So, I've encountered this problem and solved it using the hombo bombo technique. If you're not familiar with the topic, it's a technique I use to solve pretty much any math problem:
Brute force the answer.
Try to find any patterns.
However, I felt this problem is different. Here's what I did:
Usually, it ends here and I never look at the problem again. But for this particular problem, it was a little too much to say "yeah the pattern is obvious.". How counting modules for 2 seemingly different expressions have anything to do with powers of 2. So, I tried playing a little bit with binomial coefficients and no luck. So, I collected my questions:
In the bruteforce solution, I was checking different values of $$$m$$$, but I didn't know the upper bound of $$$m$$$. For example: when $$$n = 35$$$, biggest $$$m$$$ that satisfied the equation was $$$4$$$. For $$$n = 37$$$, however, biggest $$$m$$$ was $$$1919190$$$. What does this value depend on?
Also, it was sufficient to check values for $$$a$$$, $$$b$$$ up to $$$3$$$ only, this yielded the same solutions. Don't know is it because small value of $$$m$$$ or what.
Most importantly, how to start solving it and similar problems, instead of just making some bruteforce and hoping the pattern magically appears.
How is the answer related to Bernoulli numbers, let alone this sequence A086668?
I did search for the exact expression but no luck. I'm interested in the solution and some kind of a proof. Thanks in advance and appreciate any help.
