What is the smallest prime number that divides some number of the form 424242424....42+1 or the form 42424242....42-1?

We're just looking at numbers such as

4242 + 1

42424242 + 1

4242 - 1

424242 - 1

etc.

In other words, numbers where a plus (or minus) 1 is being added to a string of digits with a "42" pattern repeated at some length. I just tested a few random ones in WA to see if these sort of numbers were prime or not. Some are and some aren't. Thus...there doesn't appear to be any "smallest" factor - in general - that divides any number of this sort.

Maybe there is some algorithm that will tell us when such numbers are prime, but I'm not aware of any........

What are all the dots for?

4242+1  is prime

424242424242 +1  isn't prime 

4242 - 1 is prime

424242 -1 isn't prime

Thus....I'm not sure if there is a general rule for determining prime divisors here.

Anyone else have a general proof..... (or disproof???)

Chris, I do not understand what is being asked.  Can you explain it to me?

I think the questioner wants to know, if we add or subtract 1 from a string of digits having a pattern of 424242 etc., do such numbers  have some  prime factorization pattern???  Using WolframAlpha and choosing some at random, it seems as if some are prime numbers and some aren't. Thus, there is no "smallest" factor that I can detect. Perhaps there is some algorithm to tell when these may be prime and when they aren't???  I don't know.......

Does that help??

No I've changed my mind - I still dont understand.

What did you run through Wolfram Alpha?

Thanks Chris.