I have an idea for a system that would be outsourcing some brute force calculations to many users in hopes of finding divisors of a number. However, there is a possibility that a given number would be prime, or that it would not have a divisor in a given range.
For example, a computer might be tasked to look for a divisor of a number 2^n-1 (which would be a very large number) and whether there is a divisor in range x*10^10 to (x+1)*10^10.
Proving that one has found a divisor is easy - they just need to submit that divisor and the number can be checked against it. However, how should the users go about proving that there is no divisor to be found in a given range? Obviously one couldn't just trust a user in their claims - this would open a possibility for malicious reports messing with the system.
So far I was pondering a few options - submitting overlapping regions to various users for checking and comparing their results, challenging the users to find the smallest / biggest remainder of a check they would be performing in the range and so forth. Are there any approaches to the presented problem that are more vetted in the crypto world than "an idea that sounds okay"?