An equation of the form

(1) |

(2) |

(3) |

If is Odd, then is Positive and the solution in terms of smallest Integers is
and , where is the th Convergent. If is Even, then is Negative, but

(4) |

(5) |

The more complicated equation

(6) |

(7) |

(8) |

Call a Diophantine equation consisting of finding Powers equal to a sum of equal
Powers an `` equation.'' The 2-1 equation

(9) |

(10) |

(11) |

(12) |

If zero is counted as a square, both Positive and Negative numbers are included, and the order of the two squares is distinguished, Jacobi showed that the number of ways a number can be written as the sum of two squares is four times the excess of the number of Divisors of the form over the number of Divisors of the form .

A set of Integers satisfying the 3-1 equation

(13) |

(14) |

Solutions to an equation of the form

(15) |

(16) |

(17) |

(18) |

(19) |

**References**

Beeler, M.; Gosper, R. W.; and Schroeppel, R. *HAKMEM.* Cambridge, MA: MIT Artificial Intelligence Laboratory, Memo AIM-239, Feb. 1972.

Beiler, A. H. ``The Pellian.'' Ch. 22 in *Recreations in the Theory of Numbers: The Queen of Mathematics Entertains.*
New York: Dover, pp. 248-268, 1966.

Bell, E. T. *The Development of Mathematics, 2nd ed.* New York: McGraw-Hill, p. 159, 1945.

Chrystal, G. *Textbook of Algebra, 2 vols.* New York: Chelsea, 1961.

Degan, C. F. *Canon Pellianus.* Copenhagen, Denmark, 1817.

Dickson, L. E. ``Number of Representations as a Sum of 5, 6, 7, or 8 Squares.'' Ch. 13 in
*Studies in the Theory of Numbers.* Chicago, IL: University of Chicago Press, 1930.

Dickson, L. E. *History of the Theory of Numbers, Vol. 2: Diophantine Analysis.* New York: Chelsea, 1966.

Guy, R. K. *Unsolved Problems in Number Theory, 2nd ed.* New York: Springer-Verlag, 1994.

Lam, T. Y. *The Algebraic Theory of Quadratic Forms.* Reading, MA: W. A. Benjamin, 1973.

Rajwade, A. R. *Squares.* Cambridge, England: Cambridge University Press, 1993.

Scharlau, W. *Quadratic and Hermitian Forms.* Berlin: Springer-Verlag, 1985.

Shapiro, D. B. ``Products of Sums and Squares.'' *Expo. Math.* **2**, 235-261, 1984.

Smarandache, F. ``Un metodo de resolucion de la ecuacion diofantica.'' *Gaz. Math.* **1**, 151-157, 1988.

Smarandache, F. ``Method to Solve the Diophantine Equation .'' In *Collected Papers, Vol. 1.*
Bucharest, Romania: Tempus, 1996.

Taussky, O. ``Sums of Squares.'' *Amer. Math. Monthly* **77**, 805-830, 1970.

Whitford, E. E. *Pell Equation.* New York: Columbia University Press, 1912.

© 1996-9

1999-05-24