Problem 52834. Easy Sequences 32: Almost Pythagorean Triples

Created by Ramon Villamangca

Solve Later

{"parts":[{"partUri":"/matlab/document.xml","contentType":"application/vnd.mathworks.matlab.code.document+xml","content":"<?xml version=\"1.0\" encoding=\"UTF-8\"?><w:document xmlns:w=\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\"><w:body><w:p><w:pPr><w:pStyle w:val=\"text\"/><w:jc w:val=\"left\"/></w:pPr><w:r><w:t>An Almost Pythagorean Triple (abbreviated as \"APT'), is a set of 3 integers in which square of the largest element, which we will call as its 'hypotenuse', is </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/></w:customXmlPr><w:r><w:t>1</w:t></w:r></w:customXml><w:r><w:t> </w:t></w:r><w:r><w:rPr><w:u/></w:rPr><w:t>less</w:t></w:r><w:r><w:t> than the sum of square of the smaller elements (shorter sides). This means that if </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/></w:customXmlPr><w:r><w:t>c</w:t></w:r></w:customXml><w:r><w:t> is the hypotenuse and </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/></w:customXmlPr><w:r><w:t>a</w:t></w:r></w:customXml><w:r><w:t> and </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/></w:customXmlPr><w:r><w:t>b</w:t></w:r></w:customXml><w:r><w:t> are the shorter sides, </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/></w:customXmlPr><w:r><w:t>APT</w:t></w:r></w:customXml><w:r><w:t>, satisfies the following equation: </w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/><w:jc w:val=\"left\"/></w:pPr><w:r><w:t> </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"true\"/></w:customXmlPr><w:r><w:t>c^2 = a^2+b^2-1</w:t></w:r></w:customXml></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/><w:jc w:val=\"left\"/></w:pPr><w:r><w:t> where: </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/></w:customXmlPr><w:r><w:t>a \\le b &lt; c</w:t></w:r></w:customXml></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/><w:jc w:val=\"left\"/></w:pPr><w:r><w:t>The smallest </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/></w:customXmlPr><w:r><w:t>APT</w:t></w:r></w:customXml><w:r><w:t> is the triple </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/></w:customXmlPr><w:r><w:t>[5,5,7]</w:t></w:r></w:customXml><w:r><w:t>, with </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/></w:customXmlPr><w:r><w:t>7^2=5^2+5^2-1</w:t></w:r></w:customXml><w:r><w:rPr/><w:t> and perimeter (the sum of the 3 elements)</w:t></w:r><w:r><w:rPr><w:b/></w:rPr><w:t> </w:t></w:r><w:r><w:t>of </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/></w:customXmlPr><w:r><w:t>17</w:t></w:r></w:customXml><w:r><w:t>. Some researchers consider </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/></w:customXmlPr><w:r><w:t>[1,2,2]</w:t></w:r></w:customXml><w:r><w:t> as the smallest </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/></w:customXmlPr><w:r><w:t>APT</w:t></w:r></w:customXml><w:r><w:t>, but here, we will only look at </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/></w:customXmlPr><w:r><w:t>APT</w:t></w:r></w:customXml><w:r><w:t>'s where the hypotenuse is \"strictly\" greater than the other shorter sides. Other examples of </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/></w:customXmlPr><w:r><w:t>APT</w:t></w:r></w:customXml><w:r><w:t>'s are </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/></w:customXmlPr><w:r><w:t>[9,8,12]</w:t></w:r></w:customXml><w:r><w:t>, and </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/></w:customXmlPr><w:r><w:t>[33,64,72]</w:t></w:r></w:customXml><w:r><w:t>. </w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/><w:jc w:val=\"left\"/></w:pPr><w:r><w:t>Unfortunately, unlike Pythagorean Triples, a 'closed formula' for generating all possible </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/></w:customXmlPr><w:r><w:t>APT</w:t></w:r></w:customXml><w:r><w:t>'s, has not yet been discovered, at the time of this writing. For this exercise, we will be dealing with </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/></w:customXmlPr><w:r><w:t>APT</w:t></w:r></w:customXml><w:r><w:t>'s with a known ratio between the hypotenuse and the shortest side: </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/></w:customXmlPr><w:r><w:t>r = c/a</w:t></w:r></w:customXml><w:r><w:t>. </w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/><w:jc w:val=\"left\"/></w:pPr><w:r><w:rPr><w:b/></w:rPr><w:t>Given the value of </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/></w:customXmlPr><w:r><w:t>r</w:t></w:r></w:customXml><w:r><w:rPr><w:b/></w:rPr><w:t>, find the perimeter of the </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/></w:customXmlPr><w:r><w:t>APT</w:t></w:r></w:customXml><w:r><w:rPr><w:b/></w:rPr><w:t> with the </w:t></w:r><w:r><w:rPr><w:b/><w:u/></w:rPr><w:t>r-th smallest</w:t></w:r><w:r><w:rPr><w:b/></w:rPr><w:t> perimeter. </w:t></w:r><w:r><w:t>For example for </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/></w:customXmlPr><w:r><w:t>r = 2</w:t></w:r></w:customXml><w:r><w:t>, that is </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/></w:customXmlPr><w:r><w:t>c = 2a</w:t></w:r></w:customXml><w:r><w:t>, the smallest perimeter is </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/></w:customXmlPr><w:r><w:t>19</w:t></w:r></w:customXml><w:r><w:t> for </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/></w:customXmlPr><w:r><w:t>APT</w:t></w:r></w:customXml><w:r><w:t> </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/></w:customXmlPr><w:r><w:t>[4,7,8]</w:t></w:r></w:customXml><w:r><w:t>, while the second (r-th) smallest perimeter is </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/></w:customXmlPr><w:r><w:t>71</w:t></w:r></w:customXml><w:r><w:t>, for the </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/></w:customXmlPr><w:r><w:t>APT</w:t></w:r></w:customXml><w:r><w:t> with dimensions </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/></w:customXmlPr><w:r><w:t>[15,26,30]</w:t></w:r></w:customXml><w:r><w:t>. For </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/></w:customXmlPr><w:r><w:t>r = 3</w:t></w:r></w:customXml><w:r><w:t>, the third smallest perimeter is </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/></w:customXmlPr><w:r><w:t>1393</w:t></w:r></w:customXml><w:r><w:t> for </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/></w:customXmlPr><w:r><w:t>APT</w:t></w:r></w:customXml><w:r><w:t> </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/></w:customXmlPr><w:r><w:t>[204,577,1224]</w:t></w:r></w:customXml><w:r><w:t>. </w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/><w:jc w:val=\"left\"/></w:pPr><w:r><w:t>The output can be very large, so please present only the last 12 digits if the number of digits of the perimeter exceeds 12.</w:t></w:r></w:p></w:body></w:document>","relationship":null}],"relationships":[{"relationshipType":"http://schemas.mathworks.com/matlab/code/2013/relationships/document","target":"/matlab/document.xml","relationshipId":"rId1"}]}

<div style = "text-align: start; line-height: 20.4333px; min-height: 0px; white-space: normal; color: rgb(0, 0, 0); font-family: Menlo, Monaco, Consolas, monospace; font-style: normal; font-size: 14px; font-weight: 400; text-decoration: rgb(0, 0, 0); white-space: normal; "><div style="block-size: 369px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 407px 184.5px; transform-origin: 407px 184.5px; vertical-align: baseline; "><div style="block-size: 63px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 31.5px; text-align: left; transform-origin: 384px 31.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; "><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 380px 8px; transform-origin: 380px 8px; unicode-bidi: normal; ">An Almost Pythagorean Triple (abbreviated as "APT'), is a set of 3 integers in which square of the largest element, which we will call as its 'hypotenuse', is 1<span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 2px 8px; transform-origin: 2px 8px; unicode-bidi: normal; "> <span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 12.5px 8px; transform-origin: 12.5px 8px; unicode-bidi: normal; ">less<span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 260px 8px; transform-origin: 260px 8px; unicode-bidi: normal; "> than the sum of square of the smaller elements (shorter sides). This means that if c<span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 74px 8px; transform-origin: 74px 8px; unicode-bidi: normal; "> is the hypotenuse and a<span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 16px 8px; transform-origin: 16px 8px; unicode-bidi: normal; "> and b<span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 71.5px 8px; transform-origin: 71.5px 8px; unicode-bidi: normal; "> are the shorter sides, <img src="data:image/png;base64,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" style="width: 32.5px; height: 18px;" width="32.5" height="18"><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 104px 8px; transform-origin: 104px 8px; unicode-bidi: normal; ">, satisfies the following equation: </div><div style="block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; "><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 16px 8px; transform-origin: 16px 8px; unicode-bidi: normal; "> <img src="data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAMcAAAAmCAYAAABwHY/hAAAEBElEQVR4nO2b7ZGCMBCG3x7ogAZogAqowA7sgA5swRoowR5swRpswfux7LAiASG7RHL7zGRu7vA0m+x3IuA4juM4juM4juM4zqGpAHQAngBeAB4AmqQz2ocCwBUk7wskf5t0Rnb8J1nVqEALdgZQA7iAFu+FvBevAHAHyVuD5GfncEs4Lwv+k6yqdPiMEmcM3iVXziBlkVQYHEO1+4zs+E+yqnIDeRZJgWHhxs9yoQNQTvz9DpL7CGllBZrnUoTPQVamwLQsu8K1R67GEeKG3/emsnaISY2OICtTgJzAExQJk1GCFq1LOYlEPHAMp1AjvjY8iqxsFCxvUuM49ZM5gkfRhJ3CKfVEvqDBoCz1hv8/gqwlqFY6gaLlTxjHA3l3qkJcQdFSy5O2sIvArCxbmybaslojI2Uy42hBC3eURdOiBhWomsWepXFwzbHlvS1ktSa5cTSY7l7lTglSNm1lsTIO2YZdqyhWslpjahxF/wGh/LQCGcbRFi0EyyvrphKf8hcgubfk7UtYGQefQ73wvl8VPmWWWMpqjbpxcPvrAUqVWlA4feB9gfiUfMow1qRYvDmxI4YapAB8PeLS/85tS6mofHI81edvEd+QsDKODkOrHaD5y9Yut3flflrLao2qcTQgBRkrPRdydwwHKk/QYrajccPnyeocrIAxI+ZUnq++jJVRXomRC8uOYiz3FTrppZVxsCzcyXn2P2sMhjP+XGtZrVEzjhPCh3jy4hl7kzllXeNRpJfeOrYqEhv91P+zkso0RLYGp4ZG6LYwDtnCveC7Pd5DVmtUjEO+yVSKwtcNfj2MroE3/xF4fll4vhVZx00Nntdt4XVr6jwZBVn5x8gIrhkRUqbMKsYR0+I7ItKThu4IcXRckyJ+g9ywmLFms2VtETrAY3m1L46mTJmjjYPTqVDUyBFWlnvgOZ8EzxnPVrjDFxoyvZl73bfzkrKE5JWXRrUdZKqUGVAwDlmMpSiw9u5WSWcQ8qIy3957TbRrDtnCDcmbq4OMNg6+nBXyKtbs3a1acgbS06b4Uo+2cbC8c2vE0Sq3LzFFG8dSyLVm727VkjOQxqpdb3yDtnGwvNeFzwsV6kcm2jhk/h1anDPyOQWfcwZ8sU7WGwX2vWynaRxSOaZSKvk8p04kE20cMr8eLxD3u0Ne54jIrgwbPMvZ4T3/LvvX79nT1zQO2cIdF/A1aA1y/ppBtHEUeE81mv5NT6Co8st397cglZ87Qk8MDqAdPd/7sMvKOHgfS/H3nG9Sc8SXXbhNshYgJZB5fE6p1JgGtFh81UXKWfbPrkjTudGuOSq813Ud8t5bYLjGNFWbtsjXIWRPDdrA3CK24ziO4ziO4ziO4ziO4ziO4ziOY8QfaJ4ygEB93kYAAAAASUVORK5CYII=" style="width: 99.5px; height: 19px;" width="99.5" height="19"></div><div style="block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; "><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 41px 8px; transform-origin: 41px 8px; unicode-bidi: normal; "> where: <img src="data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAHwAAAAkCAYAAABLw14kAAAC50lEQVR4nO2a7ZGDIBCG3x7swAZswAqsIB3YQTqwhdSQEuwhLViDLeR+wA4rB56woLnMPjPMmPOyfLzLskAARVEURVEURVGUT6AHcLNF+VJaAE8AK4C3LfdLW6QcYgDQCL5/hxO8L9IipQo3AAuMUJ3AzgQnuPKBcKHfMGG5FdgjW09504rQQNafryEktDQEt8zeKLQlpYFZXtYPaMul1BCaGJldybIggQtNbRkuasufdDCDz5OmHvKQRINQS2jiaW0v3t97lOnHHiGhpwp1kkbZdnlDn/Z5xjb5yZ0toUGoITRB9TzsZ79uEkGyC/BpA/WUFrqHGbfF1kXaJPelQzgzbuAav2Y08GyhYW3zEPoA8LLPg30uub63tg7fmUoK3cBFLX6mwHOVw4dLHZwgoRlMBh+Bd3uM2Ao945z9MI9IE35n6XyQZkE9ZwgNGLHJSX0HvSHRebnYIQ/hMzz1ePKsAfGhwVntcwhqT+z9Hmf3a0bcOSmCHo5UNDixjg9wncpd785KYoDt7H1H6uBOnLJHv8KB+QwW7za4sdh2gTqYMxN8zhCe92mK/A934pQ1fMB2d7Gg/jaL6pMsPb+M+VsXgs+W2ODlEBO+xH6ZEpvY7Aa2szQH//yglvAdCo7/EWN8YGokW6E9+QyZ8OREsVDNnVh6g1ZbeH54JLb7lzG+tZGs30fxBy9HeO7EsQRzLwHKpZbw/LZPbG/PGGXuFB5p/b6h/lmwP3gpovM+hcI5vV9Qx4F7OIcqITzPR2IO3O2828ATFx7Sexix+UHMBCdE7ZlOUH0pgvMDFb+dFNFegXelKSU8X35CSxRpdTjp5TPpZQuJDfZuxblic1IyeC44fa+DG/yzb6pCwqcuUzxq3eHuAuj0MEmTBmb2zjAe5IeG0b4bUw1fCB2lzrY8YPp1ZftJoFyno+/zPn3krRvd5kjKf3G0I7T48p9a8XCWW666y1YyoOVCUvQnQYqiKIqiKIqifAk/IsB4nRPGszsAAAAASUVORK5CYII=" style="width: 62px; height: 18px;" width="62" height="18"></div><div style="block-size: 63px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 31.5px; text-align: left; transform-origin: 384px 31.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; "><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 42px 8px; transform-origin: 42px 8px; unicode-bidi: normal; ">The smallest <img src="data:image/png;base64,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" style="width: 32.5px; height: 18px;" width="32.5" height="18"><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 38.5px 8px; transform-origin: 38.5px 8px; unicode-bidi: normal; "> is the triple <img src="data:image/png;base64,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" style="width: 48.5px; height: 18.5px;" width="48.5" height="18.5"><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 18px 8px; transform-origin: 18px 8px; unicode-bidi: normal; ">, with <img src="data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAMkAAAAmCAYAAABu1L9SAAAD6ElEQVR4nO2c7ZGjMAyG3x7SAQ3QABWkAjpIB3SQFqghJdBDWqCGtLD3w7yDz3zYAdlgomdGczO7XGJblixLYgFFURRFURRFURRFUX6KO4AOwN8gHYDy0BGloQTwAvCBmXcPsxZX5Fd1LEIN4D38e4fZNFzI6sBxxaaEMYoHzDyfGOfdHDiuGPyqjsXoARTOz7iIr/TDScYL01PjATPvT/rhROVXdTyhhPEKIcJjlh7U5Q6zgO+4QxanwPKcHzCelHQAbs7/v2H0sO7vzsaavluMRnFFHW/WTY9RwT6ZWzSbCnl6GTvmduWDsMXl3eTMRnLDeI+aky7gM3LTcQFj/H/YeJeiVwgVXxxaD889tgzmIKj0JQm5ZxTIY+M02KdfIB8d3zAaB2WTkXSDuHGn+2X0kj5amGP4zN7UpcP+zV3DeOgzZ314iuzd3Dno+A7jEO4wY91sJAXCwgN6Dl+oRW+aU9ajhBnz3vRtD9nM1i7Pt8ADxkjWHKKPHHVsn56bjCRkssxm+J7tcP4j2IXHcYXtm6cZPkfSs8Ywkh7Gq1bYPtYcdbzLSEJgqOW7vD7hP2nOBr2iLTwRQg2GhTbp0ENaqYwGbGH9I3TsOeoYSGAkXNzW84y0J02Be6lzxecxS/jvc1uRVupaBvODsIRMjjoGEhgJQ62lmL3Gct1gzbDcZ0PrNGuyZbOyZtDg/wueL+XNqvvcd0psphhKrWD0+MS80cTU8ZFENRJfqMX0cTsMxJZvepl8KdhQkYiVK0yNxfWyBcyadJjOu4NMSBLV8w0wI7dWD5LS