If you use existing factoring functions this can be solved in minutes
for 200 bit random numbers. Factoring functions from Pari/GP for example.
The algorithm below is not just factoring consecutive numbers. See below.
Here is the general algorithm which has been converted into a working
program:
Generate W, a random 200 bit number
sievelimit = 250
primelimit = 2000000
precompute pr = product of first primelimit primes
For n1 = W+1 to W+sievelimit
If n1 is of the form 6k+1 or 6k+5
if gcd(n1,pr) is 1
f = factor(n1)
if number of factors of n1 is 2 and both >2^50
print Solution f
Loop
The line:
if gcd(n1,pr) is 1
is a quick way to skip numbers which have prime factors below the 2 millionth prime.
The precompute pr step is optimized in Pari/GP and only takes 7.785 seconds on slow hardware.
For the worst case 200 bit number the factor function in Pari/GP uses MPQS method and takes less than 30 seconds on old hardware.
Here is a random 200 bit number W:
1567470448908230034126591070540826459978233372650796513704199
The program only factors one number W+10 and finds the solution
p1 = 5346955435967300929
p2 = 293151956787285973328498761492202409914321
p1 is 63 bits, p2 is 138 bits, n = p1*p2
|W-n| = 10, |W-n|< 2^12