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It is impossible for the product of two prime numbers to be prime. It is impossible for the sum of two prime numbers to be prime as long as one of the numbers isn't 2.
since 1202 in a publication
Both of those lists are infinitely long.
The sum of three odd numbers will be odd, whether they are prime or not. The sum of three prime numbers can be even as long as one of them is 2.
I'm sure there are more than 2 prime numbers that are 400 digits long.