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Old 2018-03-02, 03:31   #1
MARTHA
 
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Default Let y^2=xz-x^2+1, and if value of x is known, can y and z be directly calculated? Given all variable

Let y^2=xz-x^2+1, and if value of x is known, can y and z be directly calculated? Given all variables are Integers.
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Old 2018-03-02, 03:36   #2
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Quote:
Originally Posted by MARTHA View Post
Let y^2=xz-x^2+1, and if value of x is known, can y and z be directly calculated? Given all variables are Integers.
Is this homework ?
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Old 2018-03-02, 05:12   #3
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Quote:
Originally Posted by MARTHA View Post
Let y^2=xz-x^2+1, and if value of x is known, can y and z be directly calculated? Given all variables are Integers.
Yes. There are an infinite amount of them. You can write your expression like
xz=y^2+x^2-1, or (divide both left and right by x) \(z=\frac{y^2+x^2-1}x\), or \(z=\frac{y^2-1}x+x\). Now you see that the only condition to have z integer is that y^2-1 is divisible by x. Pick any x, say x=5, then all y ending in 1, 4, 6, or 9 will have squares ending in 1 or 6, so y^2-1 will end in 0 or 5.
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Old 2018-03-02, 05:21   #4
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Wolfram Alpha link:

https://www.wolframalpha.com/input/?...%2B+1+for+y,+z
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Old 2018-03-02, 05:35   #5
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whaaa, haha
Click image for larger version

Name:	cat doctor.PNG
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Old 2018-03-02, 05:53   #6
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Quote:
Originally Posted by LaurV View Post
whaaa, haha
Attachment 17790
It's not what it seems. It's not a case of Pet-Discrimination.
Dog-Health is more Number-Theory related.
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Old 2018-03-02, 06:31   #7
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I am innocent I just followed your link. However, for the top of my head I can not imagine what kind of association wolfram alfa did (I never searched for cats and dogs on my computers, either)
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Old 2018-03-02, 06:40   #8
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There is not enough data to DIRECTLY calculate y and z when x is known, even given the fact that all are integers, because there is not a unique solution.

Jacob

Last fiddled with by S485122 on 2018-03-02 at 06:47 Reason: removed the part doing the homework
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Old 2018-03-02, 07:00   #9
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Forgot to put the integer constraint.
Solution doesn't really change but for whatever reason the engine also prints the "Implicit derivatives" Whatever that is.


https://www.wolframalpha.com/input/?...r+the+integers
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Old 2018-03-02, 09:59   #10
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Following LaurV's suggestion the following solution will be found:

\(y = k\cdot x + 1\)
\(z = 2k + (k^2+1)\cdot x\)

for all \(k\in N\)
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Old 2018-03-02, 12:37   #11
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Thank you all !!!
you guys are amazing
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