Digit sum of a Mersenne-prime exponent
 

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


[FONT=&quot]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.[/FONT]
[FONT=&quot]
[/FONT]
[FONT=&quot]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:[/FONT]
[FONT=&quot]
[/FONT]
[FONT=&quot]25 = 5^2,
[/FONT]
[FONT=&quot]32 = 2^5,
[/FONT]
[FONT=&quot]28 = 2^2 x 7, etc.
[/FONT]

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.

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.

[QUOTE=Dobri;583128]... could look at empirical observations to eliminate less probable choices for manual testing such as 35.[/QUOTE]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? :confused:

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

[QUOTE=retina;583129]
But, whatever the case, it doesn't matter what you choose as long as you have something to reach for. :tu:[/QUOTE]
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.

[B]Spurious Correlations[/B]
[QUOTE]Go to the next page of charts, and keep clicking "next" to get through all 30,000.[/QUOTE]

[url]https://www.tylervigen.com/spurious-correlations[/url]

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?

[QUOTE=Dobri;583128]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.[/QUOTE]

We can use the [URL="https://primes.utm.edu/mersenne/heuristic.html"]Lenstra-Pomerance-Wagstaff heuristic[/URL] 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:

[code]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[/code]

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

[QUOTE=charybdis;583150]We can use the [URL="https://primes.utm.edu/mersenne/heuristic.html"]Lenstra-Pomerance-Wagstaff heuristic[/URL] to estimate the probability that 2^p-1 is prime.[/QUOTE]
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.

[QUOTE=kriesel;583146]
So by that spurious measure, we should keep testing 7 digit and mostly 8 digit exponents?
[/QUOTE]
We keep doing DC for mostly 8-digit exponents indeed. :)

[QUOTE=Dobri;583158]The LPW heuristic is just another empirical observation based on the limited sample size.[/quote]

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.

[quote]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.[/QUOTE]

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.

[QUOTE=Dobri;583159]We keep doing DC for mostly 8-digit exponents indeed. :)[/QUOTE]With a LOW expectation of finding any missed primes. Error rate in LL first test is averaging ~1-2% per test depending partly on whether Jacobi check was included; error rate in PRP first test with GEC is probably <1ppm. Also a remarkably high number of 8-digit-exponent Mersenne primes were already found. If they've already been all found, we'll find no more during DC, TC, etc. Through most of recorded history, the number known lay below the cyan line representing the Lenstra-Pomerance-Wagstaff heuristic. [URL]https://www.mersenneforum.org/showpost.php?p=512669&postcount=13[/URL]. There's historical precedent for gaps of over 4:1 on exponent, and delays between successive discoveries exceeding a century. At our recent rate of progress of ~6M/year, a 4.1:1 gap from M51 in December 2018 would be ~338M, ~mid 2061. (Not a prediction, just a computation for comparison.)

[QUOTE=charybdis;583176]No it isn't. There are good reasons for believing it *might* be true, as detailed on the page I linked to.
...
Right, it does look like there's a change in the slope, but that could just be chance.
...
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.[/QUOTE]
The effort to fit the pre-GIMPS Mersenne primes with a line using e^(-gamma) from the Mertens' third theorem (which is valid for ALL primes) is based on the assumption that the Mersenne primes have a distribution similar to ALL primes.

However, the GIMPS discoveries of the last 12 Mersenne primes from M40 to M51 are hard to ignore. One should at least entertain the idea that the Mersenne primes have their own distribution which is non-linear and so far looks like two lines with distinct slopes.

Let's assume that there are several Mersenne primes within the local 9-digit exponent range. Then even if there is at least one Mp with a digit sum = 35, 40,..., it is a matter of choice to ignore such digit sums (or rarely select them) and aim at digit sums that are already a part of the digit sum histogram for 51 known Mersenne primes.

The prime number theorem does not tell us whether there are infinitely many Mersenne primes or not. There is an alternative for the slope of the tangent line to eventually become closer to zero and the saturated graph to end up with the last Mersenne prime.

[QUOTE=Dobri;583198] Let's assume that there are several Mersenne primes within the local 9-digit exponent range. Then even if there is at least one Mp with a digit sum = 35, 40,..., it is a matter of choice to ignore such digit sums (or rarely select them) and aim at digit sums that are already a part of the digit sum histogram for 51 known Mersenne primes.[/QUOTE]Why did you choose base-10? What is the mathematical reasoning behind that? Why not choose, say, base-17? Or base-6? Or base-phi?

[QUOTE=retina;583199]Why did you choose base-10? What is the mathematical reasoning behind that? Why not choose, say, base-17? Or base-6? Or base-phi?[/QUOTE]
This is an early attempt to eventually improve the competitiveness of Mersenneries with modest resources.
I am still figuring out to what extent the combination of digit sum histograms obtained for multiple prime bases could help.

[QUOTE=Dobri;583198]The effort to fit the pre-GIMPS Mersenne primes with a line using e^(-gamma) from the Mertens' third theorem (which is valid for ALL primes) is based on the assumption that the Mersenne primes have a distribution similar to ALL primes.

However, the GIMPS discoveries of the last 12 Mersenne primes from M40 to M51 are hard to ignore. One should at least entertain the idea that the Mersenne primes have their own distribution which is non-linear and so far looks like two lines with distinct slopes.

