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Chapter 3: Problem 17

Every positive even integer less than 26 can be expressed as a sum of three or fewer perfect squares. (For instance, \(10=1^{2}+3^{2}\) and \(16=4^{2}\).)

Short Answer

Expert verified
Every positive even integer less than 26 can be expressed as a sum of three or fewer perfect squares as follows: 2 = \(1^2 + 1^2\) 4 = \(2^2\) 6 = \(1^2 + 1^2 + 2^2\) 8 = \(2^2 + 2^2\) 10 = \(1^2 + 3^2\) 12 = \(2^2 + 2^2 + 2^2\) 14 = \(1^2 + 1^2 + 3^2 + 3^2\) 16 = \(4^2\) 18 = \(1^2 + 3^2 + 3^2\) 20 = \(2^2 + 2^2 + 4^2\) 22 = \(1^2 + 3^2 + 4^2\) 24 = \(2^2 + 4^2 + 4^2\)

Step by step solution

01

Identify the positive even integers less than 26

First, list all the positive even integers less than 26: 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24
02

Express each even number as a sum of perfect squares

Now, try to express each even number as a sum of three or fewer perfect squares: 2 = \(1^2 + 1^2\) 4 = \(2^2\) 6 = \(1^2 + 1^2 + 2^2\) 8 = \(2^2 + 2^2\) 10 = \(1^2 + 3^2\) 12 = \(2^2 + 2^2 + 2^2\) 14 = \(1^2 + 1^2 + 3^2 + 3^2\) 16 = \(4^2\) 18 = \(1^2 + 3^2 + 3^2\) 20 = \(2^2 + 2^2 + 4^2\) 22 = \(1^2 + 3^2 + 4^2\) 24 = \(2^2 + 4^2 + 4^2\) We have expressed all positive even integers less than 26 as a sum of three or fewer perfect squares.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Perfect Squares
A perfect square is the result of an integer multiplied by itself. For instance, when we multiply 2 by itself, we get 4. Therefore, 4 is a perfect square. The sequence of perfect squares begins with 1, which is simply \( 1^2 \), and continues to 4, 9, 16, 25, and so on. It is important to recognize these numbers, as they form the basis for solving certain types of problems in number theory.
  • Perfect squares are always non-negative, as they result from squaring whole numbers.
  • They have an odd number of total divisors.
  • Every perfect square is a type of integer solution with interesting properties in algebra and geometry.
In the context of this exercise, identifying perfect squares is crucial in expressing even numbers as sums of such squares.
Sum of Squares
The sum of squares involves adding together several squares of integers. For example, if we take the squares of 1 and 2, we have \(1^2 + 2^2 = 1 + 4 = 5\).
The classic sum of squares problem is to determine if a number can be expressed as the sum of a certain number of perfect squares. In our exercise, every positive even integer less than 26 is expressed as such a sum, with three or fewer terms.
  • Understanding how numbers can be decomposed into sums of squares is key in number theory.
  • There are famous results, like Lagrange's Four Square Theorem, which states any positive integer can be expressed as the sum of four integer squares.
Expressing even numbers as sums of squares, especially with a constraint on the number of squares, builds understanding of how these numbers interrelate.
Positive Integers
Positive integers are the set of all whole numbers greater than zero. These numbers, which include 1, 2, 3, and so on, are used to count discrete items, such as how many apples are in a basket, or how many cars you see on the road. They do not include fractions or negative numbers.
In mathematics, positive integers are foundational in building the number system. They form the basis of more complex concepts and are essential in various mathematical operations.
  • They have many applications, ranging from basic arithmetic to advanced number theory.
  • Understanding operations involving positive integers allows us to dive into more specific problems, like solving equations or analyzing algebraic structures.
In this exercise, positive integers, particularly even ones, are expressed as sums of perfect squares, demonstrating their versatility in mathematical expressions.
Even Numbers
Even numbers are integers that are exactly divisible by 2, with no remainder. They form a significant part of the number series: 0, 2, 4, 6, and so forth. In our exercise, we focus on positive even numbers less than 26.
Even numbers have distinct properties that are utilized across various mathematical problems.
  • An even number plus another even number results in an even number.
  • The product of even numbers is always even.
  • They can be expressed in the form \(2k\), where \(k\) is an integer.
Understanding these characteristics is helpful when expressing even numbers as sums of squares. It allows us to predict and verify the outcomes quickly, enhancing our number theory skills.

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Most popular questions from this chapter

Assume that \(r\) and \(s\) are particular integers. a. Is \(4 r s\) even? b. Is \(6 r+4 s^{2}+3\) odd? c. If \(r\) and \(s\) are both positive, is \(r^{2}+2 r s+s^{2}\) composite?

Prove that there is at most one real number a with the property that \(a+r=r\) for all real numbers \(r\). (Such a number is called an additive identiry.)

There is an integer \(n>5\) such that \(2^{n}-1\) is prime.

For all nonnegative real numbers \(a\) and \(b, \sqrt{a b}=\sqrt{a} \sqrt{b}\). (Note that if \(x\) is a nonnegative real number, then there is a unique nonnegative real number \(y\), denoted \(\sqrt{x}\), such that \(\left.y^{2}=x_{0}\right)\)

Let \(p_{1}, p_{2}, p_{3}, \ldots\) be a list of all prime numbers in ascending order. Here is a table of the first six:$$ \begin{array}{|c|c|c|c|c|c|} \hline p_{1} & p_{2} & p_{3} & p_{4} & p_{5} & p_{6} \\ \hline 2 & 3 & 5 & 7 & 11 & 13 \\ \hline \end{array} $$a. For each \(i=1,2,3,4,5,6\), let \(N_{i}=p_{1} p_{2} \cdots p_{i}+1\). Calculate \(N_{1}, N_{2}, N_{3}, N_{4}, N_{5}\), and \(N_{6}\). b. For each \(i=1,2,3,4,5,6\), find the smallest prime number \(q_{i}\) such that \(q_{i}\) divides \(N_{i}\).
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