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549 lines
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<h1 class="center">Pythagorean Triples - Advanced</h1>
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<p class="center">(You may like to read <a href="../pythagoras.html">Pythagoras' Theorem</a><br>
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and <a href="../pythagorean_triples.html">Introduction to Pythagorean Triples</a> first)</p>
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<p class="center"> </p>
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<p>A "Pythagorean Triple" is a set of positive <a href="../whole-numbers.html">integers</a> <b>a</b>, <b>b</b> and <b>c</b> that fits the rule:</p>
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<p class="largest" align="center">a<sup>2</sup> + b<sup>2</sup> = c<sup>2</sup></p>
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<p style="float:left; margin: 0 10px 5px 0;"><img src="../geometry/images/pythagoras-abc.svg" alt="Pythagoras a b c triangle" height="242" width="221"></p>
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<h2>Triangles</h2>
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<p>And when we make a triangle with sides <b>a</b>, <b>b</b> and <b>c</b> it will be a <a href="../right_angle_triangle.html">right angled triangle</a> (see <a href="../pythagoras.html">Pythagoras' Theorem</a> for more details):</p>
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<p class="center"><img src="../geometry/images/pythagoras-squares.svg" alt="pythagoras squares: a^2+b^2=c^2" height="109" width="375"></p>
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<div style="clear:both"></div>
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<p>Note:</p>
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<ul>
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<li><b>c</b> is the <span class="larger"> <b>longest side</b> of the triangle</span>, called the "hypotenuse"</li>
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<li><b>a</b> and <b>b</b> are the other two sides</li>
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</ul>
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<h2>Pythagorean Triples</h2>
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<p>A famous example of a Pythagorean Triples:</p>
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<table style="border: 0; margin:auto;">
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<tbody>
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<tr>
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<td style="text-align:center;"><img src="../geometry/images/triangle-3-4-5.svg" alt="3,4,5 Triangle" height="190" width="226"></td>
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</tr>
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<tr>
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<td style="text-align:center;"><span class="large">The <a href="../geometry/triangle-3-4-5.html">3,4,5 Triangle</a></span></td>
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</tr>
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<tr>
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<td style="text-align:center;">3<sup>2</sup> + 4<sup>2</sup> = 5<sup>2</sup></td>
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</tr>
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<tr>
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<td style="text-align:center;">9 + 16 = 25</td>
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</tr>
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</tbody></table>
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<p>Two more examples:</p>
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<table align="center" cellpadding="4">
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<tbody>
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<tr style="text-align:center;">
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<td><img src="../geometry/images/triangle-5-12-13.svg" alt="5,12,13 Triangle" height="87" width="176"></td>
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<td style="width:30px;"> </td>
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<td><img src="../geometry/images/triangle-9-40-41.svg" alt="9,40,41 Triangle" height="75" width="254"></td>
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</tr>
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<tr style="text-align:center;">
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<td><span class="large">5, 12, 13 </span></td>
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<td> </td>
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<td><span class="large">9, 40, 41 </span></td>
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</tr>
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<tr style="text-align:center;">
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<td>5<sup>2</sup> + 12<sup>2</sup> = 13<sup>2</sup></td>
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<td> </td>
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<td>9<sup>2</sup> + 40<sup>2</sup> = 41<sup>2</sup></td>
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</tr>
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<tr style="text-align:center;">
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<td>25 + 144 = 169</td>
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<td> </td>
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<td>(try it yourself)</td>