Let's assume that there are several Mersenne primes within the local 9-digit exponent range. Then even if there is at least one Mp with a digit sum = 35, 40,..., it is a matter of choice to ignore such digit sums (or rarely select them) and aim at digit sums that are already a part of the digit sum histogram for 51 known Mersenne primes.

The prime number theorem does not tell us whether there are infinitely many Mersenne primes or not. There is an alternative for the slope of the tangent line to eventually become closer to zero and the saturated graph to end up with the last Mersenne prime.[/QUOTE]

You haven't answered any of my questions. I'm going to assume you haven't tried generating random sets of exponents as I advised. If you had, you'd see that even when the heuristic is being used to generate the exponents, you can still get sudden apparent changes in slope or find "primes" substantially more or less often than you'd expect. I still have no idea what you mean by a "saturated graph", as it clearly isn't anything to do with [URL="https://en.wikipedia.org/wiki/Saturation_(graph_theory)"]this[/URL].

Of course the PNT doesn't tell us there are infinitely many Mersenne primes; if it did, it wouldn't be an open problem. But we in fact find many more Mersenne primes than the PNT alone would suggest. It would take a significant "phase transition" for them to stop appearing - or indeed for the slope to change suddenly. I don't know of any similar examples in number theory; if you can point me to one then please do so. (Phase transitions are common in combinatorics, but in that case we generally know that there must be some change in behaviour. For example, in the Erdos-Renyi random graph model, it's clear that for very small p the connected components will almost surely be very small, and for large p the graph will be almost surely connected. It's not surprising that we get a phase transition somewhere in between.)

All of your hypotheses are based entirely on small patterns that are not so strange that we can discount the possibility that they are due to chance alone. I'll finish by noting that 24 out of the 51 known Mersenne primes are the first with their given exponent digit sum, so if we followed your strategy we'd miss a lot of primes.

[QUOTE=Dobri;583119]The base-10 digit sums of the Mersenne exponents for the 51 known Mersenne primes are as follows...
[FONT=&quot]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[/FONT][/QUOTE]Let's call that 2[SUP]m[/SUP]k[SUP]n[/SUP] where m can be 0 or larger integer, n can be larger than 1. That accommodates 25=2[SUP]0[/SUP]5[SUP]2[/SUP] (Mp43, base10 digitsum). Such a formulation captures all the evens and quite a few of the odds.

It seems to accommodate base 10 and 16 with the small sample sizes. Some counterexamples from base 30:
Mp28, 55=5*11
Mp30, 70=2*5*7
Mp32, 84=2[SUP]2[/SUP]*3*7
Mp34, 57=3*19
Mp38, 65=5*13
Mp45, 69=3*23
Mp49, 51=3*17
Modifying it to allow for these also, to primes or products of powers of (up to 3?) small primes reminds me of Ptolemaic epicycles.

[QUOTE=Dobri;583202]This is an early attempt to eventually improve the competitiveness of Mersenneries with modest resources.
I am still figuring out to what extent the combination of digit sum histograms obtained for multiple prime bases could help.[/QUOTE]
A way of testing such attempts is to remove a data point from the set, and then evaluate whether the method would find it.
Mp51* has the first occurrence of digit sum 47 in base 10, 68 in base 16, and 108 in base 30.
Testing exponents only if they had digit sums appearing for the first 50 Mersenne primes would miss it.

Also Mp43, Mp39, Mp32, Mp27, Mp25, Mp24, Mp22, Mp21, Mp19, Mp15, Mp7, Mp1-5 produce first appearances in all 3 bases.

A more productive if selfish way might be to select p = 1 mod 4, or 1 mod 8. That leaves the less likely exponents for others to process. [URL]https://www.mersenneforum.org/showpost.php?p=517500&postcount=4[/URL]
That approach is backed by heuristic arguments in support of Mersenne numbers with exponent = 1 mod 8 having fewer factors.

There have been hundreds of documented attempts to predict or guess exponents of Mersenne primes. Also attempts made in employing base 12. No one has been successful yet. [URL]https://www.mersenneforum.org/showpost.php?p=512904&postcount=5[/URL]

Unique values, of sum of decimal digits of exponents of Mersenne primes currently known to Terrans, in sorted order:
2, 3, 4, 5, 7, 8, 10, 11, 13, 14, 16, 17, 19, 23, 25, 26, 28, 29, 31, 32, 37, 38, 41, 47
For base ten:
Leftmost digit will be >0 so contribute at least 1, up to 9, on average 5, if uniformly distributed.
Middle digits will be distributed 0-9 so ~4.5 each.
Rightmost digit is restricted for multidigit primes to be 1, 3, 7 or 9, so ~5.
The more digits, the bigger the total, on average.
We're now in 9-digit exponent territory, so 5 + 4.5*7 + 5 ~ 41.5 is what we might expect for future discoveries.
Larger exponents are less likely to lead to primes, per candidate. That shifts the average leading digits downward somewhat, moving the expected average toward or below 41.
Larger exponents are far more computationally expensive; ~tripling the exponent makes a primality test a ~10fold larger investment of computing time. That is a much larger effect on discovery probability per unit of computing time resource (GHz-decade, or some such).

I think RDS would dismiss the whole thread as an exercise in numerology.

Base 10 is an eminent human construct. The only thing interesting about the number TEN (beyond us having ten fingers and 10 toes) is that 10 is both a triangular number and a tetrahedral number.