8ZFENZK1UKvEelHqm6Iaw5a9IqkwOxSzC2w3zJ84u3PxDimMBJjOx87QSer4SKIaiS/UujrcPKEV92/wtQJRqbXnOQl4MroOQQqJcHrP5o5mJKFZrStjZ4SkPdBaG0yoSDZT8uSM0aApEU7vKfZGM5KQrNbVsZsWpY3kifXQkd/7XnlGsgWGd48YRiIRTu8p1EYzEioqRah1ZHbLBxc3xmeHfG+qNhd6+5DWo9yIYiQssqUKtc6U3bL5pmdNmtRGYmexrkYUI+ELRKkW7IzZLSC8Zy0GqY2Ed5IrJmmiGAmzOrXvwczxZU16LBcMYyOpVIazS/Ng5BAjs3UGxI3EDrWO2BypsN/VfmE61xZmDY7yrJJKtWsdDf4PoVkneQt91xkRNxKGWlf1KsR9+egDE1axknz0XwSRVKqbbmYT5xNm3rn2ZIXA10HmiqWbYWryirGpC9/ptlOqDc7xroR0DF3DzI9zZZvJVaOFAstp9hb5tfsrM7A/6qqbWFEURVEURVEURVEURVEURVEURTkZ/wCi5zysuW8z8wAAAABJRU5ErkJggg==" style="width: 100.5px; height: 19px;" width="100.5" height="19"><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 134.5px 8px; transform-origin: 134.5px 8px; unicode-bidi: normal; "> and perimeter (the sum of the 3 elements)<span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 4px 8px; transform-origin: 4px 8px; unicode-bidi: normal; "> <span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 8px 8px; transform-origin: 8px 8px; unicode-bidi: normal; ">of <img src="data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAACQAAAAkCAYAAADhAJiYAAAA6UlEQVRYhe2WQQ3DMAxFP4cwCIESGIIiKIMxKINSKIZBKIdRKIZR6A6ztWiHxkm+qkjzl6xenOQpL5UMeDyevhIBhEzPAOBmrKEFZAVwGDbZpc9SSylISEC0zoDGApgDn1syZwQwy/dpBNqk4klPkH32EpjfzAagKIfk3tiESl01QBYFD1ToqgGyRHW9kL/JS4BU19oCwwRSXWMPQDRdLCCaLhYQTRcDiKqLAUTVxQDaQNTVChRB1tUKdAdZVyuQTgpTD0CprrORpCg6VijQXLBWdW0skAXfoSutVQ7LRdfT/i6Px+P5u7wB3c96XQFb71kAAAAASUVORK5CYII=" style="width: 18px; height: 18px;" width="18" height="18"><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 24px 8px; transform-origin: 24px 8px; unicode-bidi: normal; ">. Some researchers consider <img src="data:image/png;base64,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" style="width: 48.5px; height: 18.5px;" width="48.5" height="18.5"><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 51px 8px; transform-origin: 51px 8px; unicode-bidi: normal; "> as the smallest <img src="data:image/png;base64,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" style="width: 32.5px; height: 18px;" width="32.5" height="18"><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 94px 8px; transform-origin: 94px 8px; unicode-bidi: normal; ">, but here, we will only look at <img src="data:image/png;base64,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" style="width: 32.5px; height: 18px;" width="32.5" height="18"><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 113px 8px; transform-origin: 113px 8px; unicode-bidi: normal; ">'s where the hypotenuse is "strictly" greater than the other shorter sides. Other examples of <img src="data:image/png;base64,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" style="width: 32.5px; height: 18px;" width="32.5" height="18"><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 19.5px 8px; transform-origin: 19.5px 8px; unicode-bidi: normal; ">'s are <img src="data:image/png;base64,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" style="width: 56px; height: 18.5px;" width="56" height="18.5"><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 18px 8px; transform-origin: 18px 8px; unicode-bidi: normal; ">, and <img src="data:image/png;base64,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" style="width: 71px; height: 18.5px;" width="71" height="18.5"><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 4px 8px; transform-origin: 4px 8px; unicode-bidi: normal; ">. </div><div style="block-size: 63px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 31.5px; text-align: left; transform-origin: 384px 31.