Number of digits in prime

[storm5510]

Originally Posted by odicin

Added 133982 : 993*2^1717566-1 (517042 digits) on port 2000

A little known program call mprint can generate the expression above. The order has to be different before it will accept it. "2^1717566-1*993" without the quotes.

This must be longer than 517,042 digits. Log(2) * 1717566 = 517,039. This is without the exponentiation, multiplication, and the subtraction of 1. I captured mprint's output to a file. It is 6,535,835 bytes in length. There are no extra characters beyond a CRLF at the end which accounts for only two bytes. mprint wrote it all as a single line. Only Notepad++ could open it.

[kar_bon]

Using pfgw with

pfgw64 -od -q"993*2^1717566-1" >prime.txt

puts out the number in decimal expansion in a file of 517,097 bytes:
including the expression in front and some spaces at the end.
Deleting those will give the exact 517,042 digits (using only the standard Editor in WIN).
The number of digits you can also use WolframAlpha.

[VBCurtis]

Originally Posted by storm5510

This must be longer than 517,042 digits. Log(2) * 1717566 = 517,039. This is without the exponentiation, multiplication, and the subtraction of 1.

How many digits will multiplying by 993 add? Hint: 3. How many digits will subtracting 1 add?

I mean, you did the hard work to get 517,039... it boggles my mind that you faceplant when trying to get from there to 517,042.

[retina]

Python also agrees:

>>> len(str(993*2**1717566-1))
517042

[storm5510]

OK. I will take your word for it. No dispute.
.
.
It appears mprint assumes "2^" in front. What I actually did was 2^2^.....

Running mprint without parameters produces this:

USAGE: mprint p [base] [perfect]
       Outputs 2^p-1 in base base to stdout
       If you add the third parameter, it outputs 2^(p-1)*2^(p-1)

Yes, "base" appears twice in the second line.

Oh well, more of a curiosity than anything else.

[ATH]

mprint only print Mersenne numbers 2^p-1 but can print it in different bases for example decimal, hexidecimal etc.

To get the number of digits in k*bⁿ + c:
First ignore "+ c" unless c is HUGE and/or b and n are very very small.

log (k*bⁿ) = log(k) + n*log(b)

In your case: log₁₀(993*2^1717566-1) = log₁₀(993) + 1717566*log₁₀(2) ~ 3 + 1717566*log₁₀(2)


In general: log(a*b) = log(a) + log(b) and log(a^b) = b*log(a)

[kruoli]

Originally Posted by storm5510

Yes, "base" appears twice in the second line.

This likely means in "base {value of parameter [base]}" and that's why it is written twice.

[a1call]

\\ I learned this from SM. In PARI-GP
allocatemem()
allocatemem()
allocatemem()
allocatemem()
length(Str(2^82589933 - 1))
##

(13:41) gp > \\ I learned this from SM. In PARI-GP
(13:41) gp > allocatemem()
*** Warning: new stack size = 16000000 (15.259 Mbytes).
(13:41) gp > allocatemem()
*** Warning: new stack size = 32000000 (30.518 Mbytes).
(13:41) gp > allocatemem()
*** Warning: new stack size = 64000000 (61.035 Mbytes).
(13:41) gp > allocatemem()
*** Warning: new stack size = 128000000 (122.070 Mbytes).
(13:41) gp > length(Str(2^82589933 - 1))
%1 = 24862048
(13:41) gp > ##
*** last result computed in 8,767 ms.

Google agrees:

Largest known prime number

2^82,589,933 - 1
The Great Internet Mersenne Prime Search (GIMPS) has discovered the largest known prime number, 2^82,589,933 - 1, having 24,862,048 digits.

51st known Mersenne prime found | Announce - News

[kar_bon]

Originally Posted by storm5510

USAGE: mprint p [base] [perfect]
       Outputs 2^p-1 in base base to stdout
       If you add the third parameter, it outputs 2^(p-1)*2^(p-1)

Yes, "base" appears twice in the second line.

The perfect number of a Mersenne prime 2^p-1 should read as 2^p-1*(2^p-1) instead.

[Batalov]

Originally Posted by kar_bon

Using pfgw with

pfgw64 -od -q"993*2^1717566-1" >prime.txt

pfgw64 -od -q"{this or many other forms, including U(), gcd() etc}" | awk '{print length($2)}'
and done
___________________

The perfect number of a Mersenne prime 2^p-1 should read as 2^p-1*(2^p-1) instead.

This shows care much care was put into mprint program. Simply don't use it; use better and much more developed tools: python, Pari/gp, pfgw and so on.