Prediction for the next prime

[paulunderwood]

This is a simple statistical prediction for the next 3*2^n-1 prime based on historical evidence. It assumes no primes have been missed. The statistics do not take into account the "prime number theorem" Please criticize.

"n"    Increase over the previous exponent
----   -------------------------------------
1	0.0000000
2	2.0000000
3	1.5000000
4	1.3333333
6	1.5000000
7	1.1666667
11	1.5714286
18	1.6363636
34	1.8888889
38	1.1176471
43	1.1315789
55	1.2790698
64	1.1636364
76	1.1875000
94	1.2368421
103	1.0957447
143	1.3883495
206	1.4405594
216	1.0485437
306	1.4166667
324	1.0588235
391	1.2067901
458	1.1713555
470	1.0262009
827	1.7595745
1274	1.5405079
3276	2.5714286
4204	1.2832723
5134	1.2212179
7559	1.4723413
12676	1.6769414
14898	1.1752919
18123	1.2164720
18819	1.0384042
25690	1.3651097
26459	1.0299338
41628	1.5733021
51387	1.2344336
71783	1.3969097
81330	1.1329981
85687	1.0535719
88171	1.0289892
97063	1.1008495
123630	1.2737088
155930	1.2612634
164987	1.0580838
234760	1.4229000
414840	1.7670813
584995	1.4101702
702038	1.2000752
727699	1.0365522
992700	1.3641629
1201046	1.2098781
1232255	1.0259848
2312734	1.8768307
3136255	1.3560812
------------------------
av	1.3400057
std.dev	0.2971180
std.error 0.04
-95%	1.26 3948985.29
+95%	1.42 4456213.62

[axn]

And what is the probability that no primes would be found between now and 4456213.62?

EDIT:- What happens if you take Geometric Mean (or use log of the ratios?)

[paulunderwood]

Originally Posted by axn1

And what is the probability that no primes would be found between now and 4456213.62?

EDIT:- What happens if you take Geometric Mean (or use log of the ratios?)

Statistics was not my strong subject.

The geometric mean is 1.31.

My original assumption is based on a standard error for a normal distribution, which it is not the distribution of this data. Here is a stem and leaf diagram betrig:

1.0   9,4,5,2,3,2,5,2,5,3,2
1.1   6,1,3,6,8,7,7,3,0,0
1.2   7,3,0,8,2,1,3,7,6,0
1.3   3,8,6,9,6,5
1.4   4,1,7,2,1
1.5   0,0,7,4,7
1.6   3,7
1.7   5,6
1.8   8,7
1.9
2.0   0
2.1   
2.2
2.3
2.4
2.5   7

More criticism welcome, along with possible answers to anx1's questions.

[gd_barnes]

Originally Posted by paulunderwood

This is a simple statistical prediction for the next 3*2^n-1 prime based on historical evidence. It assumes no primes have been missed. The statistics do not take into account the "prime number theorem" Please criticize.

The NPLB project double-checked k=3 from n=100K-260K and checked it against the k<300 Rieselprime.org page a few months ago. No missing or incorrect primes were found. You're probably aware that Prof. Caldwell and some coharts had already double-checked all k<300 up to n=100K several years ago so now all k=3 for n<260K has been double-checked.

In checking that list vs. your list, I see a typo. Your prime at n=81330 should be n=80330, which would also affect 2 of your ratios. Other then that, your list should be absolutely correct up to n=260K.

NPLB plans to double-check all k=3-1001 up to n=260K by late 2009. So far, we have completed all k=3-25 and 401-417.


Gary

[paulunderwood]

A well spotted typo, Gary. The typo only appears in the spreadsheet and makes little difference to the crude calculation.

[gd_barnes]

Originally Posted by paulunderwood

A well spotted typo, Gary. The typo only appears in the spreadsheet and makes little difference to the crude calculation.

Agreed since you're mostly dealing with the geometric mean. It likely would have a small 'normalizing' effect on the standard error from normal distribution but inconsequential.

[paulunderwood]

Here is the latest "crude method" for predicting the next prime based on the last prime. Any modifications such a possion distribution or criticisms are welcome

1	0.0000000	
2	2.0000000	
3	1.5000000	
4	1.3333333	
6	1.5000000	
7	1.1666667	
11	1.5714286	
18	1.6363636	
34	1.8888889	
38	1.1176471	
43	1.1315789	
55	1.2790698	
64	1.1636364	
76	1.1875000	
94	1.2368421	
103	1.0957447	
143	1.3883495	
206	1.4405594	
216	1.0485437	
306	1.4166667	
324	1.0588235	
391	1.2067901	
458	1.1713555	
470	1.0262009	
827	1.7595745	
1274	1.5405079	
3276	2.5714286	
4204	1.2832723	
5134	1.2212179	
7559	1.4723413	
12676	1.6769414	
14898	1.1752919	
18123	1.2164720	
18819	1.0384042	
25690	1.3651097	
26459	1.0299338	
41628	1.5733021	
51387	1.2344336	
71783	1.3969097	
80330	1.1190672	
85687	1.0666874	
88171	1.0289892	
97063	1.1008495	
123630	1.2737088	
155930	1.2612634	
164987	1.0580838	
234760	1.4229000	
414840	1.7670813	
584995	1.4101702	
702038	1.2000752	
727699	1.0365522	
992700	1.3641629	
1201046	1.2098781	
1232255	1.0259848	
2312734	1.8768307	
3136255	1.3560812	
4235414	1.3504686	
av	1.3401779	
std.dev	0.2943653	
std.error	0.04	
-95%	1.26	5342998.07
95%	1.42	6009418.75
		
	mean “n”	5676208.41

[davieddy]

Originally Posted by axn1

And what is the probability that no primes would be found between now and 4456213.62?

EDIT:- What happens if you take Geometric Mean (or use log of the ratios?)

This is analogous to Mersenne primes: a plot of log(n) against
the rank order of each prime fits a straight line well.
We conjecture from this that the expected number of primes between
n1 and n2 is c*ln(n2/n1).
If we take n2/n1 to be your average ratio, we choose c such that
the expected number of primes is one.
We can use this to construct a poll where the ranges represent
the 25% percentiles. The four ranges offered in the poll are:
n<a
a<=n<b
b<=n<c
c<=n
The "fair" choice of ranges has a
75% chance of no primes before a
50% chance of no primes before b
25% chance of no primes before c

The construction of this fair poll (or one with more options)
gives the clearest possible answer to "where is the next prime" IMO.

David