2. Let p be a prime number such that p> 5. Prove that p2-1 =0 (mod 24).
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
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![**Educational Website Content**
**Math Exercises and Theory**
1. **Prime Number Determination:**
- Only one prime of the form \(1 + 4n^4\) exists. Determine this prime number and prove it’s the only one of this particular form.
- *Hint*: Research Sophie Germain’s Identity to factor \(1 + 4n^4\).
2. **Congruence Proof:**
- Let \(p\) be a prime number such that \(p > 5\). Prove that \(p^2 - 1 \equiv 0 \pmod{24}\).
3. **Binomial Coefficient Condition:**
- Let \(q\) be a prime and \(n \in \mathbb{N}\) such that \(1 \leq n < q\). Prove that \(q \mid \binom{q}{n}\).
4. **Congruence Verification:**
- Use the theory of congruences to verify that:
\[
25 \mid (2^{n+4} + 3^{n+2} - 5^{n+6})
\]
for all \(n \in \mathbb{N}\).
5. **Linear Congruence Solutions:**
- Using congruence theory (not brute force), find all solutions to the following linear congruence:
\[
8x + 9y \equiv 10 \pmod{11}
\]
6. **Final Digit Determination:**
- Determine the possibilities for the final digit of a sixth power of an integer.
7. **Summation Modulo Proof:**
- Prove that if \(n\) is an odd positive integer or divisible by 4, then
\[
1^3 + 2^3 + 3^3 + \ldots + (n - 1)^3 \equiv 0 \pmod{n}
\]
- Is the statement true if \(n\) is even but not divisible by 4?
*Note: There are no graphs or diagrams associated with these exercises. Each problem focuses on the application of mathematical theory and congruences.*](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F66c7b497-dac7-4855-b923-2e60bbc73063%2F68373553-042c-48ca-a005-f0c3ea98a52d%2Fhpuk6r8_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Educational Website Content**
**Math Exercises and Theory**
1. **Prime Number Determination:**
- Only one prime of the form \(1 + 4n^4\) exists. Determine this prime number and prove it’s the only one of this particular form.
- *Hint*: Research Sophie Germain’s Identity to factor \(1 + 4n^4\).
2. **Congruence Proof:**
- Let \(p\) be a prime number such that \(p > 5\). Prove that \(p^2 - 1 \equiv 0 \pmod{24}\).
3. **Binomial Coefficient Condition:**
- Let \(q\) be a prime and \(n \in \mathbb{N}\) such that \(1 \leq n < q\). Prove that \(q \mid \binom{q}{n}\).
4. **Congruence Verification:**
- Use the theory of congruences to verify that:
\[
25 \mid (2^{n+4} + 3^{n+2} - 5^{n+6})
\]
for all \(n \in \mathbb{N}\).
5. **Linear Congruence Solutions:**
- Using congruence theory (not brute force), find all solutions to the following linear congruence:
\[
8x + 9y \equiv 10 \pmod{11}
\]
6. **Final Digit Determination:**
- Determine the possibilities for the final digit of a sixth power of an integer.
7. **Summation Modulo Proof:**
- Prove that if \(n\) is an odd positive integer or divisible by 4, then
\[
1^3 + 2^3 + 3^3 + \ldots + (n - 1)^3 \equiv 0 \pmod{n}
\]
- Is the statement true if \(n\) is even but not divisible by 4?
*Note: There are no graphs or diagrams associated with these exercises. Each problem focuses on the application of mathematical theory and congruences.*
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