Which are some equivalence propositions for the 3N+1 conjecture, sequence of Syracuse order, proof of term formula of n, and some theories about ta(n) = tc(n).

Syracuse has been maintaining political and military power strength all the time,and in pre-war and duration of the war Syracuse has im-plemented a series of positive and reasonable political,military and diplomacy policy,comparing with the cramped domestic situation and potential intense international situation of Athens that fell in the Peloponnesian War has formed the sharp contrast.

This paper studies the operating properties of Syracuse operator, gives the detachment theorem, and proves it as well, by introducing a number of new operators and concepts, such as Syracuse operator, invariable numbers, identical resource numbers.