I believe that other bases like [SIZE="3"][B]2[/B][/SIZE] should be considered. Or [B][SIZE="3"]12, 16, 100, 360[/SIZE][/B]. Perhaps pick base [SIZE="3"][B]88[/B][/SIZE] (the number of Keys in a Grand Piano) or [B][SIZE="3"]57[/SIZE][/B] (for the 57 varieties of the Heinz Sauce.)

Bases with multiple unique small prime factors, and bases small enough for unique digit symbols in ASCII are appealing. Up to 62 is accommodated by 0-9A-Za-z.

[QUOTE=rudy235;583215]Base 10 is an eminent human construct. The only thing interesting about the number TEN (beyond us having ten fingers and 10 toes) is that 10 is both a triangular number and a tetrahedral number.

I believe that other bases like [SIZE="3"][B]2[/B][/SIZE] should be considered. Or [B][SIZE="3"]12, 16, 100, 360[/SIZE][/B]. Perhaps pick base [SIZE="3"][B]88[/B][/SIZE] (the number of Keys in a Grand Piano) or [B][SIZE="3"]57[/SIZE][/B] (for the 57 varieties of the Heinz Sauce.)[/QUOTE]
It's always possible to create the robotic hands with the thumbs sticking out from both sides of 2 hands, that's the best type of hands to perform piano because the piano keys(D, E♭, E, F, F#, G, A♭, A, B♭, B, C, C#, D) are dozenal based anyway.


I've used this type of methods to guess on Mersenne exponents also such as: [dozenal] Z484Ӿ9277 which is [decimal] M[M]168433723[/M]

[dozenal] (4 + 8) + (4 + Ӿ) + (9 + 2) + (7 + 7) = (10 + 12) + (Ɛ + 12) = 22 + 21 = 43 = 15 * 3

[QUOTE=charybdis;583206]You haven't answered any of my questions. I'm going to assume you haven't tried generating random sets of exponents as I advised. If you had, you'd see that even when the heuristic is being used to generate the exponents, you can still get sudden apparent changes in slope or find "primes" substantially more or less often than you'd expect. I still have no idea what you mean by a "saturated graph", as it clearly isn't anything to do with [URL="https://en.wikipedia.org/wiki/Saturation_(graph_theory)"]this[/URL].
...
All of your hypotheses are based entirely on small patterns that are not so strange that we can discount the possibility that they are due to chance alone. I'll finish by noting that 24 out of the 51 known Mersenne primes are the first with their given exponent digit sum, so if we followed your strategy we'd miss a lot of primes.[/QUOTE]
Oscillations around the linear fit are common but to have a change in slope for a dozen consecutive Mersenne primes from M40 till M51 and assume that it is happening just by chance would be in blind defense of the linear heuristic.
This thread is about paths less traveled. It is a gray area, there are no definite answers as more factual information has to be obtained both theoretically and computationally.
Concerning the term 'saturation', it happens when the growth slows down and eventually stops. A common example is the logistic function, see <https://en.wikipedia.org/wiki/Logistic_function>.

If selecting mainly digit sums from the histogram of known Mersenne primes, obviously one would miss the discovery of Mersenne primes with digit sums observed for the first time. However, this increases the chances of discovering Mersenne primes with digits sums most frequently occurring in the digit sum histogram of the known Mersenne primes.
It is a trade-off between the risks one would take with limited resources for a prolonged period of time.
It is not like a game of poker when the results are known immediately.

Analyzing incomplete patterns is better than doing nothing. The aim is at eventually assisting the selfless volunteers who run a test for a 9-digit exponent for 5-6 consecutive years or more in a slow computer at home.
If this thread is given a chance to grow and mature, there will be more clarity and understanding whether we are dealing with a mere chance or there is more to this than meets the eye.

[QUOTE=kriesel;583212]
I think RDS would dismiss the whole thread as an exercise in numerology.[/QUOTE]
I myself am undecided to what extent incomplete patterns could produce tangible results in number theory.
The digit sum histogram is just one partial pattern characteristic among many alternative ways to analyze the known exponents.
If the forum decides to keep this thread, I may post further developments in the coming years (or decades).
We are not in a hurry after all.
I myself have just 5 modest computers and still manage to find my user name at <https://www.mersenne.org/report_top_500>.
As a volunteer, I am paying my electricity bills, and would like to increase the efficiency of my limited manual testing.
In my opinion, further sophistication aided by machine learning with the use of multiple incomplete patterns could reduce the element of chance in the manual selection of exponents.

At the risk of arrest for criminal belaboring of the obvious, I give one obvious reason to deem digit sums a poor candidate for any sort of correlation with Mersenne prime exponents.

If b > 1 is an integer, the digit sum of n to the base b exhibits noticeable oscillations. The most extreme, of course, is the base-b digit sum of b^k - 1 is k*(b-1) and the base-b digit sum of b^k is 1.

So if the largest Mersenne prime exponent with a given number k of base-b digits is only slightly less than b^k, while the next larger Mersenne prime exponent is only slightly larger than b^k, or perhaps a larger power of b, the base-b digit sum will drop noticeably. Similar oscillations can occur between consecutive exponents if the first has a large block of digits b-1 which is gone in the next.

For example, consider the consecutive Mersenne prime exponents 127 and 521. To the base b = two, the digit sums are 7 and 3 respectively. In base b = ten, the consecutive Mersenne prime exponents 19 and 31 have digit sums 10 and 4 respectively, while 9941, 11213, 19937, and 21701 have digit sums 23, 8, 29, and 11 respectively.

[QUOTE=Dobri;583230]Oscillations around the linear fit are common but to have a change in slope for a dozen consecutive Mersenne primes from M40 till M51 and assume that it is happening just by chance would be in blind defense of the linear heuristic.
This thread is about paths less traveled. It is a gray area, there are no definite answers as more factual information has to be obtained both theoretically and computationally.
Concerning the term 'saturation', it happens when the growth slows down and eventually stops. A common example is the logistic function, see <https://en.wikipedia.org/wiki/Logistic_function>.[/quote]

Yes, the deviation from the heuristic may well be statistically significant at the 2-sigma level, but I don't think that should be enough to change our views. If the new slope were to continue for another 12 Mersenne primes then I'd be more willing to accept your hypothesis. I suppose we disagree about what the prior is. I would say the prior should be heavily weighted towards the relationship being best approximated by some smooth function, as there's no precedent or plausible mathematical explanation for a sudden change in behaviour. Perhaps the relationship isn't linear and Mersenne primes do start appearing more often than the heuristic suggests as p gets larger. But I'd be far more willing to believe that the true behaviour is some smooth curve rather than two straight lines with a transition at p=20M.

Similarly, if you're at a casino and the roulette wheel comes up red 15 times in a row after behaving "normally" for a long time, you might say that is statistically significant, but the prior is heavily in favour of nothing having changed, so this would still most likely be down to chance alone. If red comes up 50 times in a row, then someone has almost certainly found a way to rig the wheel so that red comes up every time.

Saturation in the sense you mention doesn't just happen out of nowhere. It happens when the growth rate of a quantity is slowed as a result of its own growth, or when there is a natural limit to how large the quantity can grow. In other words, something is getting saturated (hence my question to you a few posts back). For example, as a population grows, it can only reach a certain level before competition for resources slows the growth down to a halt. There is only a certain amount of salt that will dissolve in water before precipitation happens just as often as dissolution. How could the existence of a particular number of Mersenne primes make higher Mersennes less likely to be prime?

[quote]If selecting mainly digit sums from the histogram of known Mersenne primes, obviously one would miss the discovery of Mersenne primes with digit sums observed for the first time. However, this increases the chances of discovering Mersenne primes with digits sums most frequently occurring in the digit sum histogram of the known Mersenne primes.
It is a trade-off between the risks one would take with limited resources for a prolonged period of time.
It is not like a game of poker when the results are known immediately.[/QUOTE]

So if you pick and choose your targets like this, you're more likely to find Mersenne primes that fit your pattern, and less likely to find ones that don't. When there's no evidence that Mersennes fitting your pattern are more likely to be prime, there's no advantage in doing this - in fact there's a disadvantage because you'd get to larger numbers more quickly. It gets worse: if we kept on following your advice, we would not only miss Mersenne primes whose digit sums haven't appeared yet, but we'd miss any more primes sharing the same digit sums as those primes. We'd be forever limiting ourselves to searching exponents with the 24 known digit sums.

[QUOTE=Dobri;583233]In my opinion, further sophistication aided by machine learning with the use of multiple incomplete patterns could reduce the element of chance in the manual selection of exponents.[/QUOTE]

We have ample data set size for exponents with factors found by TF, and set with factors found by P-1, and set with no factors found by economical factoring effort but found with additional uneconomic factoring effort, and set with no factors known but conclusive primality test indicating composite. What we don't have is a large set of training data for known Mersenne primes, especially if some get classified in categories of high, medium, or low likelihood, or finer gradations for selection. [URL]https://machinelearningmastery.com/much-training-data-required-machine-learning/[/URL] speaks of training sets in the tens, hundreds, or thousands.

I recall reading of a system that had been trained to distinguish photos of dogs from photos of wolves. After considerable training effort with many photos, it was queried to display what distinguished one from the other. The highlighted area was the snow in the background of wolf photos. It ignored the animals completely, and revealed problems with the training data.

All of this assumes that there is an additional pattern we're not yet aware of, in effect an as yet not found shortcut to identifying "good" candidates for Mersenne primes vs. "mediocre" or "poor" candidates that requires far less computational time than a primality test or additional factoring effort. And that statistical analysis or machine learning can find the pattern if it exists. And that the signal/noise or false positive and false negative rates are acceptably low for it to be useful. And that it will also apply sufficiently well to the untested range p>104M. Those assumptions can be wrong. We're already using the product of a great deal of info. Skipping composite exponents is one example. Eliminating ~2/3 of the remaining exponents by factoring is another.

If we somehow find some way via ML of pre-scoring untested exponents, how would we validate it, and would the PrimeNet server adopt prioritizing the higher likelihood exponents, or continue with a systematic exhaustive mostly-monotonic-ascent search? We could implement that prioritizing now, prioritizing p=1 mod 8. It is however based on a very small sample size. The effect would be limited. A 2:1 probability advantage only justifies testing up to about a 1.39x larger exponent. (effort ~ p[SUP]2.1[/SUP]; 2[SUP]1/2.1[/SUP] ~1.39)

There's a big difference, between generating probability estimates for each individual untested unfactored exponent, by the millions, to be stored for later lookup in the database, requiring modification to the database, and having a simple easily applied mathematical description for what to prioritize, requiring some lines of code.

