One of my colleagues came into my office the other day, and said to me:<\/p>\n
If p is prime and greater than 3, prove that p² – 1 is divisible by 24.<\/p>\n
Interested readers might want to try this for themselves: it’s not particularly difficult. Of course I started in right away.<\/p>\n
Immediately I thought: well, divisibility by 24 means divisibility by 2, by 3 and by 4. Let’s see what we can say about this considering p is odd. i.e. p = 2n + 1. So p² – 1 = (2n + 1)(2n + 1) – 1 = 4n² + 4n. Clearly divisible by 4.<\/p>\n
Then I thought: aha. p² – 1 = (p + 1)(p – 1). One of (p + 1), p, (p – 1) is clearly divisible by 3, since these are 3 consecutive numbers. Since p is prime, either (p + 1) or (p – 1) is divisible by 3. Since p is odd, (p + 1) and (p – 1) are both even. And furthermore, since they are consecutive even numbers, one of them must be divisible by 4.<\/p>\n
QED.<\/p>\n","protected":false},"excerpt":{"rendered":"
One of my colleagues came into my office the other day, and said to me: If p is prime and greater than 3, prove that p² – 1 is divisible by 24. Interested readers might want to try this for themselves: it’s not particularly difficult. Of course I started in…<\/p>\n","protected":false},"author":2,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[1],"tags":[],"class_list":["post-737","post","type-post","status-publish","format-standard","hentry","category-uncategorized"],"_links":{"self":[{"href":"https:\/\/www.elbeno.com\/blog\/index.php?rest_route=\/wp\/v2\/posts\/737","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.elbeno.com\/blog\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.elbeno.com\/blog\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.elbeno.com\/blog\/index.php?rest_route=\/wp\/v2\/users\/2"}],"replies":[{"embeddable":true,"href":"https:\/\/www.elbeno.com\/blog\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=737"}],"version-history":[{"count":1,"href":"https:\/\/www.elbeno.com\/blog\/index.php?rest_route=\/wp\/v2\/posts\/737\/revisions"}],"predecessor-version":[{"id":738,"href":"https:\/\/www.elbeno.com\/blog\/index.php?rest_route=\/wp\/v2\/posts\/737\/revisions\/738"}],"wp:attachment":[{"href":"https:\/\/www.elbeno.com\/blog\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=737"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.elbeno.com\/blog\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=737"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.elbeno.com\/blog\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=737"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}