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Chapter 3: Problem 22

Show that 13 is the largest prime that can divide two successive integers of the form \(n^{2}+3\)

Short Answer

Expert verified
The largest prime divisors of two successive numbers is 13.

Step by step solution

01

Express successive integers

To solve this problem, we need to consider two successive expressions of the form \(n^2+3\) and \((n+1)^2+3\).

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Key Concepts

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

Prime Numbers
Prime numbers are the building blocks of mathematics. These are numbers greater than 1 that have no divisors other than 1 and themselves. For example, 2, 3, 5, and 7 are prime numbers. They are divisible only by 1 and the number itself. Such numbers cannot be broken down into smaller factors, making them essential for various mathematical proofs and applications. Prime numbers are central to number theory due to their fundamental properties. As in the given exercise, determining whether a prime number like 13 can divide certain expressions is a classic problem in number theory. Understanding prime numbers helps find solutions and validate patterns in mathematical expressions.
Successive Integers
Successive integers are integers that follow each other in sequence without any gaps. For instance, if you have a number n, the successive integer would be n+1. Consider 4 and 5, or 10 and 11 鈥� these are successive. Such numbers are important in problems that require you to evaluate patterns or divisibility over continuous sets of numbers. In this particular exercise, we are looking at two expressions: one based on n and the other on n+1. This evaluation helps to understand how properties like prime factorization or common divisors apply to sequences of numbers, setting the stage for more complex proofs.
Mathematical Proof
A mathematical proof is a logical deduction that shows why a statement is true. Proofs are essential for confirming theories across mathematics. There are different forms of proof, such as direct, contradiction, and induction. In the exercise, establishing that 13 is the largest prime to divide such successive integers involves crafting a logical argument. This might involve demonstrating that no larger prime can divide both expressions by showing how the quadratic nature changes their value. Writing such proofs requires clarity and precision to ensure every step logically follows from the previous one.
Quadratic Expressions
Quadratic expressions are polynomial expressions that include terms up to the second degree, usually in the form of \(ax^2 + bx + c\). In this exercise, the expression \(n^2 + 3\) is quadratic because it involves \(n^2\), the variable squared. Quadratic expressions can be used to model a variety of real-world phenomena, from projectile motion to economic cost functions. They also play a significant role in number theory, helping understand properties like divisibility and factorization. Analyzing how these expressions behave when substituting successive integers helps determine factors or divisors, such as primes, that might divide them.

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Most popular questions from this chapter

Determine whether the integer 701 is prime by testing all primes \(p \leq \sqrt{701}\) as possible divisors. Do the same for the integer 1009 .

An integer is said to be square-free if it is not divisible by the square of any integer greater than 1. Prove the following: (a) An integer \(n>1\) is square-free if and only if \(n\) can be factored into a product of distinct primes. (b) Every integer \(n>1\) is the product of a square-free integer and a perfect square. [Hint: If \(n=p_{1}^{k_{1}} p_{2}^{k_{2}} \cdots p_{s}^{k_{1}}\) is the canonical factorization of \(n\), then write \(k_{i}=2 q_{i}+r_{i}\) where \(r_{i}=0\) or 1 according as \(k_{i}\) is even or odd.]

Prove that for every \(n \geq 2\) there exists a prime \(p\) with \(p \leq n<2 p\). [Hint: In the case where \(n=2 k+1\), then by the Bertrand conjecture there exists a prime \(p\) such that \(k

Show that any composite three-digit number must have a prime factor less than or equal to 31 .

(a) The arithmetic mean of the twin primes 5 and 7 is the triangular number 6 . Are there any other twin primes with a triangular mean? (b) The arithmetic mean of the twin primes 3 and 5 is the perfect square \(4 .\) Are there any other twin primes with a square mean?
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