The Consecutive Integer Partition Theorem, also known as the Odd Divisor Proprety of Consecutive Sums or a proprety of Sylvester’s Theorem, states that: for any positive integer N, the total number of ways to express N as the sum of one or more consecutive positive integers (including the trivial one-term case where the sum is N itself) is equal to the number of positive odd divisors of N.<\/p>\n\n\n\n
This by itself is difficult to understand. I must personally admit that trying to comprehend this monstrous statement is equilivant to swallowing an American bullfrog.<\/p>\n\n\n\n
You are given any positive integer N. Without loss of generality, assume it is 12.<\/p>\n\n\n\n
How many odd factors does 12 have? They are 1 and 3. So 12 has 2 odd factors.<\/p>\n\n\n\n
Recall that this equals to “the total number of ways to express N as the sum of one or more consecutive positive integers.” So, the total number of ways to express n as {the sum of consecutive positive integers} is 2.<\/p>\n\n\n\n
For example, we can express 12 as 3 + 4 + 5. This is indeed a sum of consecutive positive integers. 12 is also equal to 12 (Recall that 12 is also a sum of one or more consecutive positive integers.) There aren’t any more ways (see for yourself!)<\/p>\n\n\n\n
So yes, it is true that given a positive integer N, its total number of odd factors equals the number of ways to write N as the sum of one or more consecutive positive integers.<\/p>\n\n\n\n
One more example:<\/p>\n\n\n\n
14. 14 has 2 odd factors, 1 and 7. We can write 2 + 3 + 4 + 5 and just 14 by itself. This equals two ways so yes, the Consecutive Integer Partition Theorem (for these two cases, at least) is true!<\/p>\n\n\n\n
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Don’t Forget! <\/strong>When finding factors of N, don’t forget that 1 also qualifies as an odd number. <\/em><\/p>\n\n\n\n Also Don’t Forget! <\/strong>When finding ways to express N as a sum of some numbers, don’t forget that N itself is also a way. For example, 86 is also a way to express 86 as the sum of ONE (or more) consecutive positive integers.<\/em><\/p>\n","protected":false},"excerpt":{"rendered":" The Consecutive Integer Partition Theorem, also known as the Odd Divisor Proprety of Consecutive Sums or a proprety of Sylvester’s Theorem, states that: for any positive integer N, the total number of ways to express N as the sum of one or more consecutive positive integers (including the trivial one-term case where the sum is […]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[1],"tags":[76,75,77,22,54],"class_list":["post-1289","post","type-post","status-publish","format-standard","hentry","category-daily","tag-2016-amc12b-16","tag-amc12","tag-bullfrog","tag-math","tag-number-theory"],"_links":{"self":[{"href":"https:\/\/yesrenee.com\/index.php?rest_route=\/wp\/v2\/posts\/1289","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/yesrenee.com\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/yesrenee.com\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/yesrenee.com\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/yesrenee.com\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=1289"}],"version-history":[{"count":3,"href":"https:\/\/yesrenee.com\/index.php?rest_route=\/wp\/v2\/posts\/1289\/revisions"}],"predecessor-version":[{"id":1293,"href":"https:\/\/yesrenee.com\/index.php?rest_route=\/wp\/v2\/posts\/1289\/revisions\/1293"}],"wp:attachment":[{"href":"https:\/\/yesrenee.com\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=1289"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/yesrenee.com\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=1289"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/yesrenee.com\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=1289"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}