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</o:shapelayout></xml><![endif]--></head><body lang=EN-US link=blue vlink=purple><div class=WordSection1><p class=MsoPlainText><o:p> </o:p></p><p class=MsoPlainText><o:p> </o:p></p><p class=MsoPlainText>On Behalf Of Anders Sandberg<br>Subject: Re: [ExI] 2^57885161-1</p><p class=MsoPlainText><o:p> </o:p></p><p class=MsoPlainText>So, Spike, how does this fit with current GIMPS trends?<o:p></o:p></p><p class=MsoPlainText><o:p> </o:p></p><p class=MsoPlainText>-- <o:p></o:p></p><p class=MsoPlainText>Anders Sandberg,<o:p></o:p></p><p class=MsoPlainText><o:p> </o:p></p><p class=MsoPlainText><span style='color:black'><o:p> </o:p></span></p><p class=MsoPlainText><span style='color:black'><o:p> </o:p></span></p><p class=MsoPlainText><span style='color:black'><o:p> </o:p></span></p><p class=MsoPlainText><span style='color:black'>Thank you for asking, Anders. What a delightful question!<o:p></o:p></span></p><p class=MsoPlainText><span style='color:black'><o:p> </o:p></span></p><p class=MsoPlainText><span style='color:black'>I looked at doubling times of the largest known prime number, focusing on only since 1950, because the largest known prime has almost always been a Mersenne prime since those days, and will be for the remainder of human history, details available on request.<o:p></o:p></span></p><p class=MsoPlainText><span style='color:black'><o:p> </o:p></span></p><p class=MsoPlainText><span style='color:black'>In this first graph, I am showing the doubling time on the Y axis, with the date on the horizontal. The signal is chaotic at first, because there was no real systematic search for enormous prime numbers. It was almost as if no one cared. Shameful. Unacceptable behavior from our grandparent’s generation.<o:p></o:p></span></p><p class=MsoPlainText><span style='color:black'><o:p> </o:p></span></p><p class=MsoPlainText><span style='color:black'>I have applied a homemade smoothing and filtering technique I actually designed for another purpose, but the point is that as you go down into the higher series numbers, the curve gets smoother. Check it out:<o:p></o:p></span></p><p class=MsoPlainText><span style='color:black'><o:p> </o:p></span></p><p class=MsoPlainText><span style='color:black'><o:p> </o:p></span></p><p class=MsoPlainText><img width=743 height=467 id="Chart_x0020_1" src="cid:image003.png@01CE03E5.F5581E30"><span style='color:black'><o:p></o:p></span></p><p class=MsoPlainText><span style='color:black'><o:p> </o:p></span></p><p class=MsoPlainText><span style='color:black'>Very well, OK let’s ignore those early years before 1978, when the earth was inhabited by cave men and cave women, dwelling in yurts. Then we can go to a more interesting scale. I am intentionally avoiding going with semilog. Now we get this:<o:p></o:p></span></p><p class=MsoPlainText><span style='color:black'><o:p> </o:p></span></p><p class=MsoPlainText><span style='color:black'><o:p> </o:p></span></p><p class=MsoPlainText><img width=753 height=467 id="Chart_x0020_2" src="cid:image005.png@01CE03E5.F5581E30"><span style='color:black'><o:p></o:p></span></p><p class=MsoPlainText><span style='color:black'><o:p> </o:p></span></p><p class=MsoPlainText><span style='color:black'>So now we see what happened when computers started coming online in the 70s and we started using them for fun stuff like the Lucas Lehmer algorithm. We had a cluster of Mersenne discoveries. I first heard of the Lucas Lehmer in high school, which was in 1978.<o:p></o:p></span></p><p class=MsoPlainText><span style='color:black'><o:p> </o:p></span></p><p class=MsoPlainText><span style='color:black'>Let’s zoom in once more:<o:p></o:p></span></p><p class=MsoPlainText><span style='color:black'><o:p> </o:p></span></p><p class=MsoPlainText><img width=807 height=467 id="Chart_x0020_3" src="cid:image009.png@01CE03E5.F5581E30"><span style='color:black'><o:p></o:p></span></p><p class=MsoPlainText><span style='color:black'><o:p> </o:p></span></p><p class=MsoPlainText><span style='color:black'>Do ignore the oddball horizontal scale for now. I wanted to get that 1987 data point in there. What we are showing here is that with successively more data used in the smoothing technique, it is ever more clear that something dramatic was causing the time between doubling size of the largest known prime to be reduced dramatically. So now if we look at another order of magnitude closer, we see the start of the graph right around when GIMPS was started with the first Prime95 FFT algorithm, which is around when I got involved in it:<o:p></o:p></span></p><p class=MsoPlainText><span style='color:black'><o:p> </o:p></span></p><p class=MsoPlainText><span style='color:black'><o:p> </o:p></span></p><p class=MsoPlainText><span style='color:black'><o:p> </o:p></span></p><p class=MsoPlainText><img width=871 height=467 id="Chart_x0020_4" src="cid:image011.png@01CE03E5.F5581E30"><span style='color:black'><o:p></o:p></span></p><p class=MsoPlainText><span style='color:black'><o:p> </o:p></span></p><p class=MsoPlainText><span style='color:black'>Clearly, the introduction of GIMPS with the optimized Prime95 has dramatically lowered the time it takes to find cool new stuff!<o:p></o:p></span></p><p class=MsoPlainText><span style='color:black'><o:p> </o:p></span></p><p class=MsoPlainText><span style='color:black'>Another order of magnitude please:<o:p></o:p></span></p><p class=MsoPlainText><span style='color:black'><o:p> </o:p></span></p><p class=MsoPlainText><img width=865 height=467 id="Chart_x0020_6" src="cid:image015.png@01CE03E5.F5581E30"><span style='color:black'><o:p></o:p></span></p><p class=MsoPlainText><span style='color:black'><o:p> </o:p></span></p><p class=MsoPlainText><span style='color:black'><o:p> </o:p></span></p><p class=MsoPlainText><span style='color:black'>This is why I commented earlier that this isn’t exactly a record, but depending on how you filter the data, it is a record. We had an anomalous interval there in the unfiltered noisy data, but look at that gorgeous series 12! Oh beautiful is that data! If we take into account the trend in the last dozen Mersenne discoveries, we see we are getting steadily faster. You see there that weird cluster in the 2004-2006 timeframe that had blown my mind. I had expected a stretch like the last four years for a long time.<o:p></o:p></span></p><p class=MsoPlainText><span style='color:black'><o:p> </o:p></span></p><p class=MsoPlainText><span style='color:black'>Friends, my apologies. I just looked over what I have written and realize that it must be incomprehensible to those who do not follow this kind of junk. I get used to the way we write about this subject on my Mersenne Prime group; we take so much for granted. But I realize that here I am among a number of practically normal people in the area of obscure number theory. Perhaps many here are prime virgins.<o:p></o:p></span></p><p class=MsoPlainText><span style='color:black'><o:p> </o:p></span></p><p class=MsoPlainText><span style='color:black'>spike<o:p></o:p></span></p></div></body></html>