predicting approximate location of Mersenne primes

[ssybesma]

I saw a thread called "Simple graph to predict Primes" but it went into a discussion of sieves which is not what I was thinking of.

Has there ever been an attempt to plot the locations of known Mersenne primes so as to detect a pattern that would help predict an approximate location where the next one might be?

Thank you.

[a1call]

This thread might be relevant:

https://www.mersenneforum.org/showth...t=23186&page=5

[GP2]

The only thing we think we know is that the average ratio between the exponents of successive Mersenne primes is approximately 1.47576

See for instance https://primes.utm.edu/mersenne/heuristic.html

However, this doesn't tell us exactly where to expect the next Mersenne prime. No one can predict that.

For instance, you can measure the average traffic flow on a street, but that won't tell you exactly when the next car will drive by.

[Batalov]

Originally Posted by ssybesma

I saw a thread called "Simple graph to predict Primes" but it went into a discussion of sieves which is not what I was thinking of.

Has there ever been an attempt to plot the locations of known Mersenne primes so as to detect a pattern that would help predict an approximate location where the next one might be?

Thank you.

Originally Posted by https://en.wikipedia.org/wiki/Mersenne_prime

On April 12, 2009, a GIMPS server log reported that a 47th Mersenne prime had possibly been found. The find was verified on June 12, 2009. The prime is 2^42,643,801 − 1. Although it is chronologically the 47th Mersenne prime to be discovered, it is smaller than the largest known at the time, which was the 45th to be discovered.

Think about this.
In April 2009, these were known:

#    p
1    2
2    3
3    5
4    7
5    13
6    17
7    19
8    31
9    61
10    89
11    107
12    127
13    521
14    607
15    1,279
16    2,203
17    2,281
18    3,217
19    4,253
20    4,423
21    9,689
22    9,941
23    11,213
24    19,937
25    21,701
26    23,209
27    44,497
28    86,243
29    110,503
30    132,049
31    216,091
32    756,839
33    859,433
34    1,257,787
35    1,398,269
36    2,976,221
37    3,021,377
38    6,972,593
39    13,466,917
40    20,996,011
41    24,036,583
42    25,964,951
43    30,402,457
44    32,582,657
45    37,156,667
46    43,112,609

Now, plot them as much as you want and predict where the next one is, ok?

[ssybesma]

Originally Posted by GP2

The only thing we think we know is that the average ratio between the exponents of successive Mersenne primes is approximately 1.47576

See for instance https://primes.utm.edu/mersenne/heuristic.html

However, this doesn't tell us exactly where to expect the next Mersenne prime. No one can predict that.

For instance, you can measure the average traffic flow on a street, but that won't tell you exactly when the next car will drive by.

That logarithmic graph is exactly what I had in mind, thank you. The series does look very linear when plotted that way.

[ssybesma]

Originally Posted by GP2

The only thing we think we know is that the average ratio between the exponents of successive Mersenne primes is approximately 1.47576

See for instance https://primes.utm.edu/mersenne/heuristic.html

However, this doesn't tell us exactly where to expect the next Mersenne prime. No one can predict that.

For instance, you can measure the average traffic flow on a street, but that won't tell you exactly when the next car will drive by.

Was going to add to that...yes, you're right about 'exactly' but I was more interested in the odds of location being more likely at a certain point and then checking that neighborhood, which could help cut out checking some of the less likely places if you wanted to discover one more quickly.

Of course ultimately all of the prime exponents still have to be checked in case one is missed. I'm just saying that in the future as more and more of these are discovered, it may be possible to zero in on them a little more accurately.

[ssybesma]

Originally Posted by a1call

This thread might be relevant:

https://www.mersenneforum.org/showth...t=23186&page=5

Very interesting, thank you.

[VBCurtis]

Originally Posted by ssybesma

Was going to add to that...yes, you're right about 'exactly' but I was more interested in the odds of location being more likely at a certain point and then checking that neighborhood, which could help cut out checking some of the less likely places if you wanted to discover one more quickly.

Of course ultimately all of the prime exponents still have to be checked in case one is missed. I'm just saying that in the future as more and more of these are discovered, it may be possible to zero in on them a little more accurately.

No. This belief, while common, flies in the face of the concept of probability distributions and independent events.

It's the same fallacy as rolling a 6-sided die many times, observing that 6 comes up 1/6th of the time, and saying "well, we should expect every 6th roll to be a 6; I've had 5 rolls in a row that weren't 6, so my next one is more likely to be 6."

Each roll of the die is an independent event, just as each primality test is an independent event. Prior results can give you a sense for average frequency of outcomes, but you cannot use averages to refine specific probabilities for specific tests any more than you can profitably claim "I am due for a 6!" when rolling a die.

[Batalov]

Originally Posted by GP2

...No one can predict that.

For instance, you can measure the average traffic flow on a street, but that won't tell you exactly when the next car will drive by.

And, to continue the analogy, that won't tell you that the "next" car that you are waiting for has already passed but the videotapes of the surveillance cameras were not reviewed just yet - and it wasn't your shift so you didn't see it with your own eyes.

Finally, there is always a remote chance that the car has passed but was invisible to you - your sensory organs and your video cameras could not detect it. (that is: that the limitations of the computational procedure as implemented currently produce a false negative result. The chances of that are very low, but not zero.)

[a1call]

I don't think the car analogy applies here because prime numbers are not as random as passing cars. Prime numbers occur simply at miss-junctions points (vs hit points) of harmonics of all the primes less than the square-root of the point. Take the harmonics of any finite small number of the primes and the misses become absolutely predictable. What makes the prediction of primes infinitely complex is that the harmonics become infinitely numerous as you progress higher.
There are expected patterns of concentration that can be observed. Take M1279 which is a prime number next to M1277 which is the smallest Composite Mersenne number without any know factors. A confidence?
I would say not.

[science_man_88]

Originally Posted by a1call

I don't think the car analogy applies here because prime numbers are not as random as passing cars. Prime numbers occur simply at miss-junctions points (vs hit points) of harmonics of all the primes less than the square-root of the point. Take the harmonics of any finite small number of the primes and the misses become absolutely predictable. What makes the prediction of primes infinitely complex is that the harmonics become infinitely numerous as you progress higher.
There are expected patterns of concentration that can be observed. Take M1279 which is a prime number next to M1277 which is the smallest Composite Mersenne number without any know factors. A confidence?
I would say not.

I would. The word is coincidence by the way. The difference between these numbers is 3 * M1277+3 the exponents are close, but not the numbers.