Article  introduced the T3 tree and showed a number of properties of tree, including divisibility, multiples and divisors and multiplications of the nodes. Looking through the other papers that are related with the article  , such as articles  -  , one can see that the T3 tree is really a new attempt to study integers. However, one can also see that, there has not been an article that concerns the square root of a node in the T3 tree. As is known, a divisor of integer N must be no bigger than . Hence the location where lies in the T3 tree is important for finding N’s divisor. Accordingly, this article makes an investigation on the issue and presents the results.
2.1. Symbols and Notations
Symbol T3 is the T3 tree that was introduced in  and  and symbol is by default the node at position j on level k of T3, where and . An integer X is said to be clamped on level k of T3 if and symbol indicates X is clamped on level k. An odd interval is a set of consecutive odd numbers that take a as lower bound and b as upper bound, for example, . Intervals in this whole article are by default the odd ones unless particularly mentioned. 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.
Lemma 1 (See in  ). Let be the node at the jth position on the kth level of T3 with and ; then .
Lemma 2 (See in  ). For real numbers x and y, it holds
(P8) with n being a positive integer
3. Main Results and Proofs
Theorem 1. Let be an integer and be a real number; then it holds
Proof. Since , the definition immediately yields
Since a and are integers, it yields by Lemma 2 (P13)
Considering, it knows ; consequently
Theorem 1. Let n be a positive integer and
Proof. and obviously hold. Now consider that, for , it holds
This is to say that,; since n is an integer, it is sure by definition of the floor function
Corollary 1. Let n and be a positive integers and
Example 1. Take , ; then .
Theorem 2. Let n and be a positive integers and
Proof. See the following deductions.
2) By Lemma 2(P13)
Example 2. Take and by
, , then
Theorem 4 Suppose integer k satisfies and be the leftmost node on level k of T3; then is even if k is odd, whereas, it can be either odd or even if k is even.
Proof. Since , it knows by Corollary 1 for an odd k. If k is even, let it be ; then by Theorem 2 or , which indicates can be either odd or even.
Example 3. Taking and as examples results in the following results.
Theorem 5. Suppose integers k and j satisfy and ; let be the node at position j on level k of T3; then it holds
Proof. Since , it yields ; hence it holds
By Lemma 2 (P13), it yields
By Theorem 1, it holds
Hence it results in
Corollary 2. is clamped in T3 on level and or level .
Proof. Since the biggest node on level and is the smallest node on level , it knows by (8) may be clamped on levels from to , totally levels.
By Lemma 2 (P2)
By Lemma 2 (P17 & P8)
Hence the corollary holds
Example 4. Taking the smallest nodes and the biggest nodes on level 7 and level 10 respectively, it can see that is clamped on 2 levels, whereas is clamped on 1 level.
Elementary number theory shows that an integer must have a divisor smaller than the square root of the integer itself. Hence the square root is undoubtedly an important issue of an integer. Since T3 tree is considered to be a new tool to study integers, the square root of a node is certainly helpful to know the distribution of the node’s divisors. The properties proved in this article are sure to provide a know-about the square root of the nodes. We hope it will be useful in the future.
The research work is supported by the State Key Laboratory of Mathematical Engineering and Advanced Computing under Open Project Program No. 2017A01, the Youth Innovative Talents Project (Natural Science) of Education Department of Guangdong Province under grant 2016KQNCX192, 2017KQNCX230. The authors sincerely present thanks to them all.