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</tr>
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</tbody></table>
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<h2>Endless</h2>
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<p>The set of Pythagorean Triples is endless.</p>
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<p>We can prove this with the help of the first Pythagorean Triple <b>(3, 4, 5)</b>:</p>
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<div class="center80">
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<p>Let n be any <a href="../whole-numbers.html">integer</a> greater than 1, then 3n, 4n and 5n are also a set of Pythagorean Triple. This is true because:</p>
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<p class="center">(3n)<sup>2</sup> + (4n)<sup>2</sup> = (5n)<sup>2</sup></p>
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<p>Examples:</p>
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<div class="beach">
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<table style="border: 0; margin:auto;">
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<tbody>
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<tr style="text-align:center;">
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<th>n</th>
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<th width="20"> </th>
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<th>(3n, 4n, 5n)</th>
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</tr>
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<tr style="text-align:center;">
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<td>2</td>
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<td> </td>
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<td>(6,8,10)</td>
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</tr>
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<tr style="text-align:center;">
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<td>3</td>
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<td> </td>
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<td>(9,12,15)</td>
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</tr>
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<tr style="text-align:center;">
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<td>4</td>
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<td> </td>
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<td>(12,16,20)</td>
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</tr>
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<tr style="text-align:center;">
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<td>...</td>
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<td> </td>
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<td>... etc ...</td>
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</tr>
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</tbody></table>
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</div>
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<p>So we can make <b>infinitely many</b> triples just using the (3,4,5) triple.</p>
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</div>
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<h2>Primitive Triples
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</h2>
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<p>In the case above (3,4,5) is a <b>primitive triple</b>, </p>
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<p>But all its multiples, such as (6,8,10) etc, are <b>not</b>.</p>
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<p>Primitive triples have this property: <b>a, b and c share no common factors</b>.</p>
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<div class="example">
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<h3>Example: (3,4,5)</h3>
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<p>3, 4 and 5 share no common factors, so (3,4,5) <b>is </b>a primitive triple</p> </div>
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<div class="example">
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<h3>Example: (6,8,10) </h3>
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<p>6, 8 and 10 share a common factor of 2, so (6,8,10) is <b>not </b>a primitive triple</p></div>
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<h2>Euclid's Proof of Infinitely Many Pythagorean Triples</h2>
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<p>But Euclid used a different reasoning to prove the set of Pythagorean Triples is unending.</p>
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<p>The proof was based on the fact that the difference of the squares of any two <b>consecutive</b> (one after the other) whole numbers is always an odd number.</p>
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<div class="example">
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<h3>Examples:</h3>
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<ul>
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<li>2<sup>2</sup> − 1<sup>2</sup> = 4 − 1 = <b>3</b> (an odd number),</li>
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<li>3<sup>2</sup> − 2<sup>2</sup> = 9 − 4 = <b>5</b> (an odd number),</li>
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<li>4<sup>2</sup> − 3<sup>2</sup> = 16 − 9 = <b>7</b> (an odd number),</li>
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<li>etc
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</li></ul></div>