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; "><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 274px 8px; transform-origin: 274px 8px; unicode-bidi: normal; ">Unfortunately, unlike Pythagorean Triples, a 'closed formula' for generating all possible <img src="data:image/png;base64,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" style="width: 32.5px; height: 18px;" width="32.5" height="18"><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 64px 8px; transform-origin: 64px 8px; unicode-bidi: normal; ">'s, has not yet been discovered, at the time of this writing. For this exercise, we will be dealing with <img src="data:image/png;base64,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" style="width: 32.5px; height: 18px;" width="32.5" height="18"><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 105.5px 8px; transform-origin: 105.5px 8px; unicode-bidi: normal; ">'s with a known ratio between the hypotenuse and the shortest side: <img src="data:image/png;base64,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" style="width: 50px; height: 18.5px;" width="50" height="18.5"><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 4px 8px; transform-origin: 4px 8px; unicode-bidi: normal; ">. </div><div style="block-size: 63px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 31.5px; text-align: left; transform-origin: 384px 31.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; "><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 67px 8px; transform-origin: 67px 8px; unicode-bidi: normal; ">Given the value of r<span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 92px 8px; transform-origin: 92px 8px; unicode-bidi: normal; ">, find the perimeter of the <img src="data:image/png;base64,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" style="width: 32.5px; height: 18px;" width="32.5" height="18"><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 32px 8px; transform-origin: 32px 8px; unicode-bidi: normal; "> with the <span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 43.5px 8px; transform-origin: 43.5px 8px; unicode-bidi: normal; ">r-th smallest<span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 40px 8px; transform-origin: 40px 8px; unicode-bidi: normal; "> perimeter. <span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 51.5px 8px; transform-origin: 51.5px 8px; unicode-bidi: normal; ">For example for <img src="data:image/png;base64,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" style="width: 35.5px; height: 18px;" width="35.5" height="18"><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 18px 8px; transform-origin: 18px 8px; unicode-bidi: normal; ">, that is <img src="data:image/png;base64,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" style="width: 43.5px; height: 18px;" width="43.5" height="18"><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 82.5px 8px; transform-origin: 82.5px 8px; unicode-bidi: normal; ">, the smallest perimeter is <img src="data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAACQAAAAkCAYAAADhAJiYAAABLUlEQVRYhe2WUbHDIBBFjwccYKAGoiAK4qAOcBAL0RAJeKiFaoiFvg+WmX2dPEIC5fWDO7Nfu1lOLiwJdHV1fZcsYE4+M0jcaoMswCuzsQFm4Ak4iafEvQTEKJAYR0AWeMji726u0sPv5A41Et5slAVygWKt28kZYEvks+UygaaMOu34aZfOAsUt2RI1GvqyS7lAcTtSQIPqtX4aKNakgGxmXVWgo/PRDMiruvEbgO4cn4+mW2YIF2LKJT32j08DIXkN5eX5meDIqnJLCyAITk2yoBcIR9iumbxzVhUoBRrvqr1vXXMg3Wcq6FMF6MbxBDYD0tN36dejJpCeurUGjOX3GJ/5QjvCId4o/FOMIDPB4vdY/ljAEEY53jle6opduSorAMN/QnR1dXWV6Ae9sZqCc2SAGwAAAABJRU5ErkJggg==" style="width: 