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Chapter 5: Problem 118

The difference between consecutive prime numbers is always an even number, except for two particular prime numbers. What are those numbers?

Short Answer

Expert verified
The two prime numbers for which the difference is not an even number are 2 and 3.

Step by step solution

01

Identify the pattern

Prime numbers defined as a number that has only two distinct positive divisors: 1 and itself. List out the first few prime numbers: 2, 3, 5, 7, 11, 13, ...
02

Analyze the differences

Subtract each prime number from the next one in the series: The difference between 3 and 2 is 1 (an odd number), the difference between 5 and 3 is 2 (an even number), the difference between 7 and 5 is 2 (an even number), and so on. The differences between each pair of consecutive prime numbers are generally even numbers except for the pair (2,3).
03

Find the exception

The only prime numbers which the difference between them is not an even number are 2 and 3, as their difference equals 1 which is an odd number.

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Key Concepts

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

Consecutive Primes
Consecutive primes are prime numbers that appear one after the other when listed in increasing order. For instance, if you list out a few prime numbers such as 2, 3, 5, 7, 11, and 13, you will notice that they follow one another without any other numbers between them. Prime numbers are numbers greater than 1 that have no divisors other than 1 and themselves. Understanding consecutive primes helps in studying the properties and patterns among prime numbers, like the differences between them, which often lead to interesting mathematical insights.
Odd Differences
While exploring the differences between pairs of consecutive prime numbers, you will typically find that these differences are even. This is because most prime numbers are odd, and subtracting one odd number from another yields an even result. However, there is a notable exception in the world of prime numbers. The difference between the first two prime numbers, which are 2 and 3, is not even. This is because 2 is the only even prime number, and when you subtract 2 from 3, you get 1. The number 1 is an odd number, making this pair unique among consecutive primes.
Mathematical Patterns
Mathematical patterns in prime numbers, such as the one we see with consecutive primes, are a fundamental part of number theory. An interesting pattern is that, except for the pair 2 and 3, the difference between every other pair of consecutive primes is even. These patterns help mathematicians and students understand deeper properties of numbers. Recognizing such patterns is crucial in many areas of mathematics and helps to solve complex problems related to primes. By observing these sequences, mathematicians can develop conjectures and explore new territories in mathematical research.

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Most popular questions from this chapter

In Exercises 49-70, write the first six terms of the geometric sequence with the first term, \(a_{1}\), and common ratio, \(r\). \(a_{1}=4, r=2\)

Determine whether each sequence is arithmetic or geometric. Then find the next two terms. \(6,-6,-18,-30, \ldots\)

The sum, \(S_{n}\), of the first \(n\) terms of an arithmetic sequence is given by$$S_{n}=\frac{n}{2}\left(a_{1}+a_{n}\right),$$in which \(a_{1}\) is the first term and \(a_{n}\) is the nth term. The sum, \(S_{n}\), of the first \(n\) terms of a geometric sequence is given by$$S_{n}=\frac{a_{1}\left(1-r^{n}\right)}{1-r},$$in which \(a_{1}\) is the first term and \(r\) is the common ratio \((r \neq 1)\). Determine whether each sequence is arithmetic or geometric. Then use the appropriate formula to find \(S_{10}\), the sum of the first ten terms. \(4,-12,36,-108, \ldots\)

The sum, \(S_{n}\), of the first \(n\) terms of an arithmetic sequence is given by$$S_{n}=\frac{n}{2}\left(a_{1}+a_{n}\right),$$in which \(a_{1}\) is the first term and \(a_{n}\) is the nth term. The sum, \(S_{n}\), of the first \(n\) terms of a geometric sequence is given by$$S_{n}=\frac{a_{1}\left(1-r^{n}\right)}{1-r},$$in which \(a_{1}\) is the first term and \(r\) is the common ratio \((r \neq 1)\). Determine whether each sequence is arithmetic or geometric. Then use the appropriate formula to find \(S_{10}\), the sum of the first ten terms. \(7,19,31,43, \ldots\)

Write a formula for the general term (the nth term) of each geometric sequence. Then use the formula for \(a_{n}\) to find \(a_{7}\), the seventh term of the sequence. \(0.0004,-0.004,0.04,-0.4, \ldots\)
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