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142 lines
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<h2>An amazing Prime-generating Automaton</h2><div class="info" style="cursor:help;width:200px;margin-bottom:10px;"><h3>Problem 308</h3><span style="width:300px;color:#666;">Published on Saturday, 30th October 2010, 01:00 pm; Solved by 474</span></div>
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<div class="problem_content" role="problem">
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<p>A program written in the programming language Fractran consists of a list of fractions.</p>
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<p>The internal state of the Fractran Virtual Machine is a positive integer, which is initially set to a seed value. Each iteration of a Fractran program multiplies the state integer by the first fraction in the list which will leave it an integer.</p>
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<p>For example, one of the Fractran programs that John Horton Conway wrote for prime-generation consists of the following 14 fractions:<br />
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<table class="formula">
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<tr>
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<td><table class="frac">
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<tr><td>17</td></tr><tr><td class="overline">91</td></tr>
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</table></td>
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<td>,</td>
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<td><table class="frac">
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<tr><td>78</td></tr><tr><td class="overline">85</td></tr>
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</table></td>
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<td>,</td>
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<td><table class="frac">
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<tr><td>19</td></tr><tr><td class="overline">51</td></tr>
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</table></td>
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<td>,</td>
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<td><table class="frac">
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<tr><td>23</td></tr><tr><td class="overline">38</td></tr>
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</table></td>
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<td>,</td>
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<td><table class="frac">
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<tr><td>29</td></tr><tr><td class="overline">33</td></tr>
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</table></td>
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<td>,</td>
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<td><table class="frac">
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<tr><td>77</td></tr><tr><td class="overline">29</td></tr>
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</table></td>
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<td>,</td>
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<td><table class="frac">
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<tr><td>95</td></tr><tr><td class="overline">23</td></tr>
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</table></td>
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<td>,</td>
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<td><table class="frac">
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<tr><td>77</td></tr><tr><td class="overline">19</td></tr>
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</table></td>
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<td>,</td>
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<td><table class="frac">
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<tr><td>1</td></tr><tr><td class="overline">17</td></tr>
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</table></td>
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<td>,</td>
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<td><table class="frac">
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<tr><td>11</td></tr><tr><td class="overline">13</td></tr>
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</table></td>
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<td>,</td>
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<td><table class="frac">
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<tr><td>13</td></tr><tr><td class="overline">11</td></tr>
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</table></td>
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<td>,</td>
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<td><table class="frac">
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<tr><td>15</td></tr><tr><td class="overline">2</td></tr>
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</table></td>
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<td>,</td>
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<td><table class="frac">
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<tr><td>1</td></tr><tr><td class="overline">7</td></tr>
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</table></td>
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<td>,</td>
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<td><table class="frac">
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<tr><td>55</td></tr><tr><td class="overline">1</td></tr>
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</table></td>
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<td>.</td>
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</tr>
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</table>
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</p>
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<p>Starting with the seed integer 2, successive iterations of the program produce the sequence:<br />
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15, 825, 725, 1925, 2275, 425, ..., 68, <b>4</b>, 30, ..., 136, <b>8</b>, 60, ..., 544, <b>32</b>, 240, ...</p>
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<p>The powers of 2 that appear in this sequence are 2<sup>2</sup>, 2<sup>3</sup>, 2<sup>5</sup>, ...<br />
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It can be shown that <i>all</i> the powers of 2 in this sequence have prime exponents and that <i>all</i> the primes appear as exponents of powers of 2, in proper order!</p>
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<p>If someone uses the above Fractran program to solve Project Euler Problem 7 (find the 10001<sup>st</sup> prime), how many iterations would be needed until the program produces 2<sup>10001st prime</sup> ?
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