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aria-hidden="false" aria-describedby="postFooterSocialMenu" aria-labelledby="postFooterSocialMenu"><div><div class="bm" aria-hidden="false"><button aria-controls="postFooterSocialMenu" aria-expanded="false" aria-label="Share Post" data-testid="headerSocialShareButton" class="af fk ah ai aj ak al mi an ao ap ex mj mk ly ml mm mn mo mp s mq mr ms mt mu mv mw u mx my mz"><svg xmlns="http://www.w3.org/2000/svg" width="24" height="24" fill="none" viewBox="0 0 24 24"><path fill="currentColor" fill-rule="evenodd" d="M15.218 4.931a.4.4 0 0 1-.118.132l.012.006a.45.45 0 0 1-.292.074.5.5 0 0 1-.3-.13l-2.02-2.02v7.07c0 .28-.23.5-.5.5s-.5-.22-.5-.5v-7.04l-2 2a.45.45 0 0 1-.57.04h-.02a.4.4 0 0 1-.16-.3.4.4 0 0 1 .1-.32l2.8-2.8a.5.5 0 0 1 .7 0l2.8 2.79a.42.42 0 0 1 .068.498m-.106.138.008.004v-.01zM16 7.063h1.5a2 2 0 0 1 2 2v10a2 2 0 0 1-2 2h-11c-1.1 0-2-.9-2-2v-10a2 2 0 0 1 2-2H8a.5.5 0 0 1 .35.15.5.5 0 0 1 .15.35.5.5 0 0 1-.15.35.5.5 0 0 1-.35.15H6.4c-.5 0-.9.4-.9.9v10.2a.9.9 0 0 0 .9.9h11.2c.5 0 .9-.4.9-.9v-10.2c0-.5-.4-.9-.9-.9H16a.5.5 0 0 1 0-1" clip-rule="evenodd"></path></svg><div class="j i d"><p class="bf b bg z du">Share</p></div></button></div></div></div></div></div></div></div></div></div><p id="9766" class="pw-post-body-paragraph na nb hd nc b nd ne nf ng nh ni nj nk nl nm nn no np nq nr ns nt nu nv nw nx gw bk">Fermat’s theorem on sums of two squares had famously been proven in just one single sentence.</p><figure class="ob oc od oe of og ny nz paragraph-image"><div role="button" tabindex="0" class="oh oi fj oj bh ok"><div class="ny nz oa"><picture><source srcSet="https://miro.medium.com/v2/resize:fit:640/format:webp/1*fgIJy-ZZ5g7YHKfYILX7zg.jpeg 640w, https://miro.medium.com/v2/resize:fit:720/format:webp/1*fgIJy-ZZ5g7YHKfYILX7zg.jpeg 720w, https://miro.medium.com/v2/resize:fit:750/format:webp/1*fgIJy-ZZ5g7YHKfYILX7zg.jpeg 750w, https://miro.medium.com/v2/resize:fit:786/format:webp/1*fgIJy-ZZ5g7YHKfYILX7zg.jpeg 786w, https://miro.medium.com/v2/resize:fit:828/format:webp/1*fgIJy-ZZ5g7YHKfYILX7zg.jpeg 828w, https://miro.medium.com/v2/resize:fit:1100/format:webp/1*fgIJy-ZZ5g7YHKfYILX7zg.jpeg 1100w, https://miro.medium.com/v2/resize:fit:1400/format:webp/1*fgIJy-ZZ5g7YHKfYILX7zg.jpeg 1400w" sizes="(min-resolution: 4dppx) and (max-width: 700px) 50vw, (-webkit-min-device-pixel-ratio: 4) and (max-width: 700px) 50vw, (min-resolution: 3dppx) and (max-width: 700px) 67vw, (-webkit-min-device-pixel-ratio: 3) and (max-width: 700px) 65vw, (min-resolution: 2.5dppx) and (max-width: 700px) 80vw, (-webkit-min-device-pixel-ratio: 2.5) and (max-width: 700px) 80vw, (min-resolution: 2dppx) and (max-width: 700px) 100vw, (-webkit-min-device-pixel-ratio: 2) and (max-width: 700px) 100vw, 700px" type="image/webp"/><source data-testid="og" srcSet="https://miro.medium.com/v2/resize:fit:640/1*fgIJy-ZZ5g7YHKfYILX7zg.jpeg 640w, https://miro.medium.com/v2/resize:fit:720/1*fgIJy-ZZ5g7YHKfYILX7zg.jpeg 720w, https://miro.medium.com/v2/resize:fit:750/1*fgIJy-ZZ5g7YHKfYILX7zg.jpeg 