What is the smallest positive integer $k$ , such that $$42!+k$$ splits into three primes with $18$ digits ?
My currently best result is $$k=31449145975909$$ http://factordb.com/index.php?query=42%21%2B31449145975909
I found this result by choosing random primes $p$ , then choosing $k$ minimal with $p\mid 42!+k$ and factoring the rest. This gives better results than just dividing $42!$ by two random primes , but still the results are probably much too high. Is there a better way than brute force ?