As to the two-slopes argument, two rejoinders. We are designed to see patterns. We see them whether they are real patterns or the predictable product of expected statistical variation. (The same thing happens with other phenomena, such as combustion engine cylinder peak pressure cycle-to-cycle variation.) The 10M-100M interval of Mersenne primes can appear as a slope increase. Or it can be seen as staircases of short gaps interrupted by landings (longer gaps). There are modest staircases at lower exponent. Compare 19K-24K to 70M-90M.

The best available theorizing says the distribution of Mersenne primes is governed by Poisson process, and an expected outcome is that some seemingly improbable outcomes are probable. Toss enough dice, and you'll get as many sixes consecutively as you have patience to pursue. There were an enormous number of primality tests performed between 10M and 100M. That improves the chances of the statistics matching well what is theoretically predicted. And that is what the graph shows. Back at 128-512, one could have made the argument that all the Mersenne primes may have been found. Or at 220,000-750,000. Slopes there were zero over ~4:1 and ~3.4:1. But these or even longer gaps may occur. If we are at the beginning of one now, the next Mersenne prime could be >100Mdigit.

[QUOTE=kriesel;583242]I recall reading of a system that had been trained to distinguish photos of dogs from photos of wolves. After considerable training effort with many photos, it was queried to display what distinguished one from the other. The highlighted area was the snow in the background of wolf photos. It ignored the animals completely, and revealed problems with the training data.[/QUOTE]How about [URL="https://towardsdatascience.com/is-the-medias-reluctance-to-admit-ai-s-weaknesses-putting-us-at-risk-c355728e9028"]this one[/URL]? The AI being trained to determine if a skin spot was bad learned to look for the ruler in the image. Dermatologist take pictures of the bad spots with a rule to measure size. Those were knowns. The ok spots did not have rulers, as they weren't worried about the size.

Good one. Maybe we should not be concerned about AI, but about artificial stupidity. We don't need to augment the naturally occurring kind that's already a surplus.
Of dogs, wolves, snow, grass, tanks, forest, clouds, and sun: [URL]https://hackernoon.com/dogs-wolves-data-science-and-why-machines-must-learn-like-humans-do-41c43bc7f982[/URL] Now imagine trying to do the classification and subsequent analysis of multiple possibilities reliably in real time at road speed to avoid or minimize harming or killing people.

Not only Luke 23:34 but also a well-known quote from Macbeth comes to mind
[SPOILER]" ........ ........
full of sound and fury,
Signifying nothing.”[/SPOILER]

[QUOTE=Batalov;583270]Not only Luke 23:34 but also a well-known quote from Macbeth comes to mind[/QUOTE]
Maybe [URL="https://www.biblegateway.com/passage/?search=1tim+6%3A20&version=KJV"]1Tim 6:20[/URL]

[QUOTE=Uncwilly;583277]Maybe [URL="https://www.biblegateway.com/passage/?search=1tim+6%3A20&version=KJV"]1Tim 6:20[/URL][/QUOTE]... or the ever-popular [url=https://www.biblegateway.com/passage/?search=Matthew%2015%3A14&version=KJV]Matthew 15:14[/url]

Let's mention also another empirical observation which might be obvious but related to this thread nonetheless.

When comparing the digit sum histograms of the Mersenne-prime (Mp) and Mersenne-number (Mn) discrete distributions (in the interval from 2 to 71) in the 8-digit exponent range, the Mp histogram has entries mainly below the mean value of the Mn histogram.

Assuming that there will be not so many Mp discoveries in the 9-digit exponent range (in order to go back to the linear fit of the prime number heuristic), then the Mp digit sum histogram would lay almost entirely below the mean value of the Mn digit sum histogram (in the interval from 2 to 80) except for the digit sum 47 of Mp51.

I will plot and post the Mp and Mn histograms later. Meanwhile, I enjoyed reading Shakespeare and the Scriptures. It was a good distraction. Thank you!

[QUOTE=Dr Sardonicus;583294]... or the ever-popular [url=https://www.biblegateway.com/passage/?search=Matthew%2015%3A14&version=KJV]Matthew 15:14[/url][/QUOTE]
And of course this great principle -->
[QUOTE="Ecclesiastes 1:15"]That which is crooked cannot be made straight: and that which is wanting cannot be numbered.[/QUOTE]

[QUOTE=Dobri;583303]When comparing the digit sum histograms of the Mersenne-prime (Mp) and Mersenne-number (Mn) discrete distributions (in the interval from 2 to 71) in the 8-digit exponent range, the Mp histogram has entries mainly below the mean value of the Mn histogram.

Assuming that there will be not so many Mp discoveries in the 9-digit exponent range (in order to go back to the linear fit of the prime number heuristic), then the Mp digit sum histogram would lay almost entirely below the mean value of the Mn digit sum histogram (in the interval from 2 to 80) except for the digit sum 47 of Mp51.[/QUOTE]

Smaller Mersennes are much more likely to be prime, so this is about as useful as pointing out that most of the Mersenne prime exponents up to 100M are below 50M. :clap:

[QUOTE=Dr Sardonicus;583294]... or the ever-popular [url=https://www.biblegateway.com/passage/?search=Matthew%2015%3A14&version=KJV]Matthew 15:14[/url][/QUOTE]

Hoping that I haven't inadvertently fallen into the ditch...

