Let n 10. We will prove that n² +n + 1 is not prime. Since n² +n +1> 1, we'll need to prove the second part of the "or", i.e., that 3d e N, d (n² +n+1)^d #1^d #n² +n +1. Let d = 3. Part 1: we prove that d | (n²+n+1). [Proof body for Part 1...) Part 2: we prove that d 1. [Proof body for Part 2...] Part 3: we prove that dn² +n + 1. [Proof body for Part 3...]
Let n 10. We will prove that n² +n + 1 is not prime. Since n² +n +1> 1, we'll need to prove the second part of the "or", i.e., that 3d e N, d (n² +n+1)^d #1^d #n² +n +1. Let d = 3. Part 1: we prove that d | (n²+n+1). [Proof body for Part 1...) Part 2: we prove that d 1. [Proof body for Part 2...] Part 3: we prove that dn² +n + 1. [Proof body for Part 3...]
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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![Suppose we want to prove the following statement: "for every positive integer n, the value
n² +n + 1 is not prime."
What would be the appropriate proof structure for this proof? (Hint: translate the statement into
predicate logic first!)
Let n = 10. We will prove that n² +n + 1 is not prime. Since 7² +n+1> 1, we'll need to prove the
second part of the "or", i.e., that 3d € N, d | (n² +n+1)^d #1^d #n² +n+1.
Let d = 3.
Part 1: we prove that d | (n² +n + 1).
[Proof body for Part 1...]
Part 2: we prove that d # 1.
[Proof body for Part 2...]
Part 3: we prove that dn² +n + 1.
[Proof body for Part 3...)
Let n E Z. We will prove that n² +n + 1 is not prime. Since n²+n+1> 1, we'll need to prove the
second part of the "or", i.e., that 3d e N, d (n² +n+1) ⇒d=1Vd=n²+n+1.
Let d
We want to prove that d | (n²+n+1) ⇒d=1vd=n² +n + 1.
Then since d (n²+n+ 1), the implication is vacuously true.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F2ed5ea57-5dda-4ad5-9230-267038297200%2Fa5ee270c-35f4-4759-aecd-f16ecbcb7bad%2Flhyqpq_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Suppose we want to prove the following statement: "for every positive integer n, the value
n² +n + 1 is not prime."
What would be the appropriate proof structure for this proof? (Hint: translate the statement into
predicate logic first!)
Let n = 10. We will prove that n² +n + 1 is not prime. Since 7² +n+1> 1, we'll need to prove the
second part of the "or", i.e., that 3d € N, d | (n² +n+1)^d #1^d #n² +n+1.
Let d = 3.
Part 1: we prove that d | (n² +n + 1).
[Proof body for Part 1...]
Part 2: we prove that d # 1.
[Proof body for Part 2...]
Part 3: we prove that dn² +n + 1.
[Proof body for Part 3...)
Let n E Z. We will prove that n² +n + 1 is not prime. Since n²+n+1> 1, we'll need to prove the
second part of the "or", i.e., that 3d e N, d (n² +n+1) ⇒d=1Vd=n²+n+1.
Let d
We want to prove that d | (n²+n+1) ⇒d=1vd=n² +n + 1.
Then since d (n²+n+ 1), the implication is vacuously true.
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