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<p>Can you see how subtracting squares make odd numbers in this picture?</p>
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<p class="center"><img src="images/odd-square-numbers.gif" alt="odd square numbers" height="130" width="156"><br></p>
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<p>See <a href="odd-square-number.html">Squares and Odd Numbers</a>, or have a look at this table as an example:</p>
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<div class="simple">
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<table style="border: 0; margin:auto;">
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<tbody>
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<tr>
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<th align="center" width="40">n</th>
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<th align="center" width="40">n<sup>2</sup></th>
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<th align="center" width="130">n<sup>2</sup> minus<br>previous n<sup>2</sup> </th>
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</tr>
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<tr>
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<td style="text-align:center;">1</td>
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<td style="text-align:center;">1</td>
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<td style="text-align:center;"> </td>
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</tr>
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<tr>
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<td style="text-align:center;">2</td>
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<td style="text-align:center;">4</td>
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<td style="text-align:center;">4−1 = <b>3</b></td>
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</tr>
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<tr>
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<td style="text-align:center;">3</td>
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<td style="text-align:center;">9</td>
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<td style="text-align:center;">9−4 = <b>5</b></td>
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</tr>
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<tr>
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<td style="text-align:center;">4</td>
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<td style="text-align:center;">16</td>
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<td style="text-align:center;">16−9 = <b>7</b></td>
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</tr>
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<tr>
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<td style="text-align:center;">5</td>
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<td style="text-align:center;">25</td>
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<td style="text-align:center;">25−16 = <b>9</b></td>
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</tr>
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<tr>
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<td style="text-align:center;">...</td>
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<td style="text-align:center;">...</td>
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<td style="text-align:center;">...</td>
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</tr>
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</tbody></table>
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</div>
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<p>And there are an infinite number of odd numbers. Since the perfect squares form a subset of the odd numbers, and a fraction of infinity is also infinity, it follows that there must also be an infinite number of odd squares. So there are an infinite number of Pythagorean Triples.</p>
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<h2>Properties</h2>
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<p>An interesting fact: a Pythagorean Triple always consists of:</p>
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<ul>
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<li>all even numbers, or</li>
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<li>two odd numbers and an even number.</li>
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</ul>
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<p>A Pythagorean Triple can never be made of all odd numbers or two even numbers and one odd number. This is true because:</p>
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<ul>
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<li>The square of an odd number is an odd number and the square of an even number is an even number.</li>
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<li>The sum of two even numbers is an even number and the sum of an odd number and an even number is in odd number.</li>
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</ul>
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<p>So, when both a and b are even, c is even too. Similarly when one of a and b is odd and the other is even, c has to be odd!</p>
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<h2>Constructing Pythagorean Triples</h2>
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<p>It is easy to construct sets of Pythagorean Triples.</p>
