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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 ?

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    $\begingroup$ Maybe start in the other end? Multiply 25-digit primes so that you get something marginally larger than $44^{44}$. I don't know whether that is in any way easier. $\endgroup$ – Arthur Nov 21 '18 at 10:13
  • $\begingroup$ @Arthur I constructed such numbers, but they are too faw away from $44^{44}$ $\endgroup$ – Peter Nov 21 '18 at 10:14
  • $\begingroup$ Hi guys, what does $P24$ and $P25$ mean? $\endgroup$ – Lee Nov 21 '18 at 11:20
  • $\begingroup$ @Lee $P24$ just means a prime factor with $24$ decimal digits, analogue $P25$ $\endgroup$ – Peter Nov 21 '18 at 11:28
  • $\begingroup$ I can imagine coming up with a method that could nicely give some $k$ that is small but not necessarily minimal. But it seems like to find the minimal $k$, you will have to do some sort of brute force factorizing or sieving of the integers that immediately follow $44^{44}$. If you end up doing that anyway, I wonder if it's just as good to start with that. So with a computer, start factorizing/sieving $44^{44}+1, 44^{44}+2,\ldots$. When I say "sieving", that includes skipping over even numbers and other things like that. $\endgroup$ – alex.jordan Nov 24 '18 at 1:09

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