{"id":252,"date":"2019-06-26T23:11:40","date_gmt":"2019-06-26T23:11:40","guid":{"rendered":"http:\/\/funfacts.104.42.120.246.xip.io\/?page_id=252"},"modified":"2019-11-22T17:09:48","modified_gmt":"2019-11-22T17:09:48","slug":"formula-for-primes","status":"publish","type":"page","link":"https:\/\/math.hmc.edu\/funfacts\/formula-for-primes\/","title":{"rendered":"Formula for Primes?"},"content":{"rendered":"\n<p>Is there a nice cozy formula that will always spit out primes? Try this one: f(n) = n<sup>2<\/sup>\u00a0+ n + 41.<\/p>\n\n\n\n<p>Euler\u00a0discovered that this formula has a long string of prime values: it is prime for all n between 0 and 39 inclusive. However, it is not prime for all integers. In fact, it can be shown that no non-constant polynomial with integral coefficients will always spit out primes at the natural numbers.<\/p>\n\n\n\n<p>There are formulas which\u00a0<em>always<\/em>\u00a0spit out primes when you plug in a natural number&#8230; here&#8217;s one (Mills, 1947): greatest integer less than (X raised to 3<sup>n<\/sup>),<br>where X is approximately 1.3064&#8230; Surprised? See the remark below!<\/p>\n\n\n\n<p><strong>The\u00a0Math\u00a0Behind\u00a0the\u00a0Fact:<\/strong><br>It is worth pointing out that while the formula above looks nice, it is useless&#8230; it grows too quickly, and to determine X is tantamount to knowing the\u00a0primes\u00a0in its range!<\/p>\n\n\n\n<p><strong>How to Cite this Page:<\/strong>&nbsp;<br>Su, Francis E., et al. &#8220;Formula for Primes?.&#8221;&nbsp;<em>Math Fun Facts<\/em>. &lt;http:\/\/www.math.hmc.edu\/funfacts&gt;.<\/p>\n\n\n\n<p><strong>References:<\/strong><br>P. Ribenboim, The Little Book of Big Primes<\/p>\n\n\n\n<p><strong>Fun Fact suggested by:   <\/strong><br>Francis Su <\/p>\n","protected":false},"excerpt":{"rendered":"<p>Is there a nice cozy formula that will always spit out primes? Try this one: f(n) = n2\u00a0+ n +&#46;&#46;&#46;<\/p>\n","protected":false},"author":7,"featured_media":0,"parent":0,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"","meta":{"footnotes":""},"tags":[3,10,15,49,108],"class_list":["post-252","page","type-page","status-publish","hentry","tag-easy","tag-numtheory","tag-polynomial","tag-prime","tag-prime-formula"],"jetpack_sharing_enabled":true,"_links":{"self":[{"href":"https:\/\/math.hmc.edu\/funfacts\/wp-json\/wp\/v2\/pages\/252","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=252"}],"version-history":[{"count":4,"href":"https:\/\/math.hmc.edu\/funfacts\/wp-json\/wp\/v2\/pages\/252\/revisions"}],"predecessor-version":[{"id":1443,"href":"https:\/\/math.hmc.edu\/funfacts\/wp-json\/wp\/v2\/pages\/252\/revisions\/1443"}],"wp:attachment":[{"href":"https:\/\/math.hmc.edu\/funfacts\/wp-json\/wp\/v2\/media?parent=252"}],"wp:term":[{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/math.hmc.edu\/funfacts\/wp-json\/wp\/v2\/tags?post=252"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}