[QUOTE=charybdis;583305]
Smaller Mersennes are much more likely to be prime, so this is about as useful as pointing out that most of the Mersenne prime exponents up to 100M are below 50M.
[/QUOTE]
Indeed, Mn=49,999,991 (digit sum 59) could have been an Mp but it has been factored instead.

Digit sums are awesome and have predictive power. Yes they do!
 

I have discovered, after extensive and exhaustive (and exhausting) analysis of the base-2 digit sums of all the Mersenne primes that we know of today[sup]*[/sup], that ALL the digit sums for EVERY MP have a special pattern whereby they have no other divisors other than 1 or themselves.

OMG, why has no one thought of this before!!!!!!! :shock:

That must say something, right?

Am I awesome or what! :showoff:

:razz:

[size=1][sup]*[/sup] Try saying that quickly in a single breath. Phew.[/size]

[QUOTE=retina;583307]I have discovered, after extensive and exhaustive (and exhausting) analysis of the base-2 digit sums of [B][I]the exponents of[/I][/B] all the Mersenne primes that we know of today[sup]*[/sup], that ALL the digit sums for EVERY MP have a special pattern whereby they have no other divisors other than 1 or themselves.[/QUOTE]What. 89 = 1011001b, digit sum 4 = 2[SUP]2[/SUP]. 1279 = 10011111111b, digit sum 9 = 3[SUP]2[/SUP]. 2203 = 1000 1001 1011b, digit sum 6 = 2 * 3. ...

[QUOTE=kriesel;583308]Um, 89 = 1011001, digit sum 4.[/QUOTE]I got too excited from my awesome discovery and had some extra nonsense in there. I already edited it.

But can you predict M51 from the first 50? I fail to see how any digitsums are siginificant, and (maybe) higher numbers tend to have higher digitsums, since the later part looks random and it evens out, but the earlier part tends to rise steadily. (graph not related)
(was unable to use the data directly, too much for spreadsheet graph to handle)

Digit Sum Distributions of Mersenne-Number Exponents
 

The attached image contains the plots of:
- The digit sum distribution of known Mersenne-prime exponents (green dots) in the interval [2, 47];
- The digit sum distribution of 8-digit Mersenne-number exponents (red dots) in the interval [2, 71]; and
- The digit sum distribution of 9-digit Mersenne-number exponents (blue dots) in the interval [2, 80].

The three distinct distributions are normalized to unity with respect to:
- The total number of [FONT=&quot]51[/FONT] known Mersenne-prime exponents (Mp);
- The total number of [FONT=&quot][FONT=&quot]5,761,455 [/FONT][/FONT]8-digit Mersenne-number exponents (Mn); and
- The total number of [FONT=&quot]50,847,534[/FONT] 9-digit Mersenne-number exponents (Mn).

[QUOTE=thyw;583312]But can you predict M51 from the first 50? I fail to see how any digitsums are siginificant, and (maybe) higher numbers tend to have higher digitsums, since the later part looks random and it evens out, but the earlier part tends to rise steadily. (graph not related)
(was unable to use the data directly, too much for spreadsheet graph to handle)[/QUOTE]
The idea is not to predict the next Mp but rather select suitable options for manual testing.
Assuming that there are several Mp in the 9-digit range, one would expect that at least one is among the known digit sums (and probably among the digit sums with the highest frequency of occurrence, like 38, 41, etc.).
Note that small numbers also have high digit sums if the Mn exponent contains many '8' and '9' digits.
The peculiar thing is that the 51 known Mp have digit sums mainly in the lower half of the Mn digit sum distributions.

:deadhorse:

[QUOTE=Dobri;583340]The peculiar thing is that the 51 known Mp have digit sums mainly in the lower half of the Mn digit sum distributions.[/QUOTE][URL="https://mersenneforum.org/showpost.php?p=583212&postcount=17"]Well of course they do[/URL]. Known Mp of few digits exponent have small limits on digit sums. Take 1-digit, 2 3 5 7 for example. It's hard to reach digit sum 41 with a single base 10 digit. It's not peculiar at all, but more like expected or inevitable. Try plotting 8-digit exponent known Mp digit sums with 8-digit exponent Mn; 7 with 7; 6 with 6. Preferably with point labels indicating how few samples per digit sum are available, or alternately something like 95% confidence interval error bars for the Mp sum frequencies.[QUOTE=kriesel;583212]For base ten:
Leftmost digit will be >0 so contribute at least 1, up to 9, on average 5, if uniformly distributed.
Middle digits will be distributed 0-9 so ~4.5 each.
Rightmost digit is restricted for multidigit primes to be 1, 3, 7 or 9, so ~5.
The more digits, the bigger the total, on average.
We're now in 9-digit exponent territory, so 5 + 4.5*7 + 5 ~ 41.5 is what we might expect for future discoveries.
[/QUOTE]For multidigit exponents, if uniformly distributed, of d base 10 digits, average:
sum ~ 5 + (d-2)*4.5 + 5 = 10 + 4.5 * (d-2) = 1 + 4.5 d
So, 8 digits, ~37; 7 digits, ~32.5; 9 digits, ~41.5.
Minimum digit sum, ~2; Maximum ~9d; subject to the constraint that repdigits can not be prime unless their length is prime, and digits one, so for multiple digits, maximum sum is slightly lower, for a near-repdigit containing mostly digits of value base-1. For example, base ten again, max prime exponent p < 10[SUP]9[/SUP] = [M]999999937[/M] not 999999999; the next smaller prime exponent [M]999999929[/M] is a slightly higher digit sum and a [URL="https://www.mersenneforum.org/showpost.php?p=567245&postcount=4"]near-repdigit[/URL]. There are other near-repdigit primes with slightly higher digit sums. About a million lower, there's a near-repdigit [M]998999999[/M] which exponent is prime, providing the max possible digit sum in base ten 9 digit primes, 80.

