Equation : x^{n-1} + y^n = z^{n+1}

 
 
Let   n   be a positive integer.

Prove that there exist distinct positive integers   x, \; y, \; z   such that

x^{n-1} \; + \; y^n \; = \; z^{n+1}

 

One solution is :

 

x \; = \; 2^{n^2} \: 3^{n+1}
y \; = \; 2^{n^2 - n} \: 3^n
z \; = \; 2^{n^2 - 2n + 2} \: 3^{n-1}

 
 
 
 
 
 

 
 
 
 
 
 
 
 
 
 
 
 
 
 

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About benvitalis

math grad - Interest: Number theory
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2 Responses to Equation : x^{n-1} + y^n = z^{n+1}

  1. paul says:

    Here are a few solutions.
    When n=1
    {x->1,y->3,z->2}
    {x->1,y->8,z->3}
    {x->1,y->15,z->4}
    {x->1,y->24,z->5}

    When n=2
    {x->2,y->5,z->3}
    {x->4,y->2,z->2}
    {x->4,y->11,z->5}
    {x->7,y->1,z->2}

    When n=3
    {x->27,y->18,z->9}
    {x->28,y->8,z->6}

    Paul.

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