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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.

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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.

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Most popular questions from this chapter

The difference of any even integer minus any odd integer is odd.

Definition: The least common multiple of two nonzero integers \(a\) and \(b\), denoted \(\operatorname{lcm}(a, b)\), is the positive integer \(c\) such that a. \(a \mid c\) and \(b \mid c\) b. for all integers \(m\), if \(a \mid m\) and \(b \mid m\), then \(c \mid m\). Prove that for all positive integers \(a\) and \(b\). \(\operatorname{gcd}(a, b) \cdot \operatorname{lcm}(a, b)=a b\).

When expressions of the form \((x-r)(x-s)\) are multiplied out, a quadratic polynomial is obtained. For instance, \((x-2)(x-(-7))=(x-2)(x+7)=x^{2}+5 x-14 .\) \(H\) a. What can be said about the coefficients of the polynomial obtained by multiplying out \((x-r)(x-s)\) when both \(r\) and \(s\) are odd integers? when both \(r\) and \(s\) are even integers? when one of \(r\) and \(s\) is even and the other is odd? b. It follows from part (a) that \(x^{3}-1253 x+255\) cannot be written as a product of two polynomials with integer coefficients. Explain why this is so.

Consider the statement "For all real numbers \(r\), if \(r^{2}\) is irrational then \(r\) is irrational." a. Write what you would suppose and what you would need to show to prove this statement by contradiction. b. Write what you would suppose and what you would need to show to prove this statement by contraposition.

If \(n\) is any odd integer, then \((-1)^{n}=-1\).
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