This paper presents study of problem 3x+1 (Collatz conjecture) within which following research has been made: - numeral sequences of the form \( \frac{2^p}{3^q}\) - sequence of ordered numeral intervals of the form \(\frac{\left\{ 2^{a(i)} - 2^{b(i)}\right\}}{3} \) - the operation of combining and decoupling a sequence of numeral intervals - a geometric interpretation of a sequence of numeral intervals is given - fractional numeral systems \(2\cap3\) and \(4\cap3\) are considered - sequence transformation oddness \(4k+3\) and \(4k+1\) - \(3x+1\) problem solved - a connection with a prime numbers has been established