to seforo tebeg le v e b obo llo da e batainalo t sei o bo avd ba . 2. Let p be a prime number such that p> 5. Prove that p – 1 =0 (mod 24). 3. Let q be a prime and n EN such that1
to seforo tebeg le v e b obo llo da e batainalo t sei o bo avd ba . 2. Let p be a prime number such that p> 5. Prove that p – 1 =0 (mod 24). 3. Let q be a prime and n EN such that1
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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![**1. Prime Number of the Form 1 + 4n^4**
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. Prime Number Congruence**
Let \(p\) be a prime number such that \(p > 5\). Prove that \(p^2 - 1 \equiv 0 \pmod{24}\).
**3. Prime and Natural Number Divisibility**
Let \(q\) be a prime and \(n \in \mathbb{N}\) such that \(1 \leq n < q\). Prove that \(q \mid \binom{q}{n}\).
**4. Theory of Congruences Verification**
Use the theory of congruences to verify that
\[ 25 \mid (2^{n+4} + 3^{n+2} - 5^{n+6}) \text{ for all } n \in \mathbb{N}. \]
**5. Solving Linear Congruence**
Using congruence theory (not brute force), find all solutions to the following linear congruence:
\[ 8x + 9y \equiv 10 \pmod{11}. \]
**6. Sixth Power Final Digit**
Determine the possibilities for the final digit of a sixth power of an integer.
**7. Sum of Cubes Modulo Condition**
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?](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F66c7b497-dac7-4855-b923-2e60bbc73063%2Fbfe6288b-2db3-4422-912b-6149c300a49b%2F4xrd47i_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**1. Prime Number of the Form 1 + 4n^4**
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. Prime Number Congruence**
Let \(p\) be a prime number such that \(p > 5\). Prove that \(p^2 - 1 \equiv 0 \pmod{24}\).
**3. Prime and Natural Number Divisibility**
Let \(q\) be a prime and \(n \in \mathbb{N}\) such that \(1 \leq n < q\). Prove that \(q \mid \binom{q}{n}\).
**4. Theory of Congruences Verification**
Use the theory of congruences to verify that
\[ 25 \mid (2^{n+4} + 3^{n+2} - 5^{n+6}) \text{ for all } n \in \mathbb{N}. \]
**5. Solving Linear Congruence**
Using congruence theory (not brute force), find all solutions to the following linear congruence:
\[ 8x + 9y \equiv 10 \pmod{11}. \]
**6. Sixth Power Final Digit**
Determine the possibilities for the final digit of a sixth power of an integer.
**7. Sum of Cubes Modulo Condition**
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?
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