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141 lines
5.3 KiB
HTML
141 lines
5.3 KiB
HTML
<!DOCTYPE html>
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<!-- Mirrored from www.mathsisfun.com/numbers/prime-properties.html by HTTrack Website Copier/3.x [XR&CO'2014], Sat, 29 Oct 2022 00:45:20 GMT -->
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<!-- #BeginEditable "doctitle" -->
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<title>Prime Properties</title>
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<script>reSpell=[["neighbors","neighbours"]]</script>
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.prime {color:black; background-color:gold;}
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.comp {color:black; background-color:#cdf;}
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</style>
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<h1 class="center">Prime Properties</h1>
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<p class="center">A <a href="../prime-composite-number.html">Prime Number</a> is a whole number above 1<br>that cannot be made by multiplying other whole numbers.</p>
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<h2>2 is Prime</h2>
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<p>We cannot make 2 by multiplying other whole numbers, so it is prime.</p>
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<p>Click on <b>2</b> below, what happens?<span class="comp"></span> </p>
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<script>primechartMain();</script>
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<p>Every multiple of two gets eliminated, right? Because they can't be prime. So no even numbers any more:</p>
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<p class="so">(beyond 2) <b>primes are odd</b>.</p>
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<p>Note we are <b>not </b>saying "all odd numbers are prime", but that "a prime has to be an odd number"</p>
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<h2>Multiples of 6</h2>
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<p>Now go back up and hit the <b>3</b>.</p>
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<p>From here on a prime has to be odd and <b>not </b>a multiple of 3. </p>
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<p>The next two primes (click them if you want) are 5 and 7, they are either side of 6.</p>
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<p>In fact, from now on a prime must be <b>next to a multiple of 6</b>. </p>
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<p>(Being next to a multiple of 3 is not enough. Look at 9, it has <b>even </b>numbers on each side, but 12 is next to <b>odd </b>numbers, then 15 is next to <b>even </b>numbers, etc.)</p>
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<p class="so">(beyond 3) <b>primes are next to a multiple of 6<br></b></p>
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<ul>
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<li>Notice the "twin primes" 5 and 7 next to 6</li>
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<li>then the twin primes 11 and 13 next to 12</li>
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<li>and the twin primes<b> </b>17 and 19 next to 18</li>
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<li>but this lovely pattern stops because 25 has been eliminated (a multiple of 5)</li></ul>
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<p>This is often the case with primes, a nice pattern suddenly disappears!</p>
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<p>(Note: "twin primes" must differ by only 2. The next two are 29 and 31, can you find more?)</p>
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<h2>Multiples of 24</h2>
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<p>But we do get another pattern!</p>
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<p>Let's look at the numbers on either side of a prime <b>p</b>:</p>
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<p class="center">
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<img src="images/prime-24.svg" alt="p-1 and p and p+1"></p>
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<ul>
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<li>one side must be a multiple of <b>6</b></li>
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<li>the two sides are consecutive (one after the other) <b>even </b>numbers</li>
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<li>in any two consecutive even numbers one must be a multiple of <b>4</b></li></ul>
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<p>So when we multiply a prime's neighbors we get a multiple of 4x6 = <b>24</b></p>
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<p>Multiplying neighbors is simply <b>(p−1)(p+1)</b></p>
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<p>And "multiple of 24" is <b>24n</b> where <b>n</b> is some whole number:
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</p>
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<p class="center large">(p−1)(p+1) = 24n</p>
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<p>We can multiply out (p−1)(p+1) to get p<sup>2</sup> − 1:</p>
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<p class="center large">p<sup>2</sup> − 1 = 24n</p>
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<p>And we get:</p>
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<p class="so">(beyond 3) <b>a prime squared minus 1 is a multiple of 24</b></p>
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<div class="example">
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<h3>Example: 11</h3>
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<p>11<sup>2</sup> − 1 = 121 − 1 = <b>120</b> (which is a multiple of 24)</p>
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<p>Or by multiplying its neighbors: 10 × 12 = <b>120</b></p>
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</div>
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<p>Test it yourself: try 5, or 19, or ... any prime beyond 3.<br>
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</p><p><br></p>
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<p>There are many more interesting properties of primes, can you discover more?</p>
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<p> </p>
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<div class="related">
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<a href="../prime-composite-number.html">Prime and Composite Numbers</a>
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<a href="prime-color-chart.html">Prime Color Chart</a>
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<a href="prime-numbers-advanced.html">Prime Numbers - Advanced</a>
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