what prime number can be multiplied by itself to get a 400 digit number?

Answer 1

Finding large prime numbers is a difficult and specialized branch of mathematics. Because of that, you will need to find large primes that have already beed discovered. A method of using 2 primes of 200 digits each is almost correct. Be careful though since 2 small 200 digit primes will produce a 399 digit number. The easiest way to tell is to express the primes in scientific notation. Look at two hundred digit numbers: 1 x 10^199 and 2 x 10^199 (these are each 200 digits - a 1 with 199 zeroes and 2 with 199 zeroes - but not prime) multiply them together and you get 6 x 10^398 which is a 399 digit number. To get a 400 digit number from two 200 digit numbers, the mantissas must be large enough for you to carry a digit over during multiplication.

Another way to do this is to find a prime that is nearly 400 digits and multiply it by a suitably smaller prime. The Mersenne Primes are a good source for the big number. The 15th Mersenne Prime is 2^1279 - 1 = 1.041 x 10^385 at 386 digits ( http://primes.utm.edu/mersenne/index.htm...

Now you need a prime with a 10^14 exponent or a 15 digit prime to multiply, I found 112272535095293 or 1.123 x 10^14 (http://www.alpertron.com.ar/googol.pl?di... but there are plenty more around. Multiplying these together gives 1.169 x 10^399 which is a 400 digit number.

So here is one answer to your question:

58021664585639791181
18402595044024839822
61360695169382324936
87505822471836536824
29882273371034225069
77399968259382326419
40670857624514103125
98613405099769716012
73015479957884681378
87651823707102007839