Math Problem Statement

Let ( n ) be a positive integer such that ( n + 5 ) is a twin prime and ( n + 8 ) is not a prime number. Find the maximum possible value of ( n ) such that ( \frac{n+5}{n-5} > 2 ).

36 words

Solution

We are given that n+5n + 5 is a twin prime, meaning it is part of a pair of prime numbers of the form (p,p+2)(p, p+2), and that n+8n + 8 is not prime. Additionally, we are asked to find the maximum possible value of nn such that:

n+5n5>2.\frac{n+5}{n-5} > 2.

Step 1: Analyze the inequality

We start by simplifying the given inequality:

n+5n5>2.\frac{n+5}{n-5} > 2.

Multiply both sides by n5n-5 (assuming n>5n > 5):

n+5>2(n5).n + 5 > 2(n - 5).

Expand both sides:

n+5>2n10.n + 5 > 2n - 10.

Now, subtract nn from both sides:

5>n10.5 > n - 10.

Add 10 to both sides:

n<15.n < 15.

Thus, nn must be less than 15. We will now examine the conditions on nn.

Step 2: Conditions on twin primes

Since n+5n + 5 is a twin prime, it must be part of a pair of primes (p,p+2)(p, p+2). We list all twin primes less than 15:

  • 33 and 55 (so n+5=5n + 5 = 5, giving n=0n = 0, but nn must be positive)
  • 55 and 77 (so n+5=7n + 5 = 7, giving n=2n = 2)
  • 1111 and 1313 (so n+5=13n + 5 = 13, giving n=8n = 8)
  • 1717 and 1919 (so n+5=19n + 5 = 19, giving n=14n = 14)

Thus, the possible values of nn are n=2,8,14n = 2, 8, 14.

Step 3: Check the condition on n+8n + 8

We are also given that n+8n + 8 is not a prime number. We check for each possible nn:

  • If n=2n = 2, then n+8=10n + 8 = 10, which is not prime.
  • If n=8n = 8, then n+8=16n + 8 = 16, which is not prime.
  • If n=14n = 14, then n+8=22n + 8 = 22, which is not prime.

So, all these values of nn satisfy the condition that n+8n + 8 is not prime.

Step 4: Maximize nn

Since nn must be less than 15 and the maximum possible value of nn is 14, the maximum possible value of nn that satisfies all the conditions is:

14.\boxed{14}.

Further Exploration:

  1. Can you find any other twin primes for values of nn greater than 15?
  2. How would the inequality change if we had n+5n52\frac{n+5}{n-5} \leq 2 instead?
  3. What happens if n+8n + 8 is a prime number? How would the conditions change?
  4. What is the relationship between twin primes and other types of prime pairs?
  5. How do you determine whether a number is a prime or not?

Tip: Always verify the boundary conditions and the problem constraints carefully to avoid overlooking potential solutions.

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Math Problem Analysis

Mathematical Concepts

Prime Numbers
Twin Primes
Inequalities

Formulas

\( \frac{n+5}{n-5} > 2 \)

Theorems

Definition of Prime Numbers
Definition of Twin Primes

Suitable Grade Level

Grades 9-12