Special number near $5^{5^3}$

Question

I search the number nearest to $5^{5^3} = 5^{125}$, which is product of two $44$-digit primes.

The direct method is : Begin with $N = 5^{125}$. Increase $N$ until the desired number is found. Whenever a small factor is found, the number need not to be factored completely. Do the same with decreasing, beginning with $N$. The two found numbers have the form $N-a$ and $N+b$. Choose the number nearer to $5^{125}$. The method is trivial, but very time-consuming.

Is there a way to find the number faster?

Answer 1

I believe the closest number to $5^{125}$ which is product of two 44-digit primes is $5^{125}-588$:

5^125 - 588 = 43665185956168249163369594154581467144074421 *
              53841261640441238035516896393833679276609797

It was found by testing the numbers $5^{125}\pm 1$, $5^{125}\pm 2$, ...; checking if they have any factor smaller than $10^9$ first and running the general-purpose factorizer on them otherwise.

Answer 2

(This is not an answer) There are about $8\times 10^{41}$ primes between $10^{43}$ and $10^{44}$, hence there are about $3.2\times 10^{83}$ integers of the form $pq,$ the largest one is about $10^{88}$,the least one is about $10^{86},$ the average distance is about $3\times 10^4$, so you need try about $3\times 10^4$ times to find the nearest solution.