Problem 46

Question

Let $a, b, c,$ and $n$ be any positive integers and $p$ be any prime. Prove each. If $p$ and $p+2$ are twin primes, then $p$ must be odd.

Step-by-Step Solution

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Answer

If $p$ and $p+2$ are twin primes, then $p$ must be odd. Assuming $p$ is even leads to a contradiction since the only even prime number is 2, which means $p+2$ would also be even and not prime. Therefore, $p$ must be odd for $p$ and $p+2$ to be twin primes.

1Step 1: Recall definitions of prime, odd, and even numbers

To satisfy being a twin prime, a number must be a prime number and also have another prime number either 2 more or 2 less than itself. Prime numbers are integers greater than 1 that have exactly two factors, 1 and the integer itself. Examples include: 2, 3, 5, 7, 11, 13, ... Odd numbers are integers that can be written in the form $2n + 1$ where $n$ is an integer. These numbers cannot be divided by 2 without a remainder. Examples include: 1, 3, 5, 7, 9, ... Even numbers are integers that can be written in the form $2n$ where $n$ is an integer. These numbers can be divided by 2 without a remainder. Examples include: 0, 2, 4, 6, 8, ...

2Step 2: Analyze the properties of even and odd integers

Given that $p$ and $p+2$ are twin primes, we need to determine if $p$ can be even or must be odd. Since all even integers, except for the number 2, are divisible by 2, this means that they will have factors other than 1 and themselves, which contradicts the definition of a prime number. On the other hand, odd integers are not divisible by 2. Therefore, there's a possibility for them to be prime numbers.

3Step 3: Deduce that $p$ must be odd using contradiction

Let's assume that $p$ is an even number. We know that the only even prime number is 2, making $p = 2$. This implies that $p + 2 = 4$. However, 4 is not a prime number since it has factors 1, 2, and 4. This contradicts the assumption that $p$ and $p+2$ are twin primes since if $p$ is even, $p+2$ must also be even and cannot be prime. Therefore, $p$ must be an odd number. We've arrived at the conclusion that if $p$ and $p+2$ are twin primes, then $p$ must be odd.

Key Concepts

Prime NumbersOdd NumbersEven NumbersMathematical Proof

Prime Numbers

Prime numbers play a crucial role in mathematics due to their unique properties. A prime number is a natural number greater than 1 that cannot be formed by multiplying two smaller natural numbers. In simpler terms, a prime has only two divisors: one and the number itself. This characteristic makes them stand out from other numbers.

Some well-known prime numbers are 2, 3, 5, 7, and 11. Notably, 2 is unique because it is the only even prime number. All other primes are odd, which means that they cannot be evenly divided by 2. This distinction between even and odd primes is essential in understanding the concept of twin primes, where two primes are just two units apart.

Odd Numbers

An odd number is any integer that cannot be divided evenly by 2. In more technical terms, all odd numbers can be described by the formula $2n + 1$, where $n$ is an integer. For example, numbers such as 3, 5, 7, and 9 are odd.

The importance of odd numbers becomes clear when considering prime numbers. With the exception of the number 2, all prime numbers are odd. This matters when talking about twin primes. Twin primes are pairs of prime numbers that differ by 2, such as 11 and 13 or 17 and 19. In these pairs, both numbers are odd, which must be the case because if one were even, it would be divisible by 2 and not prime.

Even Numbers

Even numbers form a distinct category of numbers. They are defined as any integer that can be divided by 2, typically expressed in the form $2n$, with $n$ being an integer. Common examples of even numbers include 0, 2, 4, 6, and 8.

Even numbers are critical to understanding why twin primes must be odd, beyond the number 2. 2 is the only even prime because any other even number would have at least three divisors: 1, 2, and the number itself. This divisibility eliminates any even number larger than 2 from being prime and thus cannot participate as a twin prime in a prime pair like $p$ and $p+2$.

Mathematical Proof

Mathematical proof is essential in validating conjectures and logical statements. The exercise involving twin primes uses contradiction to prove its statement effectively.

Assume, for illustration, that $p$ could be an even number while also being part of a twin prime pair $ (p, p + 2) $. If $p$ were even, $p$ would have to be the number 2, as 2 is the only even prime number. This assumption would then require $p + 2 $ to be 4. However, 4 is not a prime number, as it is divisible by 2, among other numbers. This contradiction leads to the logical conclusion that $p$ cannot be even. Consequently, $p$ must be odd for $p$ and $p + 2$ to be twin primes. This form of reasoning showcases the power of mathematical proof in reinforcing the principles that underlie many mathematical ideas.

Problem 46

Problem 47

Other exercises in this chapter

Problem 45

Let $a, b, c,$ and $n$ be any positive integers and $p$ be any prime. Prove each. If $p$ and $p^{2}+8$ are primes, then $p^{3}+4$ is also a prime. (

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Problem 46

Consider the constant function $f(n)=k .$ Show that $f(n)=\mathrm{O}(1)$.

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Problem 47

Let $f(n)=O(h(n))$ and $g(n)=O(h(n))$ . Verify each. $$\left(f^{\prime}+g\right)(n)=O(h(n))$$

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Problem 48

Prove each, where $t_{n}$ denotes the $n$ th triangular number and $n \geq 2$. $$8 t_{n-1}+4 n=(2 n)^{2}$$

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