Representation of Integers as Sums of Five Nonzero Squares for All Values Greater Than or Equal to 170

Number Theory

1. Divisibility
- 1.1. Greatest Common Divisor
  - 1.1.1. Important Problems On Greatest Common Divisor
- 1.2. Expressing Cubes As Differences Of Squares
2. Prime Numbers
- 2.1. Important Problems On Prime Numbers
- 2.2. Prime Number Theorem
  - 2.2.1. The Prime Number Theorem Explained With Interpretation
3. Congruence
- 3.1. Linear Congruence
  - 3.1.1. Important Applications On Linear Congruence
4. Number Theoretic Functions
- 4.1. Greatest Integer Functions
  - 4.1.1. Divisibility Of Binomial Coefficients
5. Quadratic Reciprocity
- 5.1. Legendre Symbol
  - 5.1.1. Legendre Symbol And Its Properties
6. Fermats Last Theorem
- 6.1. Proof Of Fermats Last Theorem
- 6.2. Pythagorean Triples
  - 6.2.1. Important Problems On Pythagorean Triples
7. Integers As Sum Of Squares
- 7.1. Sum Of More Than Two Squares
  - 7.1.1. Representation Of Integers As Sums Of Five Nonzero Squares For All Values Greater Than Or Equal To 170
- 7.2. Sum Of Two Squares
  - 7.2.1. Perfect Squares In Pythagorean Triples
8. Encryption And Decryption
- 8.1. Linear Cipher
  - 8.1.1. Important Applications On Linear Cipher
- 8.2. Hill Cipher
  - 8.2.1. Hill Cipher Decryption In Number Theory Step By Step With Example

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Representation of Integers as Sums of Five Nonzero Squares

The topic Representation of Integers as Sums of Five Nonzero Squares explores a remarkable result in Number Theory—that every integer greater than or equal to 170 can be expressed as the sum of more than two squares, specifically five nonzero squares. This section dives into the theoretical background, historical development, and significance of this result, highlighting how it fits into broader patterns in the study of integer representations. With easy-to-understand explanations and examples, this guide helps students and enthusiasts grasp the depth and elegance behind the representation of numbers using square terms.

Prove that every integer \( n \geq 170 \) can be expressed as a sum of five nonzero squares.

Answer:

Our aim is to show that every integer \( n \geq 170 \) can be written in the form

\[ n = x_1^2 + x_2^2 + x_3^2 + x_4^2 + x_5^2 \]

where each \( x_i \in \mathbb{Z}^+ \), i.e., \( x_i \geq 1 \) (nonzero natural numbers).

Since

\[ 13^2 = 169 \]

we try to reduce the problem to a sum of four squares:

\[ n = 169 + m = 13^2 + m \]

where \( m = x_1^2 + x_2^2 + x_3^2 + x_4^2 \), each \( x_i \geq 1 \).

\begin{align*} & n \geq 170 \\ \Rightarrow \quad & n - 169 \geq 1 \end{align*}

So the number \( n - 169 \) is at least 1.

By Lagrange's Four Square Theorem, every natural number can be written as a sum of four squares:

m = x_1^2 + x_2^2 + x_3^2 + x_4^2 \quad \forall~ m \in \mathbb{N}

We consider the minimum value when all \( x_i \geq 1 \).

If each \( x_i \geq 1 \), then the minimum value of their squares sum is:

x_1^2 + x_2^2 + x_3^2 + x_4^2 \geq 1^2 + 1^2 + 1^2 + 1^2 = 4

So for all \( n \geq 169 + 4 = 173 \), we can write

\[ n = 169 + (x_1^2 + x_2^2 + x_3^2 + x_4^2) \]

with each \( x_i \geq 1 \), hence each square is nonzero. Therefore, \( n \) is the sum of 5 nonzero squares.

Verification for \( n = 170, 171, 172 \)

We now verify the remaining smaller cases individually:

\begin{align*} & \mathbf{n = 170:} \quad 170 = 144 + 25 + 1 + 1 + 1 = 12^2 + 5^2 + 1^2 + 1^2 + 1^2 \\ & \mathbf{n = 171:} \quad 171 = 121 + 25 + 16 + 4 + 5 = 11^2 + 5^2 + 4^2 + 2^2 + 1^2 \\ & \mathbf{n = 172:} \quad 172 = 100 + 49 + 9 + 9 + 5 = 10^2 + 7^2 + 3^2 + 3^2 + 1^2 \end{align*}

In each case, all five squares are of nonzero integers.

Hence, for all \( n \geq 170 \), we can always write

\[ n = x_1^2 + x_2^2 + x_3^2 + x_4^2 + x_5^2 \]

with each \( x_i \geq 1 \). Thus, every such \( n \) is a sum of five nonzero squares.