^{1}

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The article investigates some properties of square root of
**T**
_{3}
tree’s nodes. It first proves several inequalities that are helpful to estimate the square root of a node, and then proves several theorems to describe the distribution of the square root of the nodes on
**T**
_{3}
tree.

Article [_{3} 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 [_{3} 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 T_{3} tree. As is known, a divisor of integer N must be no bigger than N . Hence the location where N lies in the T_{3} tree is important for finding N’s divisor. Accordingly, this article makes an investigation on the issue and presents the results.

Symbol T_{3} is the T_{3} tree that was introduced in [_{3}, where k ≥ 0 and 0 ≤ j ≤ 2 k − 1 . An integer X is said to be clamped on level k of T_{3} if

Lemma 1 (See in [^{th} position on the k^{th} level of T_{3} with k ≥ 0 and 0 ≤ j ≤ 2 k − 1 ; then 2 k + 1 + 1 ≤ N ( k , j ) ≤ 2 k + 2 − 1 .

Lemma 2 (See in [

(P8) n ⌊ x ⌋ ≤ ⌊ n x ⌋ with n being a positive integer

(P17)

Theorem 1. Let a > 1 be an integer and x > 0 be a real number; then it holds

and

Proof. Since a > 1 , the definition

a ⌊ x ⌋ − 1 < a ⌊ x ⌋ ≤ a x < a ⌊ x ⌋ + 1

Since a and

a ⌊ x ⌋ ≤ ⌊ a x ⌋ ≤ a ⌊ x ⌋ + 1

Considering

a ⌊ x ⌋ − 1 < a ⌊ x ⌋ ≤ ⌊ a x ⌋ ≤ a ⌊ x ⌋ + 1

Theorem 1. Let n be a positive integer and

b 0 = n 2 , b 1 = n 2 + 1 , ⋯ , b i = n 2 + i , ⋯ , b 2 n = n 2 + 2 n , b 2 n + 1 = n 2 + 2 n + 1

then

⌊ b i ⌋ ( i = 1 , ⋯ , 2 n ) = ⌊ b 0 ⌋ = n , ⌊ b 2 n + 1 ⌋ = n + 1 (3)

Proof.

n = b 0 < b i = n 1 + i n 2 < b 2 n + 1 = n + 1

This is to say that,

⌊ b i ⌋ ( i = 1 , ⋯ , 2 n ) = n

Corollary 1. Let n and α be a positive integers and

b 0 = n 2 α , b 1 = n 2 α + 1 , ⋯ , b i = n 2 α + i , ⋯ , b 2 n α = n 2 α + 2 n α , b 2 n α + 1 = n 2 α + 2 n α + 1

then

Proof. (Omitted)

Example 1. Take b 0 = 2 8 , b 1 = 2 8 + 1 = 257 , ⋯ , b 2 × 2 4 = 2 8 + 2 × 2 4 = 288 , b 2 × 2 4 + 1 = 2 8 + 2 × 2 4 + 1 = 289 ; then ⌊ b 0 ⌋ = 2 4 , ⌊ b 1 ⌋ = 16 , ⋯ , ⌊ b 2 × 2 4 ⌋ = 16 , ⌊ b 2 × 2 4 + 1 ⌋ = 17 .

Theorem 2. Let n and α be a positive integers and

b 0 = n 2 α + 1 , b 1 = n 2 α + 1 + 1 , ⋯ , b i = n 2 α + 1 + i , ⋯ , b 2 n α = n 2 α + 1 + 2 n α n , b 2 n α + 1 = n 2 α + 1 + 2 n α n + 1

then

⌊ n α n ⌋ ≤ ⌊ b i ⌋ | ( i = 0 , 1 , 2 , ⋯ , 2 n α + 1 ) ≤ ⌊ n α n ⌋ + 1

Proof. See the following deductions.

1)

2) By Lemma 2(P13)

n 2 α + 1 = b 0 < b 1 = n 2 α + 1 + 1 < ⋯ < b 2 n α = n 2 α + 1 + 2 n α n < b 2 n α + 1 = ( n α n + 1 ) 2 ⇒ n α n = b 0 < b i | ( i = 1 , 2 , ⋯ , 2 n α ) < n α n + 1 ⇒ ⌊ n α n ⌋ = b 0 ≤ b i | ( i = 1 , 2 , ⋯ , 2 n α ) ≤ ⌊ n α n ⌋ + 1

Example 2. Take

⌊ b 2 4 + 1 ⌋ = ⌊ 2 9 + 17 ⌋ = ⌊ 529 ⌋ = 23 , ⋯ , ⌊ b 2 × 2 4 − 1 ⌋ = ⌊ 543 ⌋ = 23 , ⌊ b 2 × 2 4 ⌋ = ⌊ 544 ⌋ = 23 , ⌊ b 2 × 2 4 + 1 ⌋ = ⌊ 545 ⌋ = 23

Theorem 4 Suppose integer k satisfies k > 2 and N ( k , 0 ) be the leftmost node on level k of T_{3}; then

Proof. Since N ( k , 0 ) = 2 k + 1 + 1 , it knows by Corollary 1

Example 3. Taking N ( 7 , 0 ) , N ( 11 , 0 ) , N ( 19 , 0 ) , N ( 8 , 0 ) , N ( 10 , 0 ) and N ( 16 , 0 ) as examples results in the following results.

