1. Introduction
The T3 tree, which first appeared in [1] and was formerly introduced in [2] , is a perfect complete binary tree that is considered to be a new tool to study integers. The tree can reveal many new properties of integers such as the symmetric properties discovered in [3] and [4] , the genetic property found in [5] , and other properties introduced in [6] and [7] . The tree also shows its big potentiality in factorization of big semiprimes, as seen in [8] and [9] . A recent study found several inequalities related with estimation of multiplication on the tree. This article introduces the main results.
2. Preliminaries
This section lists for later sections the necessary preliminaries, which include definitions, notations and lemmas.
2.1. Definitions and Notations
Symbol T3 is the T3 tree that was introduced in [1] and [2] and symbol
is by default the node at position j on level k of T3, where
and
. Number of the level by default begins at zero and index of the position also by default begins at zero. Symbol
is the floor function, an integer function of real number x that satisfies inequality
, or equivalently
. Symbol
means conclusion B can be derived from condition A.
For convenience in deduction of a formula, comments are inserted by symbols that express their related mathematical foundations. For example, the following deduction
means that, lemma (L) supports the step from B to C, and proposition (P) supports the step from C to D.
2.2. Lemmas
Lemma 1. (See in [1] ) T3 tree has the following fundamental properties.
(P1). Every node is an odd integer and every odd integer bigger than 1 must be on the T3 tree. Odd integer N with
lies on level
.
(P2). On level k with
, there are
nodes starting by
and ending by
, namely,
with
.
(P3).
is calculated by
(P4). Multiplication of arbitrary two nodes of T3, say
and
, is a third node of T3. Let
; the multiplication
is given by
If
, then
lies on level
of T3; whereas, if
,
with
lies on level
of T3.
Lemma 2. (See in [10] ) Let
and x be a positive real numbers; then it holds
Particularly, if
is a positive integer, say
, then it yields
3. Main Results with Proofs
Proposition 1. For positive integer k and real number
, it holds
(1)
Proof. It can see by Lemma 2 that,
and
Meanwhile, when
, or
,
; thus
.
Consequently (1) holds.
Proposition 2. Let
and
be nodes of T3 with
; let
(2)
then when
and when
Proof. By Lemma 1 (P4), it knows, when
,
lies on level
of T3 and thus
; hence it holds
and
Thus
and
and thus
Similarly, when
,
lies on level
of T3 and
and it holds
and
Proposition 3. Let
be a node of T3 and n be an integer with
; then it holds
(3)
(4)
Thus for arbitrary integer
(5)
(6)
Proof. Considering that
holds for arbitrary
, it yields
(7)
and
(8)
Consider in (7)
and
it knows (3) and (4) hold and consequently (5) and (6) hold.
Proposition 4. Let
and
be nodes of T3 with
; then it holds
(9)
and thus for arbitrary integer
it holds
(10)
Consequently, it yields
(11)
(12)
and
(13)
(13*)
Proof. The condition that
is a node of T3 leads to
Then direct calculation shows
and
Hence (9) holds.
Multiplying each item in (9) by
for integer
immediately yields (10).
By definition of the floor function, it holds
By the Inequalities (3), (4) and (9) it yields
which says (11) holds.
Likewise, by definition of the floor function and referring to the Inequalities (5), (6) and (10), it yields
(14)
which is the (12).
Similarly, the Inequalities (10) and the definition of the floor function lead to
and
Then referring to the Inequalities (5) and (6), it immediately results in
(15)
which is just the (13).
Proposition 5. Let
and
be nodes of T3 with
; then it holds for integer
(16)
and
(17)
Proof. By Lemma 2 and Proposition 1, it holds when
and
That is
By (11) it holds
which is just the (16).
Similarly it holds
and by (12) it yields
4. Conclusion
The T3 tree is emerging its value in studying integers. A lot of equations and inequalities will be research objectives. Since most of the inequalities on the T3 tree are in the form of floor functions, their proofs are often skillful. The inequalities proved in this article are not only quite useful for knowing the T3 tree, but also excellent samples for proving inequalities with the floor functions. Hope it helpful to the readers of interests.
Acknowledgements
The research work is supported by the State Key Laboratory of Mathematical Engineering and Advanced Computing under Open Project Program No. 2017A01, Department of Guangdong Science and Technology under project 2015A010104011, Foshan Bureau of Science and Technology under projects 2016AG100311, Project gg040981 from Foshan University. The authors sincerely present thanks to them all.