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Chapter 1: Problem 11

Prove that there are 100 consecutive positive integers that are not perfect squares. Is your proof constructive or non-constructive?

Short Answer

Expert verified
The integers from 2501 to 2600 are 100 consecutive integers that are not perfect squares. The proof is constructive.

Step by step solution

01

- Understanding Perfect Squares

A perfect square is a number of the form \( n^2 \) where \( n \) is an integer. For example, 1, 4, and 9 are perfect squares because they can be written as \( 1^2 \), \( 2^2 \), and \( 3^2 \) respectively.
02

- Gap Between Consecutive Perfect Squares

Calculate the difference between two consecutive perfect squares. For two integers \( n \text{ and } (n+1) \), the difference between their squares is \( (n+1)^2 - n^2 = 2n + 1 \). The gap between consecutive perfect squares increases as \( n \) increases.
03

- Identifying the Sufficiently Large n

To find 100 consecutive positive integers that are not perfect squares, find an \( n \) such that the gap \( (n+1)^2 - n^2 \) is greater than or equal to 100. The inequality is \( 2n + 1 \geq 100 \).
04

- Solving the Inequality

Solve the inequality for \( n \). \( 2n + 1 \geq 100 \Rightarrow 2n \geq 99 \Rightarrow n \geq 49.5 \). Since \( n \) must be an integer, choose \( n \geq 50 \).
05

- Constructing the 100 Consecutive Integers

With \( n = 50 \), the perfect squares are \( 50^2 = 2500 \) and \( 51^2 = 2601 \). The integers between 2500 and 2601 are 2501, 2502,..., 2600, providing 100 consecutive integers that are not perfect squares.
06

- Conclusion

This proof is constructive because it explicitly shows the range of 100 consecutive integers (from 2501 to 2600) that are not perfect squares.

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Key Concepts

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

Consecutive Integers
Consecutive integers are numbers that follow each other in order without any gaps. For example, 1, 2, and 3 are consecutive integers. To find consecutive integers between two perfect squares, we need to understand the progression of numbers.

Once you identify the perfect squares bounding the sequence, you can easily count the consecutive integers in between. For example, given the perfect squares 2500 (50^2 ) and 2601 (51^2 ), the integers in between are continuous.

In our exercise, we wanted a sequence of 100 consecutive positive integers. By finding the gap between large enough perfect squares, we can fit 100 integers that are not perfect squares in this range.
Constructive Proof
A constructive proof provides an explicit example or method to show that something exists. Unlike a non-constructive proof, which might only show that an example must exist without giving it, a constructive proof builds or identifies a specific example.

In our problem, we demonstrated a constructive proof by showing an actual sequence of 100 consecutive positive integers that are not perfect squares. Here, the process involved:
  • Calculating perfect squares.
  • Finding the gap between these squares.
  • Ensuring the gap was big enough to fit 100 numbers.
  • Identifying the specific integers between 2500 and 2601 as the solution.

    • This method explicitly shows the integers 2501 to 2600 are the required sequence, providing a clear example.
Inequalities in Mathematics
Inequalities are mathematical statements that relate two expressions with inequality signs such as <, >, 鈮�, or 鈮�. They help us find ranges of values that satisfy certain conditions.

In the given problem, we used an inequality to determine the necessary gap between two consecutive perfect squares to ensure the presence of 100 non-perfect square integers. We set up the inequality:
2n + 1 鈮� 100

Solving this inequality allows us to find the minimum value of n that satisfies the condition. Here鈥檚 the process:
  • Start with 2n + 1 鈮� 100 `
  • Simplify to 2n 鈮� 99 `
  • Finally, solve for ` n鈮�49.5

Since n must be an integer, we use n = 50. This ensures the condition holds, providing the gap needed to find 100 consecutive integers that are not perfect squares. Inequalities are powerful tools for narrowing down such ranges and ensuring conditions are met.

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