750w, https://miro.medium.com/v2/resize:fit:786/1*fgIJy-ZZ5g7YHKfYILX7zg.jpeg 786w, https://miro.medium.com/v2/resize:fit:828/1*fgIJy-ZZ5g7YHKfYILX7zg.jpeg 828w, https://miro.medium.com/v2/resize:fit:1100/1*fgIJy-ZZ5g7YHKfYILX7zg.jpeg 1100w, https://miro.medium.com/v2/resize:fit:1400/1*fgIJy-ZZ5g7YHKfYILX7zg.jpeg 1400w" sizes="(min-resolution: 4dppx) and (max-width: 700px) 50vw, (-webkit-min-device-pixel-ratio: 4) and (max-width: 700px) 50vw, (min-resolution: 3dppx) and (max-width: 700px) 67vw, (-webkit-min-device-pixel-ratio: 3) and (max-width: 700px) 65vw, (min-resolution: 2.5dppx) and (max-width: 700px) 80vw, (-webkit-min-device-pixel-ratio: 2.5) and (max-width: 700px) 80vw, (min-resolution: 2dppx) and (max-width: 700px) 100vw, (-webkit-min-device-pixel-ratio: 2) and (max-width: 700px) 100vw, 700px"/><img alt="" class="bh mh ol c" width="700" height="525" loading="eager" role="presentation"/></picture></div></div><figcaption class="om ff on ny nz oo op bf b bg z du">Image from <a class="af oq" href="https://unsplash.com/photos/jR4Zf-riEjI" rel="noopener ugc nofollow" target="_blank">Unsplash</a></figcaption></figure><p id="19ba" class="pw-post-body-paragraph na nb hd nc b nd ne nf ng nh ni nj nk nl nm nn no np nq nr ns nt nu nv nw nx gw bk">Can prime numbers be written as the sum of two squares? For <em class="or">13</em> this is possible (2²<em class="or">+3²</em>), but for example, <em class="or">3</em> cannot be written in this way. Pierre de Fermat came up with a theorem on this topic, for which Don Zagier, an American mathematician gave a proof, which astonishingly is just one sentence long.</p><p id="3498" class="pw-post-body-paragraph na nb hd nc b nd ne nf ng nh ni nj nk nl nm nn no np nq nr ns nt nu nv nw nx gw bk">Numberphile <a class="af oq" href="https://www.youtube.com/watch?annotation_id=annotation_1866745269&feature=iv&src_vid=yGsIw8LHXM8&v=SyJlRUBoVp0" rel="noopener ugc nofollow" target="_blank">[1]</a> <a class="af oq" href="https://www.youtube.com/watch?v=yGsIw8LHXM8" rel="noopener ugc nofollow" target="_blank">[2]</a> made a great video on the one-sentence proof of Zagier but left out some details. I’ll give an overview of the points already covered in the video of Numberphile but will also explain the left out steps.</p><h1 id="c8a9" class="os ot hd bf ou ov ow ox oy oz pa pb pc pd pe pf pg ph pi pj pk pl pm pn po pp bk">The theorem</h1><p id="5dbe" class="pw-post-body-paragraph na nb hd nc b nd pq nf ng nh pr nj nk nl ps nn no np pt nr ns nt pu nv nw nx gw bk">Pierre de Fermat, a French mathematician of the seventeenth century, thought about under which conditions, primes could be written as the sum of two squares. For example, as already mentioned <em class="or">13</em> is a prime, which can be written as the sum of two squares, namely 2² and 3². Also, 17 is a prime satisfying this rule as we have 17<em class="or">=1²+4²</em>.