# Two-Parameter Solutions to Three Almost Fermat Equations

The problem below has been proposed by Morrey Klamkin (Crux Mathematicorum, 1960):

Determine two parameter solutions of the following "almost" Fermat Diophantine equations:

(1) | x^{ n-1} + y^{ n-1} | = z^{ n} |

(2) | x^{ n+1} + y^{ n+1} | = z^{ n} |

(3) | x^{ n+1} + y^{ n-1} | = z^{ n} |

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Copyright © 1996-2018 Alexander Bogomolny

Determine two parameter solutions of the following "almost" Fermat Diophantine equations:

(1) | x^{ n-1} + y^{ n-1} | = z^{ n} |

(2) | x^{ n+1} + y^{ n+1} | = z^{ n} |

(3) | x^{ n+1} + y^{ n-1} | = z^{ n} |

The solution is by Leo Moser (Crux Mathematicorum, September 1961, #456).

We will exhibit two parameter solutions for a more general equation:

(3) | x^{a} + y^{b} = z^{c}, (a, b, c) = 1. |

Since (a, b, c) = 1, we can first find m and n such that

(4) | abm + 1 = cn. |

Now let

^{bm}(u

^{abm}+ v

^{abm})

^{bm}and

y = v

^{am}(u

^{abm}+ v

^{abm})

^{am}.

Then

^{a}+ y

^{b}= (u

^{abm}+ v

^{abm})

^{abm + 1}= (u

^{abm}+ v

^{abm})

^{cn}.

So letting z = (u^{abm} + v^{abm})^{n} we get a 2-parameter solution to (3).

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Copyright © 1996-2018 Alexander Bogomolny

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