Digit sum of a Mersenne-prime exponent

[kriesel]

Originally Posted by Dobri

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

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. https://www.mersenneforum.org/showpo...9&postcount=13. 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.)

[Dobri]

Originally Posted by charybdis

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.

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.

[retina]

Originally Posted by Dobri

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.

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?

[Dobri]

Originally Posted by retina

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?

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.

[charybdis]

Originally Posted by Dobri

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.

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 this.

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.

[kriesel]

Originally Posted by Dobri

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

Let's call that 2^mkⁿ where m can be 0 or larger integer, n can be larger than 1. That accommodates 25=2⁰5² (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=2²*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.

Originally Posted by Dobri

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.

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. https://www.mersenneforum.org/showpo...00&postcount=4
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. https://www.mersenneforum.org/showpo...04&postcount=5

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.

[rudy235]

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 2 should be considered. Or 12, 16, 100, 360. Perhaps pick base 88 (the number of Keys in a Grand Piano) or 57 (for the 57 varieties of the Heinz Sauce.)

[kriesel]

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.

[tuckerkao]

Originally Posted by rudy235

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 2 should be considered. Or 12, 16, 100, 360. Perhaps pick base 88 (the number of Keys in a Grand Piano) or 57 (for the 57 varieties of the Heinz Sauce.)

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] M168433723

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

[Dobri]

Originally Posted by charybdis

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 this.
...
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.

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.

[Dobri]

Originally Posted by kriesel

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

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.