</p><blockquote class="pv"><p id="e296" class="pw px hd bf py pz qa qb qc qd qe nx du">In general, Fermat was interested in the question, whether or not for a given prime number, let’s call it p, two other arbitrary numbers, let’s call them a and b, can be found, such that p=a²+b².</p></blockquote></div></div></div></div></section></div></div></article></div><div class="l"></div><footer class="qf qg qh qi gv ab q qj ip c"><div class="l ae"><div class="ab cb"><div class="ci bh fz ga gb gc"><div class="ab cp qk"><div class="ab q kz"><div class="ql l"><span class="l qm qn qo e d"><div class="ab q kz la"><div class="pw-multi-vote-icon fj gt lb lc ld"><span><a class="af ag ah ai aj ak al am an ao ap aq ar as at" data-testid="footerClapButton" 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For 13 this is possible (2²+3²), but for example, 3 cannot be written in this way. Pierre de Fermat came up with a theorem on this topic, for which Don Zagier, an American mathematician gave a proof, which astonishingly is just one sentence long.","hasDropCap":null,"dropCapImage":null,"markups":[{"__typename":"Markup","type":"EM","start":60,"end":62,"href":null,"anchorType":null,"userId":null,"linkMetadata":null},{"__typename":"Markup","type":"EM","start":83,"end":86,"href":null,"anchorType":null,"userId":null,"linkMetadata":null},{"__typename":"Markup","type":"EM","start":106,"end":107,"href":null,"anchorType":null,"userId":null,"linkMetadata":null}],"codeBlockMetadata":null,"iframe":null,"mixtapeMetadata":null},"Paragraph:9283d5894262_4":{"__typename":"Paragraph","id":"9283d5894262_4","name":"3498","type":"P","href":null,"layout":null,"metadata":null,"text":"Numberphile [1] [2] made a great video on the one-sentence proof of Zagier but left out some details. I’ll give an overview of the points already covered in the video of Numberphile but will also explain the left out steps.","hasDropCap":null,"dropCapImage":null,"markups":[{"__typename":"Markup","type":"A","start":12,"end":15,"href":"https:\u002F\u002Fwww.youtube.com\u002Fwatch?annotation_id=annotation_1866745269&feature=iv&src_vid=yGsIw8LHXM8&v=SyJlRUBoVp0","anchorType":"LINK","userId":null,"linkMetadata":null},{"__typename":"Markup","type":"A","start":16,"end":19,"href":"https:\u002F\u002Fwww.youtube.com\u002Fwatch?v=yGsIw8LHXM8","anchorType":"LINK","userId":null,"linkMetadata":null}],"codeBlockMetadata":null,"iframe":null,"mixtapeMetadata":null},"Paragraph:9283d5894262_5":{"__typename":"Paragraph","id":"9283d5894262_5","name":"c8a9","type":"H3","href":null,"layout":null,"metadata":null,"text":"The theorem","hasDropCap":null,"dropCapImage":null,"markups":[],"codeBlockMetadata":null,"iframe":null,"mixtapeMetadata":null},"Paragraph:9283d5894262_6":{"__typename":"Paragraph","id":"9283d5894262_6","name":"5dbe","type":"P","href":null,"layout":null,"metadata":null,"text":"Pierre de Fermat, a French mathematician of the seventeenth century, thought about under which conditions, primes could be written as the sum of two squares. For example, as already mentioned 13 is a prime, which can be written as the sum of two squares, namely 2² and 3². 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