"hard" and computationally intensive

Mathematica digital keeps a certificate of each VERIFIED large prime number.

It's Impossible: They must be each found and then VERIFIED (the verification can require division which computers are unable to do due to the sheer length of digits of numerator and denomenator, in which case the number is NOT registered by Mathematica). To divide you need to know if a number is prime :)

It's the "not intersection" of all frequencies of all past numbers. There is no calculation possible. Only data of all past - which is far far far more than computers can ever handle when we past a certain threshhold.

I think you underestimate how much memory that would require (and how large a number this is). Let's just assume we store a list of numbers from 1, to your number, 1921773217311523519374373. We'll store each number as a 128-bit integer, as that's what most of the numbers would require (18446744073709551615 is the maximum unsigned 64-bit integer).

Storing 1921773217311523519374373 numbers, which are 128 bits each, would take ~3.0748371476984375x10^13 Terabytes of memory. I'm pretty sure this is significantly higher than current estimates for total computing memory in the world. This doesn't even factor in storing in their "PrimePi pair" along with them.

This rough calculation was done with:

In[1]:= N[1921773217311523519374373*UnitConvert[Quantity[128, "Bits"], "Terabytes"]]

Out[1]= Quantity[3.07484*10^13, "Terabytes"]

(This oversimplifies some things, while leaving out some obvious ways to make it more efficient. In any event, the suggestion simply is beyond what is practical)

Hi Kyle! Thank you very much for the explanation and for all the very helpful links. Wow, I didn't realize that this is a so difficult endeavour. But what I still do not understand, why they are trying to generate them over and over again? There were already calculated all primes up to the prime number which have 23 249 425 digits...why mathematica or wolfram alpha do not just use lists?

The Company Wolfram (which have many servers and cpu power) could just make (generate) a list up to the prime number which have (for example) 500 (or more) digits, in which the "nth's" simultaneously also been listed (like the program "primegen", it generates both at the same time and saved the results as a txt), and then simply assign the list to a function, then when a user type a prime number into these function, then must the function just only find these prime number in the list in which the nth's also been listed and finaly display the result...That would really help a lot of people.

Your best bet will be to use RiemannR to estimate PrimePi:

Round[RiemannR[1921773217311523519374373]]

(* 35007295017219325087811 *)

N[%]

(* 3.50073*10^22 *)

Based off the value for PrimePi[10^24], the relative error here is probably minuscule and result might be accurate up to the first 13 or 14 digits.

N[1 - RiemannR[10^24]/18435599767349200867866]

(* 9.22595*10^-14 *)

Whoa, is this the exact value of PrimePi[1921773217311523519374373]?

What type of machine did you use? Your timing seems much faster than the times quoted here.

which, however took about 108 hours to compute on a reasonably fast computer

Wow, that's massive! I thank you for that Ilian!

@Chip you were right, the variance is (actually) minuscule:

• 35007295017213256839290
• 35007295017219325087811

PrimePi on powers of 10 up to 10^27 have been calculated. See here and here.