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Dean,<br>
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All that I found < 10^7 are 27, 2187 and 159323.<br>
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Bob.<br>
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Dean Hickerson wrote:<br>
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<pre wrap="">Zakir F. Seidov (<a class="moz-txt-link-abbreviated" href="mailto:seidovzf@yahoo.com">seidovzf@yahoo.com</a>) wrote:<br><br></pre>
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<pre wrap="">with my misery "database" of 1111 perfect primes < 1,000,000<br>i've found only two pp: {27, 2187} such that (pp-1)/2 is prime.<br><br>can anybody provide me next 1000 pp's and/or find several next pp's in<br>subject. thanks, zak<br></pre>
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<pre wrap=""><!----><br>I asked him what he meant by "perfect primes" and he explained that it was<br>a typo for "perfect powers", i.e. numbers a^b with integers a>=1 and b>=2.<br><br>So suppose that (a^b-1)/2 is prime. Since a-1 divides a^b-1, we must<br>have a=3. Also, if b is composite, say b=c*d with c>1 and d>1, then<br>(3^c-1)/2 divides (3^b-1)/2. Hence b must be prime.<br><br>The values of b for which (3^b-1)/2 is prime are given in A028491;<br>the first several are:<br><br> 3, 7, 13, 71, 103, 541, 1091, 1367, 1627, 4177, 9011, 9551<br><br>The corresponding primes (3^b-1)/2 are:<br><br> 13, 1093, 797161, 3754733257489862401973357979128773, ...<br><br>These weren't in the OEIS, so I've submitted them.<br><br>Dean Hickerson<br><a class="moz-txt-link-abbreviated" href="mailto:dean@math.ucdavis.edu">dean@math.ucdavis.edu</a><br><br></pre>
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