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 Tomws 2020-05-02 15:45

Not the largest prime

My apologies, I dont know my way around this forum. Could someone re-direct this 'post' to the right place/ forum or do what is necessary? Or even answer it if there is a well known answer? Presumably the answer will get bigger as more primes are discovered

[U]This is not a Mersenne question. [/U]

What is the largest prime number [B]for which all smaller primes are known[/B]?

And secondly given the well known proof that there are an infinite number of primes, how many digits are there in the answer if all primes less than or equal to 'my' prime are multiplied together?

 LaurV 2020-05-02 16:36

Do you mean like they are all stored in a list, somewhere, or that we can produce them on request? If you [URL="https://primes.utm.edu/prove/prove1.html"]read a bit[/URL] about how the small primes are found, and how their primality is proven, you will find out that generating small primes is faster than reading them from an external memory support (like a hard disk) and due to that fact, nobody bother to "store" them. And about producing them, well, we can "produce" any sequence of primes in order below the limit where we can prove primality for general form (thousands of digits). So, I guess, we "know" all primes smaller than the "record" of primo or ecpp, just that they are not "stored" anywhere, and no human have seen most of them yet... (ha, is the prime "found" when a human sees it? who made this rule? :razz:)

 Prime95 2020-05-02 19:32

In other words, there is no such prime below which all primes are known.

No one bothers to find such a prime because anyone with a computer could calculate the next prime in a few milliseconds.

I guess the closest "pseudo answer" to your question is ~10^27. Someone posted on this forum a count for the exact number of primes less than 10^27.

 rudy235 2020-05-02 19:43

[QUOTE=LaurV;544460]Do you mean like they are all stored in a list, somewhere, or that we can produce them on request? If you [URL="https://primes.utm.edu/prove/prove1.html"]read a bit[/URL] about how the small primes are found, and how their primality is proven, you will find out that generating small primes is faster than reading them from an external memory support (like a hard disk) and due to that fact, nobody bother to "store" them. And about producing them, well, we can "produce" any sequence of primes in order below the limit where we can prove primality for general form (thousands of digits). So, I guess, we "know" all primes smaller than the "record" of primo or ecpp, just that they are not "stored" anywhere, and no human have seen most of them yet... (ha, is the prime "found" when a human sees it? who made this rule? :razz:)[/QUOTE]

That is a good answer but more in the spirit of the questioner Tomas Oliveira e Silva fron the University of Aveiro Portugal calculated all primes below 4*10[SUP]18[/SUP] in July 2014

The article in Mathematics of Computation is entitled [I][B][FONT="Fixedsys"]Empirical verification of the even Goldbach conjecture and computation of prime gaps up to 4·10^{18}
[/FONT][/B][/I]

As to the second question of the OP that would be Primorial (4*10[SUP]18[/SUP]) or (4x10[SUP]18[/SUP])# approximately e[SUP]4*10^18[/SUP]

 R. Gerbicz 2020-05-02 22:51

[QUOTE=Prime95;544470]
I guess the closest "pseudo answer" to your question is ~10^27. Someone posted on this forum a count for the exact number of primes less than 10^27.[/QUOTE]

False, to get pi(n) you don't need const*pi(n) operations, see [url]https://en.wikipedia.org/wiki/Meissel%E2%80%93Lehmer_algorithm[/url] .

To get a more real example: when we computed the large prime gaps on this forum up to 2^64 then we actually computed roughly 1/7 of all primes up to this bound.

 Dr Sardonicus 2020-05-03 01:46

[QUOTE=Tomws;544457]And secondly given the well known proof that there are an infinite number of primes, how many digits are there in the answer if all primes less than or equal to 'my' prime are multiplied together?[/QUOTE]
Believe it or not, this question runs you smack-dab into the Prime Number Theorem (PNT)!

Alas, the answer is not known as precisely as one might wish, but still...

What you want is a "reasonable" estimate for

log(2) + log(3) + ... + log(p),

where p is the largest prime <= X, X some "large" positive number. In PNT-related literature, it is the [i]natural[/i] log, log to the base [i]e[/i], or ln, that is used. And a statement equivalent to PNT is

$$\sum_{p\le X}\ln(p) \;\sim \; X$$

(ratio of RHS to LHS approaches 1 as X increases without bound) so that the number of decimal digits in the product of the primes up to X is something like X/ln(10).

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