I note there's still not much of an answer re why base 10 digits. "Convenience" works for me.[QUOTE=Uncwilly;583346]:deadhorse:[/QUOTE]Indeed. I was just thinking of using that.

[QUOTE=kriesel;583347]
I note there's still not much of an answer re why base 10 digits. "Convenience" works for me.[/QUOTE]
It is not a matter of convenience. Simply there is no rush to spill the beans in a single post.
For instance, attached are the base-2 distributions.
In the second image, the base-10 distributions are re-plotted to connect the dots in the graphs with lines.

Here's a comparison between the expected number of primes with each digit sum according to the LPW heuristic and the actual number observed, up to the current first testing limit of p=103580003. Doesn't look biased towards low digit sums, which I'm sure will come as a surprise to no-one except perhaps Dobri.

Feels like it's about time for a mod to close the thread.

[QUOTE=Dobri;583340]The peculiar thing is that the 51 known Mp have digit sums mainly in the lower half of the Mn digit sum distributions.[/QUOTE]I think there is an analog to or operation of Benford's law here. If Mersenne primes occur on the average, ~1.476:1 spacing ([URL]https://primes.utm.edu/mersenne/heuristic.html[/URL]) on exponent, any n:1 exponent ratio with n large compared to 1.476 will be probability weighted more toward the lower left digit values, lower second-from-left, etc. Uniformly in log space is nonuniformly, & favorable to lower left digits, in linear space. [URL]https://www.mersenneforum.org/showpost.php?p=557235&postcount=6[/URL]
[URL]https://primes.utm.edu/notes/faq/NextMersenne.html[/URL]
[URL]https://en.wikipedia.org/wiki/Benford%27s_law[/URL]

For separate digit lengths of exponents in base 10:
1 digit: 2 3 5 7; 7 is as good as it gets since 8 and 9 are composite; average 4.25
2 digit: 13 17 19 31 61 89; 89 is maximal possible sum 17; average per digit sum in 4 8 10 4 7 17; sum 50 / 6 = 8.33 / 2 digits = 4.167/digit
3 digit: 107 127 521 607; 8 10 8 13; 39 / 4 = 9.75; 3.25/digit
4 digit: 1279 2203 2281 3217 4253 4423 9689 9941; 19 7 13 13 14 13 32 23; 134 / 8 = 16.75; 4.1875/digit
5 digit: 11213 19937 21701 23209 44497 86243; 8 29 11 16 28 23; 95 / 6 = 15.833; 3.167/digit
6 digit: 110503 132049 216091 756839 859433; 10 19 19 38 32; 118 / 5 = 23.6; 3.933/digit
7 digit: 1257787 1398269 2976221 3021377 6972593; 37 38 29 23 41; 168 / 5 = 33.6; 4.8/digit
8-digit:
exponent digitsum
13466917 37
20996011 28
24036583 31
25964951 41
30402457 25
32582657 38
37156667 41
42643801 28
43112609 26
57885161 41
74207281 31
77232917 38
82589933 47
sum 452 / 13 exponent = 34.769 / 8 digits = 4.346/digit
(apologies for any lingering math errors)

So for 8-decimal-digit, the histogram of # of Mp vs. digitsum value is
25 1
26 1
28 2
31 2
37 1
38 2
41 3
47 1
Graphing that manually in black with only digitization noise +-0.5 counts atop Dobri's base 10 digit sum distributions, using the existing rulings for scale for convenience yields the attachment. The statistical sample size is terribly small.

[QUOTE=Dobri;583351]It is not a matter of convenience. Simply there is no rush to spill the beans in a single post.[/QUOTE]
So you have been intending to stretch this out. We can put an end to this very quick.

[FONT="Arial Black"][COLOR="Red"]Thread closed[/COLOR][/FONT]

Strategies for Manual Testing
 

Closing the thread at
[URL]https://mersenneforum.org/showthread.php?t=26997[/URL]
was premature because the OP was not given a chance to respond, especially at a stage when a post was submitted in their favor.

The post at
[URL]https://mersenneforum.org/showpost.php?p=583353&postcount=44[/URL]
[QUOTE=charybdis]Here's a comparison between the expected number of primes with each digit sum according to the LPW heuristic and the actual number observed, up to the current first testing limit of p=103580003. Doesn't look biased towards low digit sums, which I'm sure will come as a surprise to no-one except perhaps Dobri.

Feels like it's about time for a mod to close the thread.[/QUOTE]
contains an image showing "the expected number of primes with each digit sum according to the LPW heuristic".

Therefore, it appears one could select exponents for manual testing for a given digit sum as often as indicated by the corresponding expectancy in accordance with the LPW heuristic.

Therefore, the OP would like to respectfully request the previous thread to be reopened and/or the discussion to be continued in this thread instead.

You have misunderstood the point of my post. It was not intended to support your views. All I did was calculate the probabilities given by the LPW heuristic for each prime up to 103M and add up the probabilities for each digit sum. My graph therefore shows what we expect if digit sum has [B]no effect[/B] on the likelihood of being prime - and the actual distribution of primes is consistent with this.

