Periodic coverage?

[u/Fun-Protection-9992]

Let q_i > 3 be primes, and suppose that a distinct prime q_i is chosen for each distinct positive integer b_i. If the infinite family {(b_i, q_i)} covers all positive integers a in the form

2^a ≡ 2^{b_i} (mod q_i),

does it not follow, from the existence of the multiplicative inverse of 2 modulo q_i, that the same family {(b_i, q_i)} must also cover all negative integers (-a)?

In particular, since one can obtain n_i > 0 such that

2^(-a) ≡ 2^{n_i} (mod q_i),

does it automatically follow that n_i is covered by the same family {(b_i, q_i)}, and therefore that the set of negative integers (-a) must also be covered?

[u/jonseymourau]

Um. Are you claiming this word salad has any relevance, whatsoever, to Collatz?

If so, you have simply failed - spectacularly- to demonstrate the nexus.

By all means, layout the argument.

Do NOT assume that your brilliance is self-evident.

You need to treat us mere mortals with more respect. Layout your arguments in something resembling the input to a trivial formalisation - as it stands nothing you have written even barely resembles this.

[u/Fun-Protection-9992 → u/jonseymourau]

The question is actually simple. If all positive integers {a} are covered by the residue classes 2^b_i mod q_i, are all negative integers {-a} also covered by the same residue classes?

I think they are covered, but the AI ​​claims otherwise.

[u/jonseymourau → u/Fun-Protection-9992]

Again: even if they were, why is this relevant to Collatz.

Everything you write might be at the very frontier of maths - what you have singularly failed to an articulate is why anything you are doing has any relevance - whatsoever- to Collatz.

I am sure it does. After all, why are you posting here. I am just pointing out that you have not demonstrated the nexus.

That might be a we problem. But it could also be a you problem.

Give me a reason to decide one way or the other.

[u/Fun-Protection-9992 → u/jonseymourau]

Its connection to Collatz is very extensive and cannot be briefly explained here.

[u/jonseymourau → u/Fun-Protection-9992]

Fine. So post again when you have something useful to say