I thought I read somewhere once (I thought it was Wikipedia) that the Collatz Conjecture had been proven for the cases where numbers are of the form 12n + 3 and 12n + 7. Is this true? If so, could someone post a proof for those cases? I am especially curious about those cases, because I am confus...

Let $r(a,b)$ be a fraction depending on $a,b$ and nonnegative. For every $b$ there is an $a$ such that $r(a,b)$ is not $0$. Let $C(a,b)$ be a squarefree positive integer depending on $b$ and different for every $b$. Consider $$ S_J = \sum_j^J \sum_i^I \arcsin( r(i,j) \sqrt C(i,j) ) $$ Where $...

Let $r(a,b)$ be a fraction depending on $a,b$ and nonnegative. For every $b$ there is an $a$ such that $r(a,b)$ is not $0$. Let $C(a,b)$ be a squarefree positive integer depending on $b$ and different for every $b$. Consider for positive integers $I,J$ : $$ S_J = \sum_{j=1}^J \sum_{i=1}^I \arc...