5 is the digit sum with the highest expected number of primes because there are several very small primes with digit sum 5, namely 5, 23 and 41, and the heuristic gives high probabilities for these. For p=5 it gives the nonsensical probability of 1.748. This does not make higher exponents with digit sum 5 more likely to be prime!

Please can a mod close this thread too?

[QUOTE=charybdis;583380]

Please can a mod close this thread too?[/QUOTE]

:deadhorse: Topic exhausted. Enuf said.

[QUOTE=Dobri;583202]This is an early attempt to eventually improve the competitiveness of Mersenneries with modest resources.
I am still figuring out to what extent the combination of digit sum histograms obtained for multiple prime bases could help.[/QUOTE]
What a big collection of great, empty words, and what a waste of time and computing resources... :sad:
Thread closed.

Edit: Whhops, sorry, it was closed already. :blush:

The minimal bases [I]b[/I][SUB]min[/SUB] for which [COLOR="Blue"]the digit sums of the prime exponents of the known Mersenne primes are distinct (different from each other)[/COLOR] is given in the following list:

[I]b[/I][SUB]min[/SUB] = {2, 2, 3, 7, 7, 10, 10, 20, 20, 21, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 82, 82, 82, 82, 108, 108, 108, 108, 108, 108, 108, 108, 108, 108, 108, 108, 108, 108, 110, 110, 110, 110, 110, 110, 142, [COLOR="Red"]142[/COLOR]}

for [I]n[/I] = 1, 2, …, 51.

Therefore, [I]b[/I][SUB]min[/SUB] = 142 for [I]n[/I] = 51, and the distinct [COLOR="Blue"]base-142[/COLOR] digit sums for the 51 prime exponents of known Mersenne primes are

{2, 3, 5, 7, 13, 17, 19, 31, 61, 89, 107, 127, 98, 43, 10, 88, 25, 115, 164, 52, 101, 71, 215, 197, 128, 85, 82, 92, 100, 214, 220, 233, 179, 208, 254, 134, 170, 284, 289, 124, 172, 224, 319, 236, 347, 325, 167, 290, 250, 308, 311}.

If one [B][COLOR="Red"]assumes[/COLOR][/B] for [B][COLOR="Red"]manual selection[/COLOR][/B] that [I]b[/I][SUB]min[/SUB] remains equal to [COLOR="Red"]142[/COLOR] also for [I]n[/I] = [COLOR="red"]52[/COLOR], then prime exponents for finding M52 (if any) could be selected for further testing if [I]b[/I][SUB]min[/SUB] = 142.

[code]
(* Wolfram code *)
MpData = {2, 3, 5, 7, 13, 17, 19, 31, 61, 89, 107, 127, 521, 607, 1279, 2203, 2281, 3217, 4253, 4423, 9689, 9941, 11213, 19937, 21701, 23209, 44497, 86243, 110503, 132049, 216091, 756839, 859433, 1257787, 1398269, 2976221, 3021377, 6972593, 13466917, 20996011, 24036583, 25964951, 30402457, 32582657, 37156667, 42643801, 43112609, 57885161, 74207281, 77232917, 82589933};
nMpmax = Length[MpData]; bm = ConstantArray[0, nMpmax]; basemax = 1000; nMp = 0; While[nMp < nMpmax, nMp++; ds = ConstantArray[0, nMp]; check = 0; base = 1; While[(base < basemax) && (check == 0), base++; sum = 0; ic = 0; While[ic < nMp, ic++; ds[[ic]] = Total[IntegerDigits[MpData[[ic]], base]];]; dssort = Sort[ds]; check = 1; ic = 1; While[ic < nMp, ic++; If[(dssort[[ic]] == dssort[[ic - 1]]), check = 0;];]; If[check == 1, bm[[nMp]] = base;];];];
Print[bm]; Print["b_min = ", base]; Print[ds];
[/code]

Beyond [I]b[/I][SUB]min[/SUB] = 142, larger base-[I]b[/I] values (for which the digit sums of the prime exponents of the known Mersenne primes are distinct) are

[I]b[/I] = {[COLOR="Blue"]224[/COLOR], 230, 278, 300, 314, 330, 334, 336, 342, 352, 374, 398, 402, 404, 412, 414, 418, 424, 432, 444, 447, 450, 458, 459, 467, 468, 473, 474, 488, 490, 498, 504, 507, 512, 518, 522, 541, 543, 545, 546, 548, 552, 555, 566, 572, 576, 584, 585, 588, 594, 608, 614, 620, 627, 630, 642, 654, 656, 658, 660, 674, 682, 683, 688, 690, 696, 699, 704, 706, 713, 718, 722, 723, 726, 738, 740, 742, 746, 750, 756, 762, 768, 770, 777, 778, 793, 797, 798, 800, 811, 812, 814, 816, 820, 826, 827, 830, 833, 835, 836, 847, 852, 854, 857, 863, 866, 868, 870, 878, 880, 882, 884, 887, 890, 894, 896, 897, 900, 905, 906, 910, 914, 922, 923, 924, 926, 929, 930, 933, 935, 936, 940, 941, 942, 944, 952, 954, 958, 960, 962, 974, 975, 978, 979, 980, 982, 984, 988, 993, 994, 996, 998, 1000,...}.