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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?

Piquito
  • 29,594
  • Actually $x=13$. Also $(x=14; 19, 31, 613, 276964579)$, $(x=24; 3529, 821461, 838069, 411605923)$, $(x=36; 103, 922639, 3480881723167, 3023024732613877)$, x=40, 44, 46, 48, ... are some other solutions. – kennytm May 07 '16 at 16:23
  • @kennytm: Certainly, but also using brute force to which one example suffices. Regards. – Piquito May 07 '16 at 16:34
  • You are asking for numbers of the form $10^x+3$ which are square-free with just 4 prime factors. I cannot think of any elegant way of approaching such problems. But it is obviously easy to find solutions by direct computation. – almagest May 07 '16 at 18:38

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