I am somewhat serious about pi base e. So far as I can see, pi base e = 10.101002020<font face="Courier New">002111120020101120... as per
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<td valign="top" align="left" width="100"><a title="Pi expressed in base 1/e: Pi = Sum a(i)*exp(-i), i=-1,0,1,..." href="http://www.research.att.com/~njas/sequences/A050948">A050948</a></td>
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<td valign="top" align="left">Pi expressed in base 1/e: Pi = Sum a(i)*exp(-i), i=-1,0,1,... </td></tr></tbody></table></font>
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<div>Now, in what base should we examine substrings of this for primality? In the unnatural base 10, we have the primes:</div>
<div>101, 101010020200021111, ...</div>
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<div>The 62-digit string takes about a minute on the PC I'm using right now while to factor:</div>
<div>10 101002 020002 111120 020101 120001 010202 000111 012020 010120 002001 = <br>409 x 773 197250 487261 611198 297287 x </div>
<div>31 941172 112561 420564 909184 938047</div>
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<div>ahhh, and the 3rd prime is:</div>
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<div>101010020200021111200201011200010102020001110120200101200020011011201012100100021001</div>
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<div>Or, as the Alpertron puts it:</div>
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<div>101010 020200 021111 200201 011200 010102 020001 110120 200101 200020 011011 201012 100100 021001 </div>
<div>is prime </div>
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<div>Anyone want to extend this, or verify or find I've erred?</div>
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<div>-- Jonathan Vos Post</div>
<div>[in 3rd consectutive day with no electrical power to home in Southern California during record heat wave and expolding transformers]<br><br> </div>
<div><span class="gmail_quote">On 7/23/06, <b class="gmail_sendername">Russell Walsmith</b> <<a href="mailto:ixitol@gmail.com">ixitol@gmail.com</a>> wrote:</span>
<blockquote class="gmail_quote" style="PADDING-LEFT: 1ex; MARGIN: 0px 0px 0px 0.8ex; BORDER-LEFT: #ccc 1px solid">
<div>The idea of using the binary expansion of pi suggests another way to get a sequence of primes:<br><br>Starting with the first digit of A004601 = 1,1,0,0,1,0,0,1,0,0,0,0,1,1,1,1,1,1,0,1,1,0,1,... we move rightward until we encounter another 1. Since 11 (= 3 in decimal) is prime, we move to the next 1 and repeat the process.
<br>11 = 3<br>1001001 = 73<br>1001000011111 = 4639<br>1000011 = 67<br>11 = 3<br>11 = 3<br>11 = 3<br><br>This gives the sequence 3, 73, 4639, 67, 3, 3, 3, 3, 3, 5, 3, 5, 5, 5, 17, 17, 1069...<br><br>Can anyone extend this?
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<div><span class="sg"><br>Russell</span></div>
<div><span class="e" id="q_10c9d97ba14dcf13_2"><br><br><br><br>
<div><span class="gmail_quote">On 7/23/06, <b class="gmail_sendername">N. J. A. Sloane</b> <<a title="mailto:njas@research.att.com" onclick="return top.js.OpenExtLink(window,event,this)" href="mailto:njas@research.att.com" target="_blank">
njas@research.att.com</a>> wrote:</span>
<blockquote class="gmail_quote" style="PADDING-LEFT: 1ex; MARGIN: 0pt 0pt 0pt 0.8ex; BORDER-LEFT: rgb(204,204,204) 1px solid">Richard Guy said:<br>> It would be more natural (??) to do this in base 2.<br>><br>> So here are two new(?) sequences for people to
<br>> check and extend:<br>><br>> The first<br>> 2, 8, 14, 18, ... <br>><br>> digits in the binary representation of pi form the<br>> primes<br>> 3, 401, 25667, 410687, ...
<br>><br>> in decimal notation.<br><br>Me:<br><br>The binary expansion of Pi is A004601: <br>1,1,0,0,1,0,0,1,0,0,0,0,1,1,1,1,1,1,0,1,1,0,1,0,1,0,1,0,...<br><br>Converting first n bits to an integer gives A068425:<br>
1,3,6,12,25,50,100,201,402,804,1608,3216,...<br><br>The primes here are A117721:<br>3,6588397,1686629713,26986075409,16703571626015105435307505830654230989, ... <br><br>and they occur for these values of n (A065987):<br>2,23,31,35,124,323,2787,5717,6506 (and that's all I have)
<br>The latter sequence was computed by Bob Wilson.<br><br>Eric, can you extend it?<br><br>Richard, I seem to disagree with your results, but perhaps <br>I misunderstood your message?<br><br>Neil<br></blockquote></div><br>
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