18px; height: 18px;" width="18" height="18"><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 12.5px 8px; transform-origin: 12.5px 8px; unicode-bidi: normal; "> for <img src="data:image/png;base64,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" style="width: 32.5px; height: 18px;" width="32.5" height="18"><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 2px 8px; transform-origin: 2px 8px; unicode-bidi: normal; "> <img src="data:image/png;base64,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" style="width: 48.5px; height: 18.5px;" width="48.5" height="18.5"><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 143px 8px; transform-origin: 143px 8px; unicode-bidi: normal; ">, while the second (r-th) smallest perimeter is <img src="data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAACQAAAAkCAYAAADhAJiYAAAA6klEQVRYhe2WUQ2DMBCGfw91UAMYmIIqwMEc4AAL1TAJ9TALaJgF9sBd0i2DXQ9u4+G+5MIDBT74QgBwHOc/dAAuwumE54wAglZoAjALZxSIZForlX8hNcjMWJ7SJ0IlwqMSKjRxY02gC0wr+xOAgbb3PUKRLvKtdQ9ZLpDYLqG1BDU3bOc6TEgC53pA9taYC3GuLFxvLsS50hmEWnOZC7XmMhdqzWUqpMllKqTJZSpU0J7LTChCl8tM6ApdLjMh/mL3ZxCqc239kvxMiHMVxbH8O8NCwxFCI8m0vF2xOu59MpabdBzHcSQ8AUZlel3qgGXQAAAAAElFTkSuQmCC" style="width: 18px; height: 18px;" width="18" height="18"><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 26.5px 8px; transform-origin: 26.5px 8px; unicode-bidi: normal; ">, for the <img src="data:image/png;base64,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" style="width: 32.5px; height: 18px;" width="32.5" height="18"><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 2px 8px; transform-origin: 2px 8px; unicode-bidi: normal; "> with dimensions <img src="data:image/png;base64,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" style="width: 71px; height: 18.5px;" width="71" height="18.5"><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 17px 8px; transform-origin: 17px 8px; unicode-bidi: normal; ">. For <img src="data:image/png;base64,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" style="width: 35.5px; height: 18px;" width="35.5" height="18"><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 98.5px 8px; transform-origin: 98.5px 8px; unicode-bidi: normal; ">, the third smallest perimeter is <img src="data:image/png;base64,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" style="width: 33px; height: 18px;" width="33" height="18"><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 12.5px 8px; transform-origin: 12.5px 8px; unicode-bidi: normal; "> for <img src="data:image/png;base64,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" style="width: 32.5px; height: 18px;" width="32.5" height="18"><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 2px 8px; transform-origin: 2px 8px; unicode-bidi: normal; "> <img src="data:image/png;base64,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" style="width: 101px; height: 18.5px;" width="101" height="18.5"><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 4px 8px; transform-origin: 4px 8px; unicode-bidi: normal; ">. </div><div style="block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; "><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 382.5px 8px; transform-origin: 382.5px 8px; unicode-bidi: normal; ">The output can be very large, so please present only the last 12 digits if the number of digits of the perimeter exceeds 12.</div></div></div>

