I intuited a probably-inaccurate trick with exponentially raising a small prime (e.g 7*7*7etc), doubling the multiple (*2), and subtracting 1 (to get an odd number).
That got me two substantially-different primes and narrowed my numbers a lot, which then became guesswork.
The number you come up with in that way is at least guaranteed to not be divisible by 2 or 7. Obviously that doesn't mean it's prime but it definitely helps your odds.
Also usefully, (2(a+1)^n-1) % a = (2a(a+1)^(n-1) + 2(a+1)^(n-1) - 1) % a = 1. So if you choose one of the 3k+1 primes, it's also guaranteed not divisible by 3.
That got me two substantially-different primes and narrowed my numbers a lot, which then became guesswork.
My guesses in the end: