31:T4a2,md`# A remarkable Diophantine equation

Maybe you’ve heard about this deceptive equation:

${tex`\displaystyle\frac{a}{b+c}+\frac{b}{a+c}+\frac{c}{a+b}=4`}

Althought it looks innocent enough, the smallest solution in positive integers is, amazingly,

${tex`\begin{aligned}a&=154476802108746166441951315019919837485664325669565431700026634898253202035277999,\\b&=36875131794129999827197811565225474825492979968971970996283137471637224634055579,\\c&=4373612677928697257861252602371390152816537558161613618621437993378423467772036.\end{aligned}`}

There’s a [lovely article by Alon Amit](https://www.quora.com/How-do-you-find-the-positive-integer-solutions-to-frac-x-y+z-+-frac-y-z+x-+-frac-z-x+y-4/answer/Alon-Amit) explaining how to find the solution. If you change the 4 to something else, you can get *much* more ridiculous answers. Recently Christopher D. Long found the smallest solution to

${tex`\displaystyle\frac{a}{b+c}+\frac{b}{a+c}+\frac{c}{a+b}=37298`}

and it’s really quite something!`32:T49d,md` Here, in all its glory, is the solution that Long found. These scrollboxes load the digits incrementally as you scroll down, because it would crash your browser if we loaded the whole number; there are also download links in case you need to see all the digits at once. a =
(194,911,150 digits) [download](https://s3.amazonaws.com/s3.boskent.com/37298/a.txt) b =
(194,911,148 digits) [download](https://s3.amazonaws.com/s3.boskent.com/37298/b.txt) c =
(194,911,155 digits) [download](https://s3.amazonaws.com/s3.boskent.com/37298/c.txt) `33:[] 2f:{"title":"A remarkable Diophantine equation","cells":[{"id":0,"value":"$31","pinned":false,"hidden":false,"mode":"ojs"},{"id":17,"value":"{\n const div = document.createElement(\"DIV\");\n const twttr = await require(\"https://platform.twitter.com/widgets.js\").catch(() => window.twttr);\n await twttr.widgets.createTweet(\"1071255284327292933\", div);\n return div;\n}","pinned":false,"hidden":false,"mode":"ojs"},{"id":30,"value":"$32","pinned":false,"hidden":false,"mode":"ojs"}],"files":"$W33","stdlib":"1","titling":"auto"}