@Carl Love 

Thanks

So, the latest biggest found primenumber can be expressed as 

 10^ 24,862,048
Its lot bigger then number of atoms in our universe 10^80 

@tomleslie 
Thanks,

I do get no plot, but this..
`Standard Worksheet Interface, Maple 2021.1, Windows 10, May 19 2021 Build ID 1\
539851`
INTERFACE_PLOT(CURVES(Matrix(932,2,{(1, 1) = HFloat(1.), (2, 1) = HFloat(3.6235\
4490699999987), (2, 2) = HFloat(2.), (3, 1) = HFloat(5.90627730299999953), (3,  ..and so on.

@janhardo 
The primecount function and his two approximations graph ..

• John E.Littlewood shows in 1914 that there are  infinity intersections with the two aproximations graphs and prime count graph. 
   
• Stanley Skewes claims that first intersection point is on x= 10^(10^10^34)
• Carter Bays &Richard Hudson claiming in 2000 that first intersection lies before
  number 1,39822 x 10^316 ( 10^80 atoms in universe) 

Problem is now how to find this intersection point with computer and how far is this from the last found  Merssene exponent  prime number?

@Ronan 

Thanks,
Memorize some basic examples will help.

@Carl Love 

Thanks!

With this i can express a Mersenne prime number as number with base 10^MersenneDigits 

@tomleslie 
Thanks,

Yes, amazing comparison.

It are big numbers, but compared with infinity ?
Distribution graph of Mersenne Primes found by GIMPS,can give perhaps a direction for the search interval for the next new Mersenne prime ?
List of known Mersenne prime numbers - PrimeNet

You know the number on numberline from the latest Mersenne prime found in comparison with a 1000 number segment : position ?

Is there a faster algorithm to find for calculating the number of primes given to a certain number ?

@janhardo 
Impact has also the used SDD on the calculation speed : read 560 Mb/s and write 510 Mb/s is now used on my computer.
There are two slots on motherboard for M.2-sdd and  using a read 3500 Mb/s and write 2300 Mb/s sdd can make a difference?
Using IOPS as number of read/write performance..
IOPS read 360.000 and write : 390.000 as example here ( WD blue for 60 USD) ( see here the kingston fury(new) m.2-sdd 4 TB  : cost more then 1000 USD and 1,000,000 IOPS read /write) 
The cheapest way for now to get performance improvement is upgrading to a faster M.2-sdd
Maple must be then be installed on this faster SDD. 

Corsair MP600 PRO XT, 4 TB SSD Zwart, CSSD-F4000GBMP600PXT, M.2 2280, PCIe 4.0 x4 (alternate.nl)
A PCIe Gen4 x4 controller for exceptional data performance delivers phenomenal read, write and response speeds that make standard M.2 SSDs pale in comparison.

Well, using asus fast(est) gaming motherboard and a amd fast cpu  and fast sdd and fast memory can give a improvement for the Mersenn prime calculation.
Now i know why it take so much time to discover a new biggest unknown primenumber and knowing where to start for searching for sure ( gap estimation ) ? 
 

@mmcdara Thanks! There is a procedure what can calculate the latest known Mersenne prime, although it takes some time. I did calculate Mersenne prime _number 40 and it took roughly 7,5 hour ( i must check this ). Dividing the numberline  in 1000 number sections to the latest known mersenne prime is also probably impossible, because the number is very big. At the moment i am trying to express this latest mersenn prime number in a power of 10 ?

@janhardo 

restart;
BitMod:= (N::posint, n::posint)->
local B:= Bits:-Split(N, n);
    `if`(nops(B) < 2, `if`(ilog2(N+1) = n, 0, N), thisproc(add(B), n))
:
(*---
Check whether (2^p-1) is prime by Lucas-Lehmer algorithm.
-----*)
IsMersennePrimeExponent:= proc(p::And(prime, Not(2)))
local s:= 4;
    to p-2 do s:= BitMod(`*`(s,s), p) - 2 od;
    evalb(s=0)
end proc;
IsMersennePrimeExponent(2):= true
:
Typesetting:-mprintslash([(IsMersennePrimeExponent := proc (p::And(prime,Not(2)
)) local s; s := 4;  to p-2 do s := BitMod(`*`(s,s),p)-2; end do; evalb(s = 0);
end proc)],[proc (p::And(prime,Not(2))) local s; s := 4;  to p-2 do s := BitMod
(`*`(s,s),p)-2; end do; evalb(s = 0); end proc])

CodeTools:-Usage(IsMersennePrimeExponent(2));
memory used=0.77KiB, alloc change=0 bytes, cpu time=0ns, real time=0ns, gc time=0ns
true

CodeTools:-Usage(IsMersennePrimeExponent(44497));
memory used=2.58GiB, alloc change=-4.81MiB, cpu time=3.73s, real time=3.73s, gc time=171.88ms
true

#This Mersenne prime was discovered by computer in 1979:
CodeTools:-Usage(IsMersennePrime(44497));
memory used=2.65GiB, alloc change=0 bytes, 
cpu time=10.58s, real time=12.60s, gc time=6.48s

                              true
-----------------------------------------------------------------
The second procedure seems to be much faster ....
-------------------------------------------------------------------
CodeTools:-Usage(IsMersennePrimeExponent(110503));
memory used=15.73GiB, alloc change=0 bytes, cpu time=25.58s, real time=25.62s, gc time=609.38ms
true

;
The first million Mersenne exponent found on 3 sept 1996 : 1,257,787, second : see calculation
How long does it take 

CodeTools:-Usage(IsMersennePrimeExponent(1398269));
memory used=4.30TiB, alloc change=0 bytes, cpu time=92.42m, real time=92.44m, gc time=61.94s
true

CodeTools:-Usage(IsMersennePrimeExponent(2976221));
memory used=19.62TiB, alloc change=27.47MiB, cpu time=7.38h, real time=7.39h, gc time=6.09m
true

-----------------------------------------------------------

2^2,976,221 -1  Mersenne prime took 7.38 h
How big the number is, as expression of a power of 10 ?
Interesting is the latest known Mersenne prime as power of 10. 

@janhardo 

Good news ..your estimation :
-------------------------------------------------------------------------------------------------

I predict that your prime 1398269 would take 4 hours 12 minutes on my computer, an Intel Core I-7 @ 2.7 GHz. Extra processors have little effect on this (but can be used to process multiple primes simultaneously).

--------------------------------------------------------------------------------------------------

It takes about 1 1/2 hour on my computer ! ..wow ( a fast motherboard )

CodeTools:-Usage(IsMersennePrimeExponent(1398269));
memory used=4.30TiB, alloc change=0 bytes, cpu time=92.42m, real time=92.44m, gc time=61.94s
true
There is a list of primes for 1000 billione  List of prime numbers up to 1000000000000 (free.fr)
Its 10^9  numbers ,compare this with say 2^(10^6) numbers 

@Carl Love 

Your improved procedure IsMersennePrimeExponent()  has now becomes much faster !
Prime 110503 :real time=25.62 s 
I am calculating now the 1398269 prime : how long does it take? 

Suppose you do have the fastest prime test algorithm possible, then there is the question : where lies the greatest primenumber on the numberline?  
82,589,933 (Mersenne prime)     2^ 85,589,933 - 1  =  prime number.(but not always)
This the largest primenumber known now on the numberline.
For Pi(X) (generally , how many primes are there less than any given number X ?)
What could be a strategy to seek further for a bigger prime?
for 10^8 =100 million , there are 5,761,455 primes (quess?) and prime number 88,589,933 is one of them

I do see that for this number segment : 1000 numbers, in old studymaterial 10^7-1000  to 10^7 on the numberline there are 53 primenumbers to find.
The number of primes goes from  0-1000 = 168 from steps of 1000 numbers to  10^7-1000- 10^7 = 53  
The 1000 number segment where (2^82,589,933-1 ) in is positioned : how many primes are here in to find? 

Meanwhile now ,the computer is calculating further for prime 1,398,269 : how long does it take ? 
Its going to the 3 hours now ...

@janhardo 

Tried the Mersenne prime , but memory allocation error from Maple. 

This is the first in the green table on internet. 

Using  amd 6 core 5 serie 1600 CPU ( a slow one)  and 16 Mb memory.
( i made always myself a computersystem )

Is there a quess how much memory is needed for the prime calculation?

@Carl Love 

Amazing...thanks , don't tell me that we find now  the largest primenumber now with Maple..
I will make some calculations with it and look how far it goes 

see List of known Mersenne prime numbers - PrimeNet

I am interested in the Riemann hypothese and want to know more about advanced mathematics  
Someone who has made something out of the primes :

Is This the Most Maple Prime Post Ever on MaplePrimes? - MaplePrimes

@acer 
Thanks

I do see a basic do loop example here what not difficult now to understand is

@Carl Love 

Thanks
Well, still no idea why this is assignment operators are introduced in Maple, but that question cannot maybe easy  be answered?