# What is the smallest $k$ , such that $44^{44}+k$ has the desired property?

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 ?

• 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. – Arthur Nov 21 '18 at 10:13
• @Arthur I constructed such numbers, but they are too faw away from $44^{44}$ – Peter Nov 21 '18 at 10:14
• Hi guys, what does $P24$ and $P25$ mean? – Lee Nov 21 '18 at 11:20
• @Lee $P24$ just means a prime factor with $24$ decimal digits, analogue $P25$ – Peter Nov 21 '18 at 11:28
• 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. – alex.jordan Nov 24 '18 at 1:09