Large prime numbers in encryption? I have recently been reading about encryption and the importance of prime numbers and I have some questions that I would really appreciate some answers to, if possible:


Is it correct that when creating encryption keys you take one large prime number, and then multiply it by another prime number to leave you with an even larger prime number? 
If 1 is correct, then is it correct to say "the reason for the large prime number calculation is it is very difficult and time consuming to work out what the initial prime numbers were used in the original calculation"?
What constitutes as a large prime number?


The reason for this question is I have been doing some reading about encryption as stated above, and assuming the above statements are correct I think I have a really simple formula for working out what the original prime numbers used were. I have tested this prime numbers all the way up to $299993$. And I must be doing this wrong because it can not be this simple, I assume someone will point out where I am going wrong.
Is it correct that when creating encryption keys you take one large prime number multiple it by another prime number to leave you with a even larger prime number?


Any number that is a multiple of two primes is by definition not prime. This creates a semiprime: a number that has only two prime factors.

This approach is only used in a few cryptosystems, the most common of which is RSA. Many other cryptosystems exist that do not rely on integer factorization. Some of these systems (e.g., AES, ChaCha20) are symmetric algorithms unlike RSA, and some (e.g., ECC) are asymmetric like RSA. RSA is gradually being phased out in favor of modern systems based on elliptic curves.


  If 1 is correct, then is it correct to say "the reason for the large prime number calculation is it is very difficult and time consuming to work out what the initial prime numbers were used in the original calculation"?


Yes. As far as we know, integer factorization is a hard problem.


  What constitutes as a large prime number?


2048-bit RSA is a typical current recommendation. In this case, the modulus (e.g., the semiprime) is the part that is 2048 bits, so each prime is roughly 1024 bits long. For scale, a 1024-bit prime will be over 150 decimal digits long, so they are quite large.