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

@A Former Brilliant Member
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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.

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## Comments

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

Take random positive integers $x,y,z$ then $x^y\in Z^+; y^z\in Z^+; z^{x-1}\in Z^+\Rightarrow (x^y+y^z+z^{x-1})\in Z^+$ $\Rightarrow (k-2)\in Z^+$ As $2\in Z^+$ $\Rightarrow k\in Z^+;k>2$

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

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What about $k \leq 2$? @Zakir Husain

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

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You're solving for $k$, actually, @Vinayak Srivastava

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

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@Vinayak Srivastava, @Alak Bhattacharya, @Mahdi Raza

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@Zakir Husain

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@Chris Sapiano

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@Sahar Bano

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@Hamza Anushath

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@Gandoff Tan

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@Yashvardhan Pattanashetti

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

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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

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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.

There are two broad branches of number theory-Log in to reply

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