I search the smallest positive integer $k$, such that $44^{44}+k$ splits into three distinct prime factors each having $25$ decimal digits.
The $21$-digit number $k=621725397145122340237$ does the job, but it is hard to imagine that there are no smaller solutions.
Enzo Creti found the very near miss :
$$44^{44}+202693 = P24\cdot P25\cdot P25$$
Is there any better way than just factoring all numbers until the desired number is found ?