee Poussin, Atle, Selberg, although the theorem was suspected by Gauss (1791).<\/p>\n\n\n\n
Su, Francis E., et al. “Prime Number Theorem.” Math Fun Facts<\/em>. <http:\/\/www.math.hmc.edu\/funfacts>.<\/p>\n\n\n\n
Any book on analytic number theory<\/p>\n\n\n\n
Francis Su <\/p>\n","protected":false},"excerpt":{"rendered":"