Diophantine Equations (Problem 99)

xy+yz+zx1=k2x^y + y^z + z^{x - 1} = k - 2

Find solutions where x,y,z,kx, y, z, k are all positive or negative integers.

Give your answer as the number of solutions.

You're solving for kk.

Note by A Former Brilliant Member
1 week, 5 days ago

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@Yajat Shamji- For positive integers x,y,z,kx,y,z,k there are infinite solutions:

Take random positive integers x,y,zx,y,z then xyZ+;yzZ+;zx1Z+(xy+yz+zx1)Z+x^y\in Z^+; y^z\in Z^+; z^{x-1}\in Z^+\Rightarrow (x^y+y^z+z^{x-1})\in Z^+ (k2)Z+\Rightarrow (k-2)\in Z^+ As 2Z+2\in Z^+ kZ+;k>2\Rightarrow k\in Z^+;k>2

Zakir Husain - 1 week, 5 days ago

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So you're saying I have made a Diophantine equation that has infinite integer solutions? @Zakir Husain

A Former Brilliant Member - 1 week, 5 days ago

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What about k2k \leq 2? @Zakir Husain

A Former Brilliant Member - 1 week, 4 days ago

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What I usually do is code these equations. In my output, I got the fact that if you put x=1,y=1,k=5,x=1,y=1,k=5, there are infinite values for zz. Also, I haven't studied these in school, so I am not at all comfortable in solving using algebra. But there are infinite solutions.

Vinayak Srivastava - 1 week, 5 days ago

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You're solving for kk, actually, @Vinayak Srivastava

A Former Brilliant Member - 1 week, 4 days ago

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But it's not stated. Then we need solutions of the form (x,y,z,k)(x,y,z,k).

Vinayak Srivastava - 1 week, 4 days ago

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@Vinayak Srivastava Ok. I'll add that.

A Former Brilliant Member - 1 week, 4 days ago

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@Vinayak Srivastava Added.

A Former Brilliant Member - 1 week, 4 days ago

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@Yashvardhan Pattanashetti

A Former Brilliant Member - 1 week, 5 days ago

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Sorry, but Algebra is my mortal enemy, I'm very bad at it.

A Former Brilliant Member - 1 week, 5 days ago

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It is not an algebra but a number theory problem. It is from a branch of number theory called Algebraic number theory

Zakir Husain - 1 week, 5 days ago

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@Zakir Husain 'Algebraic' Number theory, how's it different?

A Former Brilliant Member - 1 week, 5 days ago

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@A Former Brilliant Member There are two broad branches of number theory- Analytic number theory and Algebraic number theory. The difference is that Algebraic number theory uses algebra as a way to get answers to number theory problems like this one. In algebra you will not discriminate between integer, rational, real or complex solutions to the equation but for number theory it matters.

Zakir Husain - 1 week, 5 days ago

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@Zakir Husain See algebra is used, so I'm out.

A Former Brilliant Member - 1 week, 5 days ago

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