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This answer presumes you require exactly a 400-digit number.

This answer treats a 40-digit number as example.

Primes often can be found adjacent to the prime factorials. ref OEIS sequence A002110 (below).

Looking at that list, the 17th entry is a 20-digit number, P=32589158477190044730 . I first tried 1 less, which was composite. I next tried 1 more. [I used the established PariGP pgm's ispseudoprime function to test primality]. If this lacks success, progress upward from the start primorial until you find a prime, call it Pr1, by adding odd prime factors.

There are other similar maneuvers that could be used ... If the 2 primes need to be distinct, work up from P above by adding prime factors and checking primality. So, check (P+3), then (P+5), etc until you hit a 2nd prime. This is a bit more sophisticated than just checking higher odds ending in 1,3,7 or 9. In this case, adding 71 to our starting primorial gives the prime 32589158477190044803 another slightly higher (adding 179) is:


the product of the 2 is a 40-digit number.

The process leading to a 400-digit results mimics the example just shown.

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Q: Find two prime numbers if multiplied would generate a 400-digit number?
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