A community for students.

Here's the question you clicked on:

55 members online
  • 0 replying
  • 0 viewing

anonymous

  • 4 years ago

Given that n is a positive integer which contains an odd factor greater than one,prove that: x^n + y^n = p has no solutions for any prime p>2.

  • This Question is Closed
  1. Mr.Math
    • 4 years ago
    Best Response
    You've already chosen the best response.
    Medals 2

    I have to go now, I will come back to it in a bit.

  2. anonymous
    • 4 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    Take Values And Substitute!!

  3. anonymous
    • 4 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    I suppose \( x,y \in \mathbb{N} \)

  4. amistre64
    • 4 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    i remember there being a trig equation to test for primes ....

  5. anonymous
    • 4 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    Really amsitre? I never heard of it...

  6. amistre64
    • 4 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    its in one of the boks upstairs about primes i believe

  7. Mr.Math
    • 4 years ago
    Best Response
    You've already chosen the best response.
    Medals 2

    You need to know that \(a^n+b^n\), where n is odd can be factorized as: \[a^n+b^n=(a+b)(a^{n-1}-a^{n-2}b+a^{n-3}b^2-\cdots -ab^{n-2}+b^{n-1})\] Assuming that \(x, y \in \mathbb{N}\), let \(x^n+y^n=(x^m)^q+(y^m)^q\), where q is an odd number. Let \(x^m=a\) and \(y^m=b\). By using the factorization above we can write \[(a^q+b^q)=(a+b)(a^{q-1}-a^{q-1}b+a^{q-3}b^2-\cdots -ab^{q-2}+b^{q-1}).\] For this quantity to be prime i) \(a+b\) has to be \(1\) or ii) \(a^{q-1}-a^{q-1}b+a^{q-3}b^2-\cdots -ab^{q-2}+b^{q-1}\) has to be \(1\). But happens only for \(a=b=1\), which gives the prime 2. But since we're looking for \(p>2\), no such solution exists.

  8. Not the answer you are looking for?
    Search for more explanations.

    • Attachments:

Ask your own question

Sign Up
Find more explanations on OpenStudy
Privacy Policy

Your question is ready. Sign up for free to start getting answers.

spraguer (Moderator)
5 → View Detailed Profile

is replying to Can someone tell me what button the professor is hitting...

23

  • Teamwork 19 Teammate
  • Problem Solving 19 Hero
  • You have blocked this person.
  • ✔ You're a fan Checking fan status...

Thanks for being so helpful in mathematics. If you are getting quality help, make sure you spread the word about OpenStudy.

This is the testimonial you wrote.
You haven't written a testimonial for Owlfred.