I have resolved, brute force, the following problem someone asked me:
Solve the Diophantine equation $$10^x=yzwt-3\space \text{where}\space \space y,z,w,t \space \text {are distinct primes}$$ $$ **********$$
I found the solution $$(x,y,z,w,t)=(12,13,29,547,48492137)$$ Note that the prime $t$ is rather large.
Is there any method to solve this with mathematical deduction? In particular to calculate another solution or ensure or deny that there are finitely many solutions?