Primes from concatenated perfects +/- 1

[Dr Sardonicus]

Originally Posted by axn

Here are the statistics for the first 20 digits:
[snip]

As you can see, the number of primes just keep going up. So we don't expect there to be any further gaps. This is not a rigorous mathematical proof, of course, but...

Hmm. Proof may be difficult, but it might be possible to estimate the number of patterns, and then the fraction of these patterns that one might expect to be prime. I'm guessing that concatenations of the small ones might account for most of the patterns.

Let's see, there's 1 pattern of all sixes. The number of patterns formed of 6's and 28's is... I don't know. It's obviously greater than the number of integer solutions (x1, x2) to x1 + 2*x2 = d, where d is the number of digits. Use 496 as well, and you're obviously greater than the number of integer solutions (x1, x2, x3) to x1 + 2*x2 + 3*x3 = d.

At any rate, if N = N(d) is the number of d-digit patterns, then you can start with the fraction of d-digit numbers that are prime, approximately 1/(d*ln(10)), which is based on the "assumption of ignorance" about the d-digit numbers you're looking at. Then, multiply by a "knowledge factor" based on the non-divisibility of your candidates by small primes. The candidates are all odd, so you can multiply by 2 right off the bat. You might be able to estimate the fraction of candidates divisible by 3 or 5, and maybe 7, and adjust the 'knowledge factor" accordingly. Then, see if the result is anywhere close to the fraction of digit patterns that are actually prime.

[science_man_88]

Originally Posted by davar55

Really good, and convincing. The number of primes
seems to approximately double for each digit, so
maybe there's a proof possible in there somewhere...

my first thoughts are maybe think about mod 6 and which ones can be in the proper modular remainders mod 6 first. edit: @Sardonicus the number of patterns is related to the number of partitions of d using only the number that represent the lengths of the perfect numbers obviously.

[axn]

Originally Posted by Dr Sardonicus

Hmm. Proof may be difficult, but it might be possible to estimate the number of patterns,

We can calculate the exact number of patterns without actually generating the patterns. Assuming no odd perfect numbers exist, we know all the perfect numbers < about 22 million digits. So we can calculate the number of patterns with up to 22 million digits. I expect the calculation to take a few minutes.

[Dr Sardonicus]

Originally Posted by axn

We can calculate the exact number of patterns without actually generating the patterns. Assuming no odd perfect numbers exist, we know all the perfect numbers < about 22 million digits. So we can calculate the number of patterns with up to 22 million digits. I expect the calculation to take a few minutes.

Yes, I'm sure the calculation is doable. I thought that there might even be a formula for the number of patterns involving concatenations of strings of several given lengths. However, I'm not so hot at combinatorics, and failed to come up with such a formula even for concatenations of a string with 1 character and a string with 2 characters.

The reason I thought of estimation, was to determine in general how many primes with d digits we might reasonably expect to find. If there are N = N(d) patterns, then a first stab at estimating the number of primes is

2*K*N(d)/(d*ln(10))

where the factor 2 is for the +/- 1 (two candidates for each pattern), and K is the "knowledge factor" based on how often the candidates are indivisible by small primes. Since all the candidates are odd (as far as we know), K is at least 2.

If we have a large enough lower bound for N(d), we can say with some confidence that there are likely primes of the required form for all sufficiently large d, even if we can't prove it.

[science_man_88]

Originally Posted by Dr Sardonicus

Yes, I'm sure the calculation is doable. I thought that there might even be a formula for the number of patterns involving concatenations of strings of several given lengths. However, I'm not so hot at combinatorics, and failed to come up with such a formula even for concatenations of a string with 1 character and a string with 2 characters.

The reason I thought of estimation, was to determine in general how many primes with d digits we might reasonably expect to find. If there are N = N(d) patterns, then a first stab at estimating the number of primes is

2*K*N(d)/(d*ln(10))

where the factor 2 is for the +/- 1 (two candidates for each pattern), and K is the "knowledge factor" based on how often the candidates are indivisible by small primes. Since all the candidates are odd (as far as we know), K is at least 2.

If we have a large enough lower bound for N(d), we can say with some confidence that there are likely primes of the required form for all sufficiently large d, even if we can't prove it.

as order is important it's permutations, not combinations. in fact it's restricted compositions

[VBCurtis]

Originally Posted by science_man_88

as order is important it's permutations, not combinations. in fact it's restricted compositions

Your reading comprehension could use some help. He said combinatorics, not combinations. You've been reading the good Dr's posts for a few months now; surely you could have reached the conclusion by now that you and I are unlikely to have corrections to make to the math content of his posts!
Your link even has combinatorics in parens....

[Batalov]

"Isn't he a bit like you and me"
A true Socratic theory, that.

[ET_]

Originally Posted by Batalov

"Isn't he a bit like you and me"
A true Socratic theory, that.

No, that sentence leads you nowhere, man...

[Dr Sardonicus]

Originally Posted by VBCurtis

Your reading comprehension could use some help. He said combinatorics, not combinations. You've been reading the good Dr's posts for a few months now; surely you could have reached the conclusion by now that you and I are unlikely to have corrections to make to the math content of his posts!
Your link even has combinatorics in parens....

Lucky for me, "combinatorics" is what I meant. I say "lucky" because, while I'm often able to to avoid mathematical errors, I am very prone to make typing mistakes. My posts have been littered with these.

As an indication of how bad my typing skills are, I offer the following anecdote based on alternative facts: I tried using a typing training program, but one day it stopped, displayed a message that it was awarding me the Black Belt in typographical errors, and, with a wail of despair, removed itself from my system.

[Batalov]

Originally Posted by Dr Sardonicus

... I am very prone to make typing mistakes. My posts have been littered with these.

Oh, in the world of modern computerized devices, it gets even worse. The program you are using to type will attempt to "correct" what you typed into what 90%+ of people type. The trouble with this is that what 90%+ of people type is absolute garbage, but computer assumes that you are one of those people and "corrects" a rare word into some banality.

Don't even get me started on typing on genomics-related messages at work. Every other word is rare and the email message becomes totally garbled. One can never type a message and hit "Send". Oh no. That would be too productive. One has to re-read and revert most spellings (and that is even with updating the local dictionary; when spell-checking: "add to dictionary", not "ignore / ignore rule"). Bleh-bleh-bleh. And even when you hit "send" the message, you are not sure that it will be received; the hospital server will silently quarantine it (and you will only get a summary message after the day is over) - because some combination of letters looked to yet another computer program like a PHI violation.

[LaurV]

Haha, this reminds me of Ellen...