Hey, I need a bit of help with a proof, please

In summary, the conversation discusses proving that for every odd prime number, there exists exactly one positive integer that, when multiplied by itself and added to the prime number, results in a perfect square. The conversation also mentions showing that this positive integer and the prime number are relatively prime, and the potential solution provides a proof for this. The solution concludes that the only possible value for the positive integer is equal to half of the prime number minus one.
  • #1
Morbid Steve
14
0
Here's the item needing to be proofed (this is not homework, but I'm very interested in it).

Show that for each odd prime number y, there is exactly one positive integer x such that x(x+y) is a perfect square.


Thanks for any help/leads, etc..

-Steve
 
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  • #2
If x(x+y) is a perfect square, show that x and x+p are relatively prime. What can you then say about x?
 
  • #3
hmm...this looks stalled, so I'll write the whole proof :smile:
Since [tex]x(x+y)[/tex] is a perfect square, [tex]x[/tex] must be a perfect square.

Proof: [tex]x(x+y)[/tex] can be of [tex]a^2b^2[/tex] form or [tex]a^2 b^2 c^2[/tex] form. If [tex]x[/tex] isn't a perfect squre, [tex]x[/tex] can be of one of forms: [tex]x=a[/tex] or [tex]x=a^2 b[/tex], and then [tex]x+y = a b^2[/tex] or [tex]x+y=b[/tex] or [tex]x+y=b c^2[/tex] respectively, but in any case, [tex]x[/tex] and [tex]x+y[/tex] has a common measure. This contradicts the fact [tex]y[/tex] is a prime number.

Now we can write [tex]x = a^2[/tex] and so [tex]x+y=d^2[/tex]. Then [tex]y =(x+y) - x = d^2 - a^2 = (d+a)(d-a)[/tex]. Now that [tex]y[/tex] is a prime number, [tex]d-a[/tex] must be 1 (notice [tex]y[/tex] cannnot be factorized)! So [tex]d = a+1[/tex] and [tex]y = d+a = 2a+1[/tex]. So [tex]x = a^2 = ((y-1)/2)^2[/tex] and this is the only possible value of [tex]x[/tex].

This is just my solution, so there may be better solutions.
 
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1. How do I begin a proof?

To begin a proof, it is important to understand the statement or problem you are trying to prove. Start by restating the statement in your own words and identifying any key definitions or theorems that may be relevant to the proof. Then, determine the approach you will take to prove the statement, such as using direct proof, indirect proof, or proof by contradiction.

2. What are the steps to writing a proof?

The steps to writing a proof will vary depending on the approach you are taking. However, some common steps include stating the given information or assumptions, making logical deductions or using previously proven theorems, and reaching a conclusion that follows logically from the given information. It is also important to clearly label and explain each step in your proof.

3. How do I know if my proof is correct?

The best way to know if your proof is correct is to check it over carefully, making sure each step follows logically from the previous one and leads to the desired conclusion. You can also ask a colleague or mentor to review your proof and provide feedback. Additionally, it can be helpful to compare your proof to other existing proofs of the same statement to see if there are any major differences or flaws.

4. What should I do if I get stuck on a proof?

If you get stuck on a proof, it is important to take a step back and reassess your approach. You can also try breaking the proof down into smaller parts or considering different approaches. If you are still having trouble, don't be afraid to ask for help from a colleague or mentor. Sometimes, discussing the problem with someone else can help you see it in a new light and find a solution.

5. How can I improve my proof-writing skills?

Improving your proof-writing skills takes practice and patience. It may be helpful to read and study proofs written by others, as well as to work on a variety of different types of proofs. Additionally, seeking feedback and guidance from experienced mathematicians or scientists can help you identify areas for improvement and learn new techniques for writing clear and concise proofs.

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