Digit sum of a Mersenne-prime exponent

[Dobri]

The base-10 digit sums of the Mersenne exponents for the 51 known Mersenne primes are as follows:


2, 3, 5, 7, 4, 8, 10, 4, 7, 17, 8, 10, 8, 13, 19, 7, 13, 13, 14, 13, 32, 23, 8, 29, 11, 16, 28, 23, 10, 19, 19, 38, 32, 37, 38, 29, 23, 41, 37, 28, 31, 41, 25, 38, 41, 28, 26, 41, 31, 38, 47.


An empirical observation shows that the digit sums are either prime numbers or have only 1 or 2 prime factors where the second factor is always 2, for instance:


25 = 5^2,

32 = 2^5,

28 = 2^2 x 7, etc.

[charybdis]

This is a meaningless observation, for the same reason that it's meaningless to note that the digit sums are all below 50.

Digit sums of Mersenne prime exponents (apart from 3 itself) can't be divisible by 3, since then the exponent would also be divisible by 3 and therefore not prime.

Taking that into account, the smallest possible digit sums which don't satisfy your condition are 35, 55, 65, 70, 77. Given the size of the Mersenne exponents searched so far, all of these apart from 35 would be unlikely to have appeared. It shouldn't be a massive surprise that 35 hasn't come up yet. After all, neither has 34. In the long run, most numbers don't satisfy your condition, so eventually we should expect most exponents of Mersenne primes not to satisfy it either.

[Dobri]

Applied statistics is not just a science but also an art. When one attempts to estimate what would be the next Mersenne prime on the basis of the current small sample size of known Mersenne primes, it depends what are the risks a Mersennery player could take.
Players having a large number of fast computers would not hesitate to include exponents with digit sums equal to 35, 55, etc. Players with a small number of modest computers could look at empirical observations to eliminate less probable choices for manual testing such as 35.
Also, the digit sum histogram of all possible digit sums for the entire testing range has a maximum at around 41. While digit sums 38 and 41 appear frequently in the list of discovered Mersenne primes, the digit sum 40 never occurred and it is a less probable choice for manual testing.
Therefore, Mersenneries with modest resources could seek a trade-off between risks to be taken for manual testing and available computational resources.

[retina]

Originally Posted by Dobri

... could look at empirical observations to eliminate less probable choices for manual testing such as 35.

This is just the Gambler's fallacy is disguise. X has occurred in the past so X will occur again. But there is no proof that the two events are related.

What if the reverse Gambler's fallacy is the case here? You should then be choosing digit sums that are not in the above set because those outputs are already "used up" so different values will be more likely, right?

How would you know which position to choose?

But, whatever the case, it doesn't matter what you choose as long as you have something to reach for.

[Dobri]

Originally Posted by retina

But, whatever the case, it doesn't matter what you choose as long as you have something to reach for.

Indeed, there is definitely something to reach for in this Mersennery game. Happy hunting, retina. :)
One could go wild and select digit sum 35 for manual testing, but preferably not if the test on a modest computer would take a year.

[a1call]

Spurious Correlations

Go to the next page of charts, and keep clicking "next" to get through all 30,000.

https://www.tylervigen.com/spurious-correlations

[kriesel]

In that vein:

Number of GIMPS Mersenne prime discoveries versus exponent digit count
1-6 digits: 0
7 digits: 4
8 digits: 13
9 digits or higher: 0
So by that spurious measure, we should keep testing 7 digit and mostly 8 digit exponents?

Perhaps move this thread to Misc. Math?

[charybdis]

Originally Posted by Dobri

Players with a small number of modest computers could look at empirical observations to eliminate less probable choices for manual testing such as 35.
Also, the digit sum histogram of all possible digit sums for the entire testing range has a maximum at around 41. While digit sums 38 and 41 appear frequently in the list of discovered Mersenne primes, the digit sum 40 never occurred and it is a less probable choice for manual testing.

We can use the Lenstra-Pomerance-Wagstaff heuristic to estimate the probability that 2^p-1 is prime. Using this, we can calculate the probability of finding no prime with exponent digit sum 35 up to the current first-time testing limit of p=103580003:

gp > c=exp(Euler());prob=1;forprime(p=3,103580003,if(sumdigits(p)==35,if(p%4==1,pprob=1-c*log(6*p)/(p*log(2));prob*=pprob,pprob=1-c*log(2*p)/(p*log(2));prob*=pprob)));print(prob)
0.29152692322560544242179474578006423121

29% isn't especially unlikely. You certainly shouldn't be concluding from this that there's something wrong with the heuristic and that exponents with digit sum 35 are fundamentally less likely to yield Mersenne primes. If I tossed a coin twice and got two heads, I wouldn't conclude that the coin was biased.

For digit sum 40, we get

gp > c=exp(Euler());prob=1;forprime(p=3,103580003,if(sumdigits(p)==40,if(p%4==1,pprob=1-c*log(6*p)/(p*log(2));prob*=pprob,pprob=1-c*log(2*p)/(p*log(2));prob*=pprob)));print(prob)
0.48796250668372976034521929359700885331

so it's not at all surprising that we haven't found any primes yet, it was basically a coin toss.

[Dobri]

Originally Posted by charybdis

We can use the Lenstra-Pomerance-Wagstaff heuristic to estimate the probability that 2^p-1 is prime.

The LPW heuristic is just another empirical observation based on the limited sample size.
There is an alternative interpretation for the graph of log2(log2(Mn)) versus n.
Perhaps another line with a smaller slope is formed after M40 or even the graph starts to reach saturation that could indicate that there is no M52 at all.

[Dobri]

Originally Posted by kriesel

So by that spurious measure, we should keep testing 7 digit and mostly 8 digit exponents?

We keep doing DC for mostly 8-digit exponents indeed. :)

[charybdis]

Originally Posted by Dobri

The LPW heuristic is just another empirical observation based on the limited sample size.

No it isn't. There are good reasons for believing it *might* be true, as detailed on the page I linked to. That's what it means to be a heuristic. Lenstra and Pomerance didn't pull the factor e^gamma out of their asses. I'm not saying I believe it's true, but it's plausible and fits the numerical evidence so far. If you want people to believe your hypothesis of some digit sums being likelier than others, you'll at least need to provide a similar heuristic argument.

There is an alternative interpretation for the graph of log2(log2(Mn)) versus n.
Perhaps another line with a smaller slope is formed after M40 or even the graph starts to reach saturation that could indicate that there is no M52 at all.

Right, it does look like there's a change in the slope, but that could just be chance. If Mersenne primes continue to appear more frequently than the heuristic suggests, then the search will be on for a mathematical explanation. But a sudden change in the slope would be odd. Why would Mersenne numbers behave differently above 2^20000000? It would be instructive for you to use the LPW heuristic to randomly generate some sets of "fake Mersenne primes" and see what the resulting graphs look like. How closely do they actually follow the straight line?

And what on earth is "the graph starts to reach saturation" supposed to mean? What object is getting saturated in order to prevent any more Mersenne primes existing? For what it's worth, even if Mersenne numbers were no more likely to be prime than random numbers of their size, the Prime Number Theorem would still tell us that there should be infinitely many Mersenne primes.