N ( 7 , 0 ) = 2 7 + 1 + 1 = 257 ⇒ ⌊ N ( 7 , 0 ) ⌋ = 16 N ( 11 , 0 ) = 2 11 + 1 + 1 = 4097 ⇒ ⌊ N ( 11 , 0 ) ⌋ = 64 N ( 19 , 0 ) = 2 19 + 1 + 1 = 1048577 ⇒ ⌊ N ( 19 , 0 ) ⌋ = 1024

N ( 8 , 0 ) = 2 8 + 1 + 1 = 513 ⇒ ⌊ N ( 8 , 0 ) ⌋ = 22 N ( 10 , 0 ) = 2 10 + 1 + 1 = 1025 ⇒ ⌊ N ( 10 , 0 ) ⌋ = 45 N ( 16 , 0 ) = 2 16 + 1 + 1 = 131073 ⇒ ⌊ N ( 16 , 0 ) ⌋ = 362

Theorem 5. Suppose integers k and j satisfy k > 2 and 0 ≤ j ≤ 2 k − 1 ; let N ( k , j ) be the node at position j on level k of T_{3}; then it holds

Proof. Since 2 k + 1 + 1 ≤ N ( k , j ) ≤ 2 k + 2 − 1 , it yields 2 k + 1 < N ( k , j ) < 2 k + 2 ; hence it holds

2 k + 1 2 < N ( k , j ) < 2 k 2 + 1

By Lemma 2 (P13), it yields

⌊ 2 k + 1 2 ⌋ ≤ ⌊ N ( k , j ) ⌋ ≤ ⌊ 2 k 2 + 1 ⌋ (6)

By Theorem 1, it holds

2 ⌊ k + 1 2 ⌋ ≤ 2 k + 1 2

and

2 k 2 + 1 < 2 ⌊ k 2 ⌋ + 2

Hence it results in

2 ⌊ k + 1 2 ⌋ = ⌊ 2 ⌊ k + 1 2 ⌋ ⌋ ≤ ⌊ 2 k + 1 2 ⌋ ≤ ⌊ N ( k , j ) ⌋ ≤ ⌊ 2 k 2 + 1 ⌋ ≤ ⌊ 2 ⌊ k 2 ⌋ + 2 ⌋ = 2 ⌊ k 2 ⌋ + 2

That is

2 ⌊ k + 1 2 ⌋ ≤ ⌊ N ( k , j ) ⌋ ≤ 2 ⌊ k 2 ⌋ + 2 (7)

or equivalently

2 ⌊ k + 1 2 ⌋ − 1 < ⌊ N ( k , j ) ⌋ < 2 ⌊ k 2 ⌋ + 2 + 1 (8)

Corollary 2. ⌊ N ( k , j ) ⌋ is clamped in T_{3} on level ⌊ k + 1 2 ⌋ − 1 and or level ⌊ k 2 ⌋ .

Proof. Since 2 ⌊ k + 1 2 ⌋ − 1 the biggest node on level ⌊ k + 1 2 ⌋ − 2 and 2 ⌊ k 2 ⌋ + 2 + 1 is the smallest node on level ⌊ k 2 ⌋ + 1 , it knows by (8) ⌊ N ( k , j ) ⌋ may be clamped on levels from ⌊ k + 1 2 ⌋ − 1 to ⌊ k 2 ⌋ , totally ⌊ k 2 ⌋ − ( ⌊ k + 1 2 ⌋ − 1 ) + 1 levels.

By Lemma 2 (P2) ⌊ k 2 ⌋ − ( ⌊ k + 1 2 ⌋ − 1 ) + 1 ≥ 2 + ⌊ k 2 − k + 1 2 ⌋ = 2 − 1 = 1

By Lemma 2 (P17 & P8) ⌊ k 2 ⌋ − ( ⌊ k + 1 2 ⌋ − 1 ) + 1 = 2 + ⌊ k 2 ⌋ − ( ⌊ k ⌋ − ⌊ k 2 ⌋ ) = 2 + 2 ⌊ k 2 ⌋ − ⌊ k ⌋ ≤ 2

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 ⌊ N ( 7 , * ) ⌋ is clamped on 2 levels, whereas ⌊ N ( 10 , * ) ⌋ is clamped on 1 level.

N ( 7 , 0 ) = 2 7 + 1 + 1 = 257 ⇒ ⌊ N ( 7 , 0 ) ⌋ = 16 ≙ 2 N ( 7 , 2 7 − 1 ) = 2 7 + 2 − 1 = 511 ⇒ ⌊ N ( 7 , 2 7 − 1 ) ⌋ = 22 ≙ 3 N ( 10 , 0 ) = 2 11 + 1 = 2047 ⇒ ⌊ N ( 10 , 0 ) ⌋ = 45 ≙ 4 N ( 10 , 2 10 − 1 ) = 2 12 − 1 = 4095 ⇒ ⌊ N ( 10 , 2 10 − 1 ) ⌋ = 63 ≙ 4

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 T_{3} 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.

Chen, G.H. and Li, J.H. (2018) Brief Investigation on Square Root of a Node of T_{3} Tree. Advances in Pure Mathematics, 8, 666-671. https://doi.org/10.4236/apm.2018.87039