proud depot
Hey everyone! I saw the other day a post that talks about 'irregular' divsibility tets like 7-divisibility and 13-divisibility and the methods where checking whether b-2a and b+4a were multiples of the respective divisors where the treated number is a+10b. I sought to prove and particular cases like these are simple to prove.
eg:
let a+10b≡[13] => 40a+10b≡39a[13] => 10(b+4a)≡[13] = Gauss lemma => b+4a≡[13]
It's a recursive test
But i sought to find a generalized form and i did actually find the final condition but the problem if that the latter involves a map from {n∈ℤ* | nʌ10=1} onto ℤ but the problem is it doesn't have a single expression but divided into 4 cases (because φ(10)=4). I tried using trigonometric functions to make it a single expression but I failed. I looked up the pattern of the first positive terms of this map on On-Line Encyclopedia of Integer Sequences to hopefully find a closed form of an already discovered sequence but uggh i didn't understand what the results were.
This is not the only problem. I aimed to generalize this test even further so much so it's not only decimal number-related but for any base. The only problem tho is the expression of that one map because the bigger the base number, the more cases we have to account for (exactly φ(p) cases where the base number is p) and this is no practical.
I'm sure someone already discovered this but don't seem to find the sources for their work haha.
dim acorn
sage dagger