Q:

What is the biggest computation one should use for an induction base

First of all sorry for me (informal) language here - i am not a native english speaker. I have tried to look up my question here but couldnt find anything. I also tried to look it up on google but i think its due to my lack of using the right words for it. I would appreciate any editing on my question.
I was prooving that a really big number (which was about the size of $10^{4033}$) is not a prime, altough one could just simply use e.g. Wolfram Alpha to verify it by computing the remainder when it is divided by one of its factors.
In my proof i used mathematical induction to show that numbers of that form have two certain factors. In my induction base i realised it would end up being something like:
$$
111 \cdot91=10101
$$
Which is true - and therefore i showed that my assumption holds for an $n \in \mathbb{N}$.
Normally my induction bases would end up being statements which would be easy to see. But this time one may need a calculator to verify if the statement above is true.
Now my question is:
If i use mathematical induction for a proof - how "complicated" or "large" should my induction bases are? Should one be able to recognize the true statement without the use of a calculator e.g.? How do i know that the computation which has to be done is fair enough?

A:

The induction base proof can be as complicated as it needs to be. It is as much a part of your proof as the induction. You need to supply enough detail to convince your audience. If that takes pages (or hundreds of pages) so be it.

