We know there are\u00a0infinitely many primes, so are many interesting questions you can ask about the\u00a0distribution of primes, i.e., how they spread themselves out. Here is something to ponder: are there arbitrarily large “gaps” in the sequence of primes?<\/p>\n\n\n\n
At first this may seem like a tough question to tackle, since it is sometimes tedious to determine whether a number is prime. But it may help to look at the problem a different way: can I find long sequences of successive integers which are all composite?<\/p>\n\n\n\n
Yes, and now it is easy to see why. Suppose I want to find (N-1) consecutive integers that are composite. The number N! has, as factors, all numbers between 1 and N. Therefore:
N!+2 is composite, since it is divisible by 2.
N!+3 is composite, since it is divisible by 3.<\/p>\n\n\n\n
In fact, for similar reasons, N!+k is composite for all k between 2 and N. This is a string of (N-1) successive integers which are all composite.<\/p>\n\n\n\n
Presentation Suggestions:<\/strong> The Math Behind the Fact:<\/strong> How to Cite this Page:<\/strong> Fun Fact suggested by: <\/strong> We know there are\u00a0infinitely many primes, so are many interesting questions you can ask about the\u00a0distribution of primes, i.e., how...<\/p>\n","protected":false},"author":7,"featured_media":0,"parent":0,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"","meta":{"footnotes":""},"tags":[3,10,49],"class_list":["post-315","page","type-page","status-publish","hentry","tag-easy","tag-numtheory","tag-prime"],"jetpack_sharing_enabled":true,"_links":{"self":[{"href":"https:\/\/math.hmc.edu\/funfacts\/wp-json\/wp\/v2\/pages\/315","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=315"}],"version-history":[{"count":3,"href":"https:\/\/math.hmc.edu\/funfacts\/wp-json\/wp\/v2\/pages\/315\/revisions"}],"predecessor-version":[{"id":1460,"href":"https:\/\/math.hmc.edu\/funfacts\/wp-json\/wp\/v2\/pages\/315\/revisions\/1460"}],"wp:attachment":[{"href":"https:\/\/math.hmc.edu\/funfacts\/wp-json\/wp\/v2\/media?parent=315"}],"wp:term":[{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/math.hmc.edu\/funfacts\/wp-json\/wp\/v2\/tags?post=315"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}
It may be good to warm up by asking is what the largest prime gap less than 100.<\/p>\n\n\n\n
Sometimes simple deductions can lead to surprising results!<\/p>\n\n\n\n
Su, Francis E., et al. “Gaps in Primes.” Math Fun Facts<\/em>. <http:\/\/www.math.hmc.edu\/funfacts>.<\/p>\n\n\n\n
Lesley Ward <\/p>\n","protected":false},"excerpt":{"rendered":"