The smallest natural number n
such that 8n3+27n2+1
is a prime is:

To find the smallest natural number \( n \) such that \( 8n^3 + 27n^2 + 1 \) is a prime number, we need to check values of \( n \) starting from the smallest natural number (i.e., 1) and increase incrementally, verifying the primality of the expression for each \( n \).

The expression is:
\[ P(n) = 8n^3 + 27n^2 + 1 \]

Let's evaluate \( P(n) \) for the smallest natural numbers \( n \) and check for primality.

1. For \( n = 1 \):
   \[
   P(1) = 8(1)^3 + 27(1)^2 + 1 = 8 + 27 + 1 = 36
   \]
   36 is not a prime number.

2. For \( n = 2 \):
   \[
   P(2) = 8(2)^3 + 27(2)^2 + 1 = 8 \cdot 8 + 27 \cdot 4 + 1 = 64 + 108 + 1 = 173
   \]
   173 is a prime number.

Therefore, the smallest natural number \( n \) such that \( 8n^3 + 27n^2 + 1 \) is a prime is \( n = 2 \).

Would you like details or have any questions?

Here are 8 related questions to explore further:
1. What is a prime number?
2. How do you check if a number is prime?
3. What are other methods to verify the primality of a number?
4. How does the expression \( 8n^3 + 27n^2 + 1 \) change as \( n \) increases?
5. Are there other forms of polynomials that can generate prime numbers?
6. What is the significance of prime numbers in number theory?
7. How can we find more values of \( n \) for which \( 8n^3 + 27n^2 + 1 \) is prime?
8. How do prime numbers relate to cryptography?

**Tip:** When verifying the primality of a large number, consider using efficient algorithms like the Miller-Rabin primality test for faster results.

Explore how to find the smallest natural number n such that 8n^3 + 27n^2 + 1 is a prime number. Follow a step-by-step solution and learn about prime numbers, polynomials, and number theory. Suitable for advanced high school students and beyond.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation