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400-Digit Product of Two Primes

Date: 11/04/2002 at 15:24:38
From: Scott Slovak
Subject: Prime numbers


My question is: What two prime numbers when multiplied together will 
equal a 400-digit number?

I've tried researching, and I don't even know where to begin.  

Thank you very much,

Date: 11/05/2002 at 15:20:32
From: Doctor Wilkinson
Subject: Re: Prime numbers

Hi, Scott.  I can tell you how to find this.  

First of all, you want to go to the Dr. Math FAQ page, 

and click on "Prime Numbers." Then click on "Mersenne Primes: History, 
Theorems and Lists," and you will find a list of primes of the form 
2^ - 1 giving the number of digits. You should see one with 386 

Now if you multiply this by a number with 15 digits whose first two 
digits are 10, you will get a number with 400 digits, because the 
first digit of the big prime can easily be estimated to be no greater 
than 6. 

To find the 15-digit prime, you can start with 100000000000001 and
proceed to test the odd numbers one at a time until you find a prime. 
To do the testing, go back to the "Prime Numbers" page and click on
"The Prime Pages"; then click on "Check a Number's Primality".  

Good luck!

- Doctor Wilkinson, The Math Forum 

Date: 11/19/2005 at 22:58:06
From: Carmen
Subject: Prime Numbers Mega Multiplication

What is the product of (2^1279 - 1) * 100000000000031 = ??

I couldn't find the equipment to do the multiplication.

I found what I had to do from your prime page, as well as finding 
the 15 digit prime, but I can't find a calculator that can accomodate 
these huge numbers.

Date: 11/20/2005 at 09:51:12
From: Doctor Vogler
Subject: Re: Prime Numbers Mega Multiplication

Hi Carmen,

Thanks for writing to Dr Math.  I recommend the program GNU Pari,
which you can download from

By the way, if you are trying to make a 400-digit product of two
primes, you can also pick any two numbers x and y whose product is 
400 digits (such as two 200-digit numbers) and ask Pari for


For example, you could take

  x = random(10^200)
  y = random(10^200)

If you have any questions about this or need more help, please write
back and show me what you have been able to do, and I will try to
offer further suggestions.

- Doctor Vogler, The Math Forum 

Associated Topics:
High School Number Theory
Middle School Prime Numbers

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