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<p>When <b>m</b> and <b>n</b> are any two positive integers (m > n):</p>
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<ul>
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<li>a = m<sup>2</sup> − n<sup>2</sup></li>
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<li>b = 2mn</li>
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<li>c = m<sup>2</sup> + n<sup>2</sup></li>
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</ul>
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<p>Then a, b and c form a Pythagorean Triple. This is known as "Euclid's formula".</p>
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<div class="example">
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<h3>Example: m=2 and n=1</h3>
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<ul>
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<li>a = 2<sup>2</sup> − 1<sup>2</sup> = 4 − 1 = <b>3</b></li>
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<li>b = 2 × 2 × 1 = <b>4</b></li>
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<li>c = 2<sup>2</sup> + 1<sup>2</sup> = 4 + 1 = <b>5</b></li>
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</ul>
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<p>And we get the first Pythagorean Triple <b>(3,4,5)</b>.</p>
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</div>
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<div class="example">
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<p>Similarly, when m=3 and n=2 we get the next Pythagorean Triple <b>(5,12,13)</b>.</p>
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</div>
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<p>This method creates all primitive triples, but we may need to swap a and b to see:</p>
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<div class="example">
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<p>Example: when m=4 and n=1 we get <b>(15,8,17)</b>, which is also (8,15,17)</p>
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</div>
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<p>It also creates some non-primitive triples (that are multiples of primitive triples):</p>
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<div class="example">
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<p>Example: when m=3 and n=1 we get <b>(8,6,10)</b>, which is also (6,8,10) by swapping.</p>
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<p>But (6,8,10) is just (3,4,5) times 2</p>
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</div>
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<h2>List of the First Few</h2>
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<p>Here is a list of all primitive Pythagorean Triples for a, b, and c less than 1000.</p>
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<p>The list has only primitive triples, so (3,4,5) is there, but (6,8,10) etc are not</p><br>
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<div class="simple">
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<table width="100%" border="0">
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<tbody>
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<tr style="text-align:center;">
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<td>(3,4,5)</td>
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<td>(5,12,13)</td>
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<td>(7,24,25)</td>
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<td>(8,15,17)</td>
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<td>(9,40,41)</td>
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</tr>
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<tr style="text-align:center;">
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<td>(11,60,61)</td>
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<td>(12,35,37)</td>
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<td>(13,84,85)</td>
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<td>(15,112,113)</td>
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<td>(16,63,65)</td>
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</tr>
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<tr style="text-align:center;">
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<td>(17,144,145)</td>
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<td>(19,180,181)</td>
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<td>(20,21,29)</td>
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<td>(20,99,101)</td>
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<td>(21,220,221)</td>
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</tr>
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<tr style="text-align:center;">
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<td>(23,264,265)</td>
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<td>(24,143,145)</td>
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<td>(25,312,313)</td>
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<td>(27,364,365)</td>
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<td>(28,45,53)</td>
|
||
</tr>
|
||
<tr style="text-align:center;">
|
||
<td>(28,195,197)</td>
|
||
<td>(29,420,421)</td>
|
||
<td>(31,480,481)</td>
|
||
<td>(32,255,257)</td>
|
||
<td>(33,56,65)</td>
|
||
</tr>
|
||
<tr style="text-align:center;">
|
||
<td>(33,544,545)</td>
|
||
<td>(35,612,613)</td>
|
||
<td>(36,77,85)</td>
|
||
<td>(36,323,325)</td>
