/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 30 Prove that for all integers \(n\... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 3: Problem 30

Prove that for all integers \(n\), if \(n>2\) then there is a prime number \(p\) such that \(n

Short Answer

Expert verified
To prove that for all integers \(n > 2\), there exists a prime number \(p\) such that \(n < p < n!\), consider the number \(x = (n! - 1)\) which lies in the desired range. Every integer greater than 1 has at least one prime divisor, so \(x\) must have at least one prime divisor, say \(p\). If we assume, for contradiction, that \(p 鈮� n\), we find that \(p\) divides both \(n!\) and \(x\), leading to a contradiction since no prime can divide 1. Thus, our assumption is false and \(p > n\). We have shown the existence of a prime number \(p\) in the range \(n < p < n!\), proving the statement.

Step by step solution

01

Find a number x in the range n < x < n!

Consider the number x = (n! - 1). This number lies in the desired range because n! > n > 1, so n! - 1 > n - 1 > 0. Hence, we have n < x < n! as desired.
02

Recognize that any number greater than 1 has at least one prime divisor

Every integer greater than 1 must have at least one prime number divisor. We know x = (n! - 1) > 1 since n > 2. Therefore, x must have at least one prime divisor, say p.
03

Show that the prime divisor p > n

We will now show that the prime divisor p cannot be less than or equal to n. Suppose, for the sake of contradiction, that p 鈮� n. Since x = (n! - 1), x + 1 = n!. We know that the prime p divides n!, because p is less than or equal to n. However, p also divides x, as we said earlier. This means that p also divides (x + 1 - x) = 1, which is a contradiction because no prime can divide 1. So, our assumption that p 鈮� n is false, and hence, p must be greater than n.
04

Conclude that there exists a prime number p in the range n < p < n!

We have found a prime number p that is a divisor of the number x = (n! - 1), and we have proven that p > n. Therefore, we have shown the existence of a prime number p such that n < p < n!. Thus, the given statement is proven.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

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

Prime Numbers
Prime numbers hold a unique spot in mathematics. Understanding them is key. A prime number is a natural number greater than 1 that cannot be formed by multiplying two smaller natural numbers. This means a prime has exactly two distinct positive divisors: 1 and itself. Examples include 2, 3, 5, 7, and so on.
Primes are fundamental in number theory due to their role as the 'building blocks' of numbers. Any natural number greater than 1 can be expressed uniquely as a product of primes.
  • 1 is not a prime number because it has only one divisor.
  • 2 is the only even prime number. Every other even number can be divided by 2 and thus can't be prime.
Prime numbers are extensively used in fields such as cryptography, showing their importance in both theoretical and applied mathematics.
Factorials
Factorials pose a fascinating realm in mathematics. The factorial of a non-negative integer \( n \) is denoted by \( n! \) and is the product of all positive integers less than or equal to \( n \). For instance, \( 5! = 5 \times 4 \times 3 \times 2 \times 1 = 120 \).
The factorial function grows impressively fast. Multiplying a sequence of increasing numbers, even a modest number like 5 can yield large results. Factorials showcase the importance of product sequences, especially when dealing with permutations and combinations.
  • Factorials express the number of ways to arrange \( n \) objects.
  • In dealing with large values, it provides a ground for approximation techniques.
Factorial calculation is handy in diverse areas like probability, algebra, and even computer science for algorithm efficiency.
Mathematical Proofs
The beauty of mathematical proofs lies in their logical precision and clarity. A proof is a logical argument verifiable through deductive reasoning to convince others of truth in mathematics. Proofs establish the certainty of mathematical statements.
A common method of proof is by contradiction. Assume the opposite of what needs to be proven and demonstrate that this leads to an inconsistency or contradiction. This exercise used such a method in step 3.
  • Proofs are essential in validating results within mathematics.
  • They provide structured solutions, offering clarity and verification.
Developing skills in constructing mathematical proofs aids in critical thinking and is invaluable in problem-solving across scientific disciplines.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Boxes, each capable of holding 36 units, are used to ship a product from the manufacturer to a wholesaler. Express the number of boxes that would be required to ship \(n\) units of the product using either the floor or the ceiling notation. Which notation is more appropriate?

The difference of any two even integers is even.

Prove that if \(n\) is any nonnegative integer whose decimal representation ends in 5 , then \(5 \mid n\).

State a necessary and sufficient condition for the floor of a real number to equal that number.

Prove that a necessary and sufficient condition for a nonnegative integer \(n\) to be divisible by a positive integer \(d\) is that \(n\) mod \(d=0\).
See all solutions

Recommended explanations on Math Textbooks

Calculus

Read Explanation

Statistics

Read Explanation

Mechanics Maths

Read Explanation

Geometry

Read Explanation

Decision Maths

Read Explanation

Theoretical and Mathematical Physics

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.