Digit sum of a Mersenne-prime exponent

[Dobri]

Originally Posted by charybdis

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.

Indeed, Mn=49,999,991 (digit sum 59) could have been an Mp but it has been factored instead.

[retina]

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^*, 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!!!!!!!

That must say something, right?

Am I awesome or what!



^* Try saying that quickly in a single breath. Phew.

[kriesel]

Originally Posted by retina

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

What. 89 = 1011001b, digit sum 4 = 2². 1279 = 10011111111b, digit sum 9 = 3². 2203 = 1000 1001 1011b, digit sum 6 = 2 * 3. ...

[retina]

Originally Posted by kriesel

Um, 89 = 1011001, digit sum 4.

I got too excited from my awesome discovery and had some extra nonsense in there. I already edited it.

[thyw]

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)

[Dobri]

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 51 known Mersenne-prime exponents (Mp);
- The total number of 5,761,455 8-digit Mersenne-number exponents (Mn); and
- The total number of 50,847,534 9-digit Mersenne-number exponents (Mn).

[Dobri]

Originally Posted by thyw

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)

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.

[Uncwilly]

[kriesel]

Originally Posted by Dobri

The peculiar thing is that the 51 known Mp have digit sums mainly in the lower half of the Mn digit sum distributions.

Well of course they do. 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.

Originally Posted by kriesel

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.

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 < 10⁹ = 999999937 not 999999999; the next smaller prime exponent 999999929 is a slightly higher digit sum and a near-repdigit. There are other near-repdigit primes with slightly higher digit sums. About a million lower, there's a near-repdigit 998999999 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.

Originally Posted by Uncwilly

Indeed. I was just thinking of using that.

[Dobri]

Originally Posted by kriesel

I note there's still not much of an answer re why base 10 digits. "Convenience" works for me.

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.

[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.