|
||
<td>(37,684,685)</td>
|
||
</tr>
|
||
<tr style="text-align:center;">
|
||
<td>(39,80,89)</td>
|
||
<td>(39,760,761)</td>
|
||
<td>(40,399,401)</td>
|
||
<td>(41,840,841)</td>
|
||
<td>(43,924,925)</td>
|
||
</tr>
|
||
<tr style="text-align:center;">
|
||
<td>(44,117,125)</td>
|
||
<td>(44,483,485)</td>
|
||
<td>(48,55,73)</td>
|
||
<td>(48,575,577)</td>
|
||
<td>(51,140,149)</td>
|
||
</tr>
|
||
<tr style="text-align:center;">
|
||
<td>(52,165,173)</td>
|
||
<td>(52,675,677)</td>
|
||
<td>(56,783,785)</td>
|
||
<td>(57,176,185)</td>
|
||
<td>(60,91,109)</td>
|
||
</tr>
|
||
<tr style="text-align:center;">
|
||
<td>(60,221,229)</td>
|
||
<td>(60,899,901)</td>
|
||
<td>(65,72,97)</td>
|
||
<td>(68,285,293)</td>
|
||
<td>(69,260,269)</td>
|
||
</tr>
|
||
<tr style="text-align:center;">
|
||
<td>(75,308,317)</td>
|
||
<td>(76,357,365)</td>
|
||
<td>(84,187,205)</td>
|
||
<td>(84,437,445)</td>
|
||
<td>(85,132,157)</td>
|
||
</tr>
|
||
<tr style="text-align:center;">
|
||
<td>(87,416,425)</td>
|
||
<td>(88,105,137)</td>
|
||
<td>(92,525,533)</td>
|
||
<td>(93,476,485)</td>
|
||
<td>(95,168,193)</td>
|
||
</tr>
|
||
<tr style="text-align:center;">
|
||
<td>(96,247,265)</td>
|
||
<td>(100,621,629)</td>
|
||
<td>(104,153,185)</td>
|
||
<td>(105,208,233)</td>
|
||
<td>(105,608,617)</td>
|
||
</tr>
|
||
<tr style="text-align:center;">
|
||
<td>(108,725,733)</td>
|
||
<td>(111,680,689)</td>
|
||
<td>(115,252,277)</td>
|
||
<td>(116,837,845)</td>
|
||
<td>(119,120,169)</td>
|
||
</tr>
|
||
<tr style="text-align:center;">
|
||
<td>(120,209,241)</td>
|
||
<td>(120,391,409)</td>
|
||
<td>(123,836,845)</td>
|
||
<td>(124,957,965)</td>
|
||
<td>(129,920,929)</td>
|
||
</tr>
|
||
<tr style="text-align:center;">
|
||
<td>(132,475,493)</td>
|
||
<td>(133,156,205)</td>
|
||
<td>(135,352,377)</td>
|
||
<td>(136,273,305)</td>
|
||
<td>(140,171,221)</td>
|
||
</tr>
|
||
<tr style="text-align:center;">
|
||
<td>(145,408,433)</td>
|
||
<td>(152,345,377)</td>
|
||
<td>(155,468,493)</td>
|
||
<td>(156,667,685)</td>
|
||
<td>(160,231,281)</td>
|
||
</tr>
|
||
<tr style="text-align:center;">
|
||
<td>(161,240,289)</td>
|
||
<td>(165,532,557)</td>
|
||
<td>(168,425,457)</td>
|
||
<td>(168,775,793)</td>
|
||
<td>(175,288,337)</td>
|
||
</tr>
|
||
<tr style="text-align:center;">
|
||
<td>(180,299,349)</td>
|
||
<td>(184,513,545)</td>
|
||
<td>(185,672,697)</td>
|
||
<td>(189,340,389)</td>
|
||
<td>(195,748,773)</td>
|
||
</tr>
|
||
<tr style="text-align:center;">
|
||
<td>(200,609,641)</td>
|
||
<td>(203,396,445)</td>
|
||
<td>(204,253,325)</td>
|
||
<td>(205,828,853)</td>
|
||
<td>(207,224,305)</td>
|
||
</tr>
|
||
<tr style="text-align:center;">
|
||
<td>(215,912,937)</td>
|
||
<td>(216,713,745)</td>
|
||
<td>(217,456,505)</td>
|
||
<td>(220,459,509)</td>
|
||
<td>(225,272,353)</td>
|
||
</tr>
|
||
<tr style="text-align:center;">
|
||
<td>(228,325,397)</td>
|
||
<td>(231,520,569)</td>
|
||
<td>(232,825,857)</td>
|
||
<td>(240,551,601)</td>
|
||
<td>(248,945,977)</td>
|
||
</tr>
|
||
<tr style="text-align:center;">
|
||
<td>(252,275,373)</td>
|
||
<td>(259,660,709)</td>
|
||
<td>(260,651,701)</td>
|
||
<td>(261,380,461)</td>
|
||
<td>(273,736,785)</td>
|
||
</tr>
|
||
<tr style="text-align:center;">
|
||
<td>(276,493,565)</td>
|
||
<td>(279,440,521)</td>
|
||
<td>(280,351,449)</td>
|
||
<td>(280,759,809)</td>
|
||
<td>(287,816,865)</td>
|
||
</tr>
|
||
<tr style="text-align:center;">
|
||
<td>(297,304,425)</td>
|
||
<td>(300,589,661)</td>
|
||
<td>(301,900,949)</td>
|
||
<td>(308,435,533)</td>
|
||
<td>(315,572,653)</td>
|
||
</tr>
|
||
<tr style="text-align:center;">
|
||
<td>(319,360,481)</td>
|
||
<td>(333,644,725)</td>
|
||
<td>(336,377,505)</td>
|
||
<td>(336,527,625)</td>
|
||
<td>(341,420,541)</td>
|
||
</tr>
|
||
<tr style="text-align:center;">
|
||
<td>(348,805,877)</td>
|
||
<td>(364,627,725)</td>
|
||
<td>(368,465,593)</td>
|
||
<td>(369,800,881)</td>
|
||
<td>(372,925,997)</td>
|
||
</tr>
|
||
<tr style="text-align:center;">
|
||
<td>(385,552,673)</td>
|
||
<td>(387,884,965)</td>
|
||
<td>(396,403,565)</td>
|
||
<td>(400,561,689)</td>
|
||
<td>(407,624,745)</td>
|
||
</tr>
|
||
<tr style="text-align:center;">
|
||
<td>(420,851,949)</td>
|
||
<td>(429,460,629)</td>
|
||
<td>(429,700,821)</td>
|
||
<td>(432,665,793)</td>
|
||
<td>(451,780,901)</td>
|
||
</tr>
|
||
<tr style="text-align:center;">
|
||
<td>(455,528,697)</td>
|
||
<td>(464,777,905)</td>
|
||
<td>(468,595,757)</td>
|
||
<td>(473,864,985)</td>
|
||
<td>(481,600,769)</td>
|
||
</tr>
|
||
<tr style="text-align:center;">
|
||
<td>(504,703,865)</td>
|
||
<td>(533,756,925)</td>
|
||
<td>(540,629,829)</td>
|
||
<td>(555,572,797)</td>
|
||
<td>(580,741,941)</td>
|
||
</tr>
|
||
<tr style="text-align:center;">
|
||
<td>(615,728,953)</td>
|
||
<td>(616,663,905)</td>
|
||
<td>(696,697,985)</td>
|
||
<td><br>
|
||
</td>
|
||
<td><br>
|
||
</td>
|
||
</tr>
|
||
</tbody></table>
|
||
</div><br>
|
||
<table width="100%" border="0">
|
||
<tbody>
|
||
<tr>
|
||
<td style="text-align:right;"><img src="../images/ganesh.jpg" alt="by ganesh" height="60" width="250"></td>
|
||
</tr>
|
||
</tbody></table>
|
||
<p> </p>
|
||
<div class="questions">2245, 2246, 2247, 9039, 9040, 9041, 9042, 9043, 9044, 9045</div>
|
||
|
||
<div class="related">
|
||
<a href="../pythagorean_triples.html">Pythagorean Triples - Basics</a>
|
||
<a href="index.html">Numbers Index</a>
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||
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|
||
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