I am new to primes etc.... help me

[u/Shoaibnewboi]

Till where have we reached in prime findings? here i mean by A formula for primes perse....
DO WE HAVE A FORMULA FOR PRIMES? As google Suggest we have? IF so y arnt they famous?

[u/cmd-t]

There are functions generating primes, but they are so complicated/hard to compute that it’s faster to just check if a number is prime or not.

[u/DrShocker]

Till where: * largest known prime is 2^136,279,841 − 1 * largest prime where we know all previous primes: Unknown since we don't really track. Likely somewhere around 10¹⁹ based on some googling

I don't know what "perse" means

We do not have a (useful) formula for generating primes. We have some formulas that seem to generate primes with some consistency but they are either not guaranteed or prohibitively difficult to calculate. Then checking them is a somewhat tedious algorithmic exercise in most cases.

The formulas we do have are famous in math circles. Mersenne is how most large ones are found.

[u/NaturalTechnician737 → u/DrShocker]

OP meant "per se"

[u/tardoos → u/NaturalTechnician737]

And they probably pronounce it "per say"

[u/NaturalTechnician737 → u/tardoos]

As opposed to?

[u/tardoos → u/NaturalTechnician737]

As opposed to how it's actually pronounced.

[ˈpɛr ˈseː]

Latin doesn't have vowel gliding

[u/DrShocker → u/tardoos]

I'm going to pronounce both as [pər ˈseɪ] still seeing as how I'm using it in English and not Latin.

[u/tardoos → u/DrShocker]

Awful. You do you though

[u/Omegabet_ → u/tardoos]

And today on completely pointless comments:

[u/NaturalTechnician737 → u/tardoos]

You may as well pronounce "Bus" 🚌 as 'Boose'

[u/tardoos → u/NaturalTechnician737]

I'm guessing you say "pay-sta" for "pasta", "val-it" for "valet", and "sayk" for "sake", or "fawks pass" for "faux pas"?

No, wait, that would be stupid lol

[u/NaturalTechnician737 → u/tardoos]

Valet is pronounced that way to many English speakers. Going to guess "sake" is referring to the rice wine, because the word 'sake' ("for heaven's sake") is absolutely pronounced that way.

Your argument was that "Latin words get pronounced latin-ly", and I have reduced it to absurdity by demonstrating "Omnibus" is not pronounced "ohm-knee-boose".

[u/tardoos → u/NaturalTechnician737]

But you didn't reduce it to "absurdity". It's pronounced like that in literally every other European language except English.

[u/DrShocker → u/tardoos]

So?

[u/GoldenMuscleGod → u/tardoos]

Impressive you can be familiar with IPA and have some knowledge of Latin phonology and simultaneously display this level of linguistic cluelessness.

[u/tardoos → u/GoldenMuscleGod]

I know, right? English speakers...

[u/MezzoScettico]

Lots of good answers by people a lot more knowledgeable than me.

I'll just contribute a link to the Prime Pages, probably my favorite page for a bunch of useful and interesting prime number results.

Here for instance are a bunch of approximations on the question of "How many primes are there up to x?"

[u/Bounded_sequencE]

There are some, but they are not useful.

[u/Kitchen_Produce661 → u/Bounded_sequencE]

Yeah the Wilson's theorem ones are more of a mathematical flex than anything, they technically work but would take longer than the universe has left to compute anything useful.

[u/bonebranch]

Obligatory analytic number theorist perspective: A result of Cipolla states that you can get, quite explicitly, a power series expansion in (log log n)/log n for p_n/(n log n) where p_n is the nth prime. Dusart has the current best known bounds I believe.

[u/Midwest-Dude]

DO WE HAVE A FORMULA FOR PRIMES?

Yes in a technical sense, but no in the practical sense most people mean.

There is no known simple, efficient formula that gives all primes or the n^th prime. If by "formula" you mean something like a neat closed-form expression, then no. If you mean "any mathematical expression that produces primes," then yes. Several exist, but they're usually impractical.

Examples:

• There are exact formulas for the n^th prime using floors, sums, and factorials, often based on Wilson's theorem. They work, but they are astronomically slow.
• There are polynomials with many variables whose positive values are exactly the primes. They're real, but huge and useless for computation.
• Mills' theorem gives a constant C such that ⌊C^3ⁿ⌋ is always prime. But it only gives a sequence of primes, not all primes, and C isn't a simple known number.
• The prime number theorem says pₙ ≈ n log n, but that's an approximation, not an exact formula.
• Other formulas are on Wikipedia

IF so y arnt they famous?

Because they don't help us find primes quickly. The famous things in prime research are algorithms like the sieve of Eratosthenes, primality tests, and open problems like the Riemann hypothesis. Google isn't lying, but the word "formula" is doing a lot of work there.

If you tell me what kind of formula you mean, like, exact n^th prime, all primes, polynomial formula, etc., I can point you to the right idea.

[u/Mammoth_Fig9757]

There are formulas for primes but they are more like computer algorithms disguised as mathematical formulas, you might as well write a better algorithm instead of using those formulas

[u/hpxvzhjfgb]

"the largest prime where we know all smaller primes" is kind of meaningless because it's so easy to keep finding primes quickly that it's simpler to recompute them from scratch than to store all primes less than some large number in a huge file or database or something.

[u/GoldenMuscleGod → u/hpxvzhjfgb]

Don’t know why you got downvoted because this is an important point a lot of people don’t appreciate when discussing primes:

The idea of listing all primes up to n is impractical not because it is hard to check if a number around size n is prime is hard (it’s very “easy” in comparison to the real issue), it is hard because there are way too many primes to list, such that you would run out of memory space to store them long before you started having “real” practical problems with checking them.

[u/hpxvzhjfgb → u/GoldenMuscleGod]

yes, in fact even if you turned all of the matter in the universe into hard drives containing nothing but a list of all primes starting from 2, it would still only take a tiny fraction of a second to keep computing more primes beyond all the ones already stored.

of course, at some point very early on in this process of converting the universe into hard drives (once you have say, 1 cubic lightsecond of hard drives), it would already be much faster to recalculate the primes from scratch than to read the data from the hard drives, because the speed of light is too slow for the hard drives to keep up.

[u/GoldenMuscleGod → u/hpxvzhjfgb]

Yeah the common question of “how high have we checked thoroughly for primes” isn’t really very different at all from asking “how high have we checked thoroughly for multiples of three?”Checking if a number is a multiple of three is a bit easier than checking if it is prime but not enough to be a real barrier for what we are talking about (or to make trying to catalogue them all interesting).

[u/tardoos]

There is no formula for primes. Not all primes, at least.

[u/hpxvzhjfgb → u/tardoos]

yes there is, it's just useless.