**Problem 1:** Let \( p \) be a prime number (integer). Show that \( p^2 + 5 \) cannot be a prime number.---This statement involves exploring properties of prime numbers and the effects of their squares on primality when a constant is added. The task is to provide a proof or explanation demonstrating why \( p^2 + 5 \), where \( p \) is a prime number, cannot result in another prime number.To approach this problem: 1. Consider all possible scenarios, including specific types of prime numbers (such as odd primes) since 2 is the only even prime, and evaluate the expression.2. Use properties of numbers (even, odd) and congruences to determine \( p^2 + 5 \)'s divisibility and primality.3. Demonstrate mathematically whether there is any exception or if the statement holds for all prime numbers \( p \).This topic tests understanding in basic algebraic manipulation and number theory concepts.

Name: **Problem 1:** Let \( p \) be a prime number (integer). Show that \(
Uploaded: 2024-03-20T07:07:26.000Z
Channel: Bartleby
Description: **Problem 1:** Let \( p \) be a prime number (integer). Show that \( p^2 + 5 \) cannot be a prime number.---This statement involves exploring properties of prime numbers and the effects of their squares on primality when a constant is added. The tas