20190403, 05:15  #1 
"Green Skull"
Apr 2019
31_{8} Posts 
"Hidden Number" malarky
I assume there are 5..6 hidden numbers in the interval between exponents 43112609 and 57885161.
7..10 hidden numbers in 57885161..74207281, just 1 hidden number in 74207281..77232917, 2..4 hidden number in 77232917..82589933. So, totally there are 15..21 hidden numbers in the interval between 43112609 and 82589933 (I do not consider these three already uncovered exponents 57885161, 74207281, 77232917). I will explain the calculation method later. Unfortunately, it will take years to test the hypothesis :) 
20190403, 05:35  #2  
Undefined
"The unspeakable one"
Jun 2006
My evil lair
5·1,223 Posts 
Quote:


20190403, 09:53  #3 
"Green Skull"
Apr 2019
11001_{2} Posts 
If we look at List of Known Mersenne Prime Numbers (https://www.mersenne.org/primes/) then we'll see that not all numbers between M43,112,609 and M82,589,933 have been tested and eliminated. There are probably several "hidden" numbers that will eventually be identified as Mersenne Prime Numbers.
My Hidden numbers hypothesis says that totally 15..21 such "hidden" numbers are not identified at present in the interval between M43,112,609 and M82,589,933. In other words, the sequence number M82 589 933 could actually be 66..72 when all numbers up to M82 589 933 will be tested. 
20190403, 10:21  #4 
Einyen
Dec 2003
Denmark
2·29·53 Posts 
All the numbers have been tested once. The chance of finding 1521 missed Mersenne Primes during double checking is infinitesimal.
Also considering GIMPS have never found a single prime with double checking, and so many new primes would completely change the expected distribution of Mersenne Primes. We already found more than the "expected" number of primes, for example 12 primes between 20M and 85M while the expected number was 3.72. Last fiddled with by ATH on 20190403 at 10:23 
20190403, 11:13  #5  
Undefined
"The unspeakable one"
Jun 2006
My evil lair
5×1,223 Posts 
Quote:


20190403, 14:04  #6  
"TF79LL86GIMPS96gpu17"
Mar 2017
US midwest
29×173 Posts 
Quote:
The best mathematicians arrive at conjectures consistent with ~6 asyet undiscovered Mersenne Primes in the interval 46M1000M and inconsistent with your assumption. Assuming a 2% probability of wrong firsttest, supported by years of empirical data, and summing the number of singlytested exponents 46M83M, at https://www.mersenne.org/primenet/, I estimate 11011 wrong firsttests, out of 550546 (so 539535 estimated correct). In those tests, 4 were found and subsequently verified prime, for a rate of 7.41e6. 7.41e6 * 11011 = 0.0816. The most likely number to be found prime during double check of this span is zero. Per the binomial calculator at https://stattrek.com/onlinecalculator/binomial.aspx the probability of just one found in the estimated 11011 wrong firsttests when retested is 0.075; the probability of more than one is 0.0032; the probability of zero is 0.922. See also https://primes.utm.edu/notes/faq/NextMersenne.html Last fiddled with by kriesel on 20190403 at 14:53 

20190403, 15:40  #7 
"Green Skull"
Apr 2019
5^{2} Posts 
Here I will set out the premises of the hypothesis.
If we draw a curve Cumulative amount of Mersenne Primes depending on Exponents for all M2..M43112609, it shows us a smooth growth after 6972593. The extrapolation of the curve to the exponents 57885161, 74207281, 77232917, 82589933 gives us estimations of Cumulative amount of Mersenne Primes for these exponents about 52, 58, 59, 62. https://ibb.co/YPtwT2v Blue dotted lines  my rough approximations that I calculated yesterday and described in comments above. They are not quite accurate, sorry for this try. Pink dotted lines  more accurate approximations, that I made today. https://ibb.co/233Vbb4 So, we can assume now, there could be 3..5 hidden numbers in the interval between exponents 43112609 and 57885161, 5..6 hidden numbers in 57885161..74207281, probably 0 hidden numbers in 74207281..77232917, 2..4 hidden number in 77232917..82589933. But the hidden numbers can be distributed differently, I am no sure. Totally there are probably 10..15 hidden numbers in the interval between 43112609 and 82589933 (not 15..21 as I assumed before in comments above). Looks good, no? :) 
20190403, 16:06  #8 
Einyen
Dec 2003
Denmark
2·29·53 Posts 
Good luck with your hypothesis, I would not bet any money on it if I were you. We will see in a few years who was correct.
I bet on 0 or 1 new prime in the interval 43112609 to 82589933, but most likely 0. Last fiddled with by ATH on 20190403 at 16:11 
20190403, 16:42  #9 
"TF79LL86GIMPS96gpu17"
Mar 2017
US midwest
29·173 Posts 
No. At least not yet. How are the approximating curves obtained? What distinguishes one from another? There appear to be about 7 total approximating curves. What evidence is there that any one curve is "better" than another? (One way of testing any strategy of making such predictions is to subtract out a randomly selected data point or two, use the remaining to make the prediction for the subtracted data points, and see how it fares in predicting the already known data. This is a quick way to disprove prediction methods.)
There's no reason to think that sequence number versus p of the sorted Mp series should be a smooth function, and there are adjacent Mps that show it is not. Consider for example the M~10k, M~3M, and M~43M neighbors There are very wide variations in exponent ratio from one to the next, from 1.01 (43112609 / 42643801) to about 4.1 (521/127), in the exhaustively verified range. Last fiddled with by kriesel on 20190403 at 17:03 
20190403, 17:07  #10  
"Green Skull"
Apr 2019
5^{2} Posts 
Quote:
And two other curves I've got by quite complex regression with 3 and 4 freedom degrees, that gave me the smallest error comparing with a couple of dozens curves in the library. All calculated curves look similar although they have a completely different nature. Two of them are completely similar and indistinguishable in the graph. 

20190403, 17:15  #11 
Undefined
"The unspeakable one"
Jun 2006
My evil lair
5×1,223 Posts 
So it would seem your "hidden" numbers are really just numbers we already know about and have already tested as composite, but you are claiming are prime. Is that what you mean?

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