An Almost Pythagorean Triple (abbreviated as "APT'), is a set of 3 integers in which square of the largest element, which we will call as its 'hypotenuse', is  less than the sum of square of the smaller elements (shorter sides). This means that if  is the hypotenuse and  and  are the shorter sides, , satisfies the following equation: 
        
        where:  
The smallest  is the triple , with  and perimeter (the sum of the 3 elements)  of . Some researchers consider  as the smallest , but here, we will only look at 's where the hypotenuse is "strictly" greater than the other shorter sides. Other examples of 's are , and . 
Unfortunately, unlike Pythagorean Triples, a 'closed formula' for generating all possible 's, has not yet been discovered, at the time of this writing. For this exercise, we will be dealing with 's with a known ratio between the hypotenuse and the shortest side: . 
Given the value of , find the perimeter of the  with the r-th smallest perimeter. For example for , that is , the smallest perimeter is  for  , while the second (r-th) smallest perimeter is , for the  with dimensions . For , the third smallest perimeter is  for  . 
The output can be very large, so please present only the last 12 digits if the number of digits of the perimeter exceeds 12.

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Last Solution submitted on Oct 07, 2022

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William on 29 Sep 2021

I think the inequalities at the top of the problem must be backwards. In the example, c = 2*a, so c > a.

Ramon Villamangca on 29 Sep 2021

Hi William, Your right. I'd already changed. This should not affect the solution, though.

Are Mjaavatten on 27 Oct 2021

Hi Ramon,
Many of your "Easy" sequence problems have been challenging and fun. But here I am confused. I get the correct solutions for Tests 1 through 3. For Test 4 (and higher) I have problems. My result for Test 4 is p = 209737287485879. When I try to calculate a, b and c from r and the given value of p_correct (using Python3 and math.isqrt to handle large integers) I fail to find integer values.

Have I missed some important point here?

Tim on 27 Oct 2021

The p_correct values contain only the last 12 digits.

Are Mjaavatten on 1 Nov 2021

It seems I stopped reading (or absorbing what I read) before the last line of the problem description. Thanks to Tim for the clarification.

Ramon Villamangca on 2 Nov 2021

Hi, Are,
Sorry, I haven't look at the comments lately. I thought no one's gonna answer this one. Thanks Tim, for the clarification...

GeeTwo on 25 Oct 2022

Like David Hill, my answer matches the test cases where n<=10, but not for larger ones. I've been driving myself batty the last couple of days trying to figure out where I went wrong. Then I realized that the test cases have it wrong. I can't prove I'm right, but I can prove them wrong:
Consider the defining condition: c² = a² + b² - 1.
Rearrange: c² - a² - b² = -1.
Modulo 2: c² - a² - b² ≡ 1 (mod2).
As squaring and negation preserve parity (value mod 2),
c + a + b ≡ 1 (mod2)
And all of the answers in the test cases are even for n>10.

GeeTwo on 26 Oct 2022

My "double" solution matches my int64 solution up through 1000 but not 10,000, which is what I expected as flintmax is ~9e15.. My double solution (generates even perimeters for 10k+) is size 59, my int64 is 63.
I didn't run calculateSize() before trying to figure out what was up, but what I learned through that definitely shaved some points. The largest single value test case(~12e5) runs on my laptop in just under 200ms, and should run in O(n) time and constant memory. It will start overflowing when n hits about 45e5. uint64 will take it to about 9e6.

GeeTwo on 12 Jan 2023 at 11:59

@Ramon Villamangca, there's definitely an issue with the larger test cases in this problem. From the definition and tracking parity, the perimeter of an APT MUST be odd.

Solution Comments

Solution 7282125

2 Comments

2 Comments

David Hill on 21 Dec 2021

I know I can't use symbolics, but I was just checking to see if my algorithm was correct. It matches the test suite for r <=10 (before needing to mod for the last 12 digits), but not for larger r's. It seems like it should be correct. I must be missing something. Not sure if you want to provide me with a hint.

Ramon Villamangca on 3 Jan 2022

HINT: An optimized solution can be obtained via Pell's equation.

Solution 6607144

1 Comment

1 Comment

Ramon Villamangca on 29 Sep 2021

Hi William, the output should be the perimeter, not the triangle sides…

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Problem 52834. Easy Sequences 32: Almost Pythagorean Triples

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