{"id":349,"date":"2019-06-26T23:42:33","date_gmt":"2019-06-26T23:42:33","guid":{"rendered":"http:\/\/funfacts.104.42.120.246.xip.io\/?page_id=349"},"modified":"2019-08-07T17:19:23","modified_gmt":"2019-08-07T17:19:23","slug":"sums-of-two-squares-ways","status":"publish","type":"page","link":"https:\/\/math.hmc.edu\/funfacts\/sums-of-two-squares-ways\/","title":{"rendered":"Sums of Two Squares Ways"},"content":{"rendered":"\n

In the Fun Fact Sums of Two Squares<\/a>, we’ve seen which numbers can be written as the sum of two squares. For instance, 11 cannot, but 13 can (as 32<\/sup>+22<\/sup>). A related question, with a surprising answer, is: on average, how many ways can a number can be written as the sum of two squares?<\/p>\n\n\n\n

We should clarify what we mean by average<\/em>. Let W(N) is the number of ways to write N as the sum of two squares. Thus W(11)=0, and W(13)=8 (as sums of squares of all possible combinations of +\/-3 and +\/-2 , in either order).<\/p>\n\n\n\n

So if A(N) is the average of the numbers W(1), W(2), …, W(N), then A(N) is the average number of ways the first N numbers<\/em> can be written as the sum of two squares. Then it makes sense to take the limit<\/a>of A(N) as N goes to infinity to get the “average” number of ways to write a number as the sum of two squares, over all<\/em> positive whole numbers.<\/p>\n\n\n\n

A surprising fact is that this limit exists, and it is Pi!<\/p>\n\n\n\n

Presentation Suggestions:<\/strong>
This might be presented after a discussion of lattice points in Pick’s Theorem<\/a>.<\/p>\n\n\n\n

The Math Behind the Fact:<\/strong>
The proof is as neat as the result! Every solution (x,y) to x2<\/sup>+y2<\/sup>=N can be thought of as a lattice point<\/a> in the plane, i.e., a point with integer coordinates. Such a lattice point lies on a circle of radius Sqrt(N).<\/p>\n\n\n\n

Therefore, the sum of W(1) through W(N) counts the number of lattice points in the plane inside<\/em> or on a circle of radius Sqrt(N) (except for the origin), and the average A(N) is this number of lattice points divided by N. But as N goes to infinity, the number of lattice points inside this circle is approximately the area of the circle, hence Pi times the radius squared: Pi*Sqrt(N)2<\/sup> or Pi*N. Therefore A(N) is approximately Pi, and this approximation gets better and better as N goes to infinity!<\/p>\n\n\n\n

Counting the the number of lattice points inside a circle is known as Gauss’ circle problem<\/em>. Also, see several other Fun Facts about sums of two squares<\/a> or formulas for pi<\/a>.<\/p>\n\n\n\n

How to Cite this Page:<\/strong> 
Su, Francis E., et al. “Sums of Two Squares Ways.” Math Fun Facts<\/em>. <http:\/\/www.math.hmc.edu\/funfacts>.<\/p>\n\n\n\n

Fun Fact suggested by:<\/strong>
Darryl Yong<\/p>\n","protected":false},"excerpt":{"rendered":"

In the Fun Fact Sums of Two Squares, we’ve seen which numbers can be written as the sum of two squares....<\/p>\n","protected":false},"author":7,"featured_media":0,"parent":0,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"","meta":{"footnotes":""},"tags":[8,4,10,12,172,171,170],"class_list":["post-349","page","type-page","status-publish","hentry","tag-geometry","tag-medium","tag-numtheory","tag-other","tag-pi-formulas","tag-picks-theorem","tag-sums-of-squares"],"jetpack_sharing_enabled":true,"_links":{"self":[{"href":"https:\/\/math.hmc.edu\/funfacts\/wp-json\/wp\/v2\/pages\/349","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=349"}],"version-history":[{"count":2,"href":"https:\/\/math.hmc.edu\/funfacts\/wp-json\/wp\/v2\/pages\/349\/revisions"}],"predecessor-version":[{"id":1017,"href":"https:\/\/math.hmc.edu\/funfacts\/wp-json\/wp\/v2\/pages\/349\/revisions\/1017"}],"wp:attachment":[{"href":"https:\/\/math.hmc.edu\/funfacts\/wp-json\/wp\/v2\/media?parent=349"}],"wp:term":[{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/math.hmc.edu\/funfacts\/wp-json\/wp\/v2\/tags?post=349"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}