A special case of the quadratic Diophantine Equation having the form

(1) |

(2) |

Pell equations, as well as the analogous equation with a minus sign on the right, can be solved by finding the Continued
Fraction
for . (The trivial solution , is ignored in all subsequent discussion.)
Let denote the th Convergent
, then we are looking for a convergent which obeys the
identity

(3) |

(4) |

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

(5) |

(6) |

Given one solution (which can be found as above), a whole family of solutions can be found by taking
each side to the th Power,

(7) |

(8) |

(9) | |||

(10) |

which gives the family of solutions

(11) | |||

(12) |

These solutions also hold for

(13) |

The following table gives the smallest integer solutions to the Pell equation with constant (Beiler 1966, p. 254).
Square are not included, since they would result in an equation of the form

(14) |

2 | 3 | 2 | 54 | 485 | 66 |

3 | 2 | 1 | 55 | 89 | 12 |

5 | 9 | 4 | 56 | 15 | 2 |

6 | 5 | 2 | 57 | 151 | 20 |

7 | 8 | 3 | 58 | 19603 | 2574 |

8 | 3 | 1 | 59 | 530 | 69 |

10 | 19 | 6 | 60 | 31 | 4 |

11 | 10 | 3 | 61 | 1766319049 | 226153980 |

12 | 7 | 2 | 62 | 63 | 8 |

13 | 649 | 180 | 63 | 8 | 1 |

14 | 15 | 4 | 65 | 129 | 16 |

15 | 4 | 1 | 66 | 65 | 8 |

17 | 33 | 8 | 67 | 48842 | 5967 |

18 | 17 | 4 | 68 | 33 | 4 |

19 | 170 | 39 | 69 | 7775 | 936 |

20 | 9 | 2 | 70 | 251 | 30 |

21 | 55 | 12 | 71 | 3480 | 413 |

22 | 197 | 42 | 72 | 17 | 2 |

23 | 24 | 5 | 73 | 2281249 | 267000 |

24 | 5 | 1 | 74 | 3699 | 430 |

26 | 51 | 10 | 75 | 26 | 3 |

27 | 26 | 5 | 76 | 57799 | 6630 |

28 | 127 | 24 | 77 | 351 | 40 |

29 | 9801 | 1820 | 78 | 53 | 6 |

30 | 11 | 2 | 79 | 80 | 9 |

31 | 1520 | 273 | 80 | 9 | 1 |

32 | 17 | 3 | 82 | 163 | 18 |

33 | 23 | 4 | 83 | 82 | 9 |

34 | 35 | 6 | 84 | 55 | 6 |

35 | 6 | 1 | 85 | 285769 | 30996 |

37 | 73 | 12 | 86 | 10405 | 1122 |

38 | 37 | 6 | 87 | 28 | 3 |

39 | 25 | 4 | 88 | 197 | 21 |

40 | 19 | 3 | 89 | 500001 | 53000 |

41 | 2049 | 320 | 90 | 19 | 2 |

42 | 13 | 2 | 91 | 1574 | 165 |

43 | 3482 | 531 | 92 | 1151 | 120 |

44 | 199 | 30 | 93 | 12151 | 1260 |

45 | 161 | 24 | 94 | 2143295 | 221064 |

46 | 24335 | 3588 | 95 | 39 | 4 |

47 | 48 | 7 | 96 | 49 | 5 |

48 | 7 | 1 | 97 | 62809633 | 6377352 |

50 | 99 | 14 | 98 | 99 | 10 |

51 | 50 | 7 | 99 | 10 | 1 |

52 | 649 | 90 | 101 | 201 | 20 |

53 | 66249 | 9100 | 102 | 101 | 10 |

The first few minimal values of and for nonsquare are 3, 2, 9, 5, 8, 3, 19, 10, 7, 649, ... (Sloane's A033313) and 2, 1, 4, 2, 3, 1, 6, 3, 2, 180, ... (Sloane's A033317), respectively. The values of having , 3, ... are 3, 2, 15, 6, 35, 12, 7, 5, 11, 30, ... (Sloane's A033314) and the values of having , 2, ... are 3, 2, 7, 5, 23, 10, 47, 17, 79, 26, ... (Sloane's A033318). Values of the incrementally largest minimal are 3, 9, 19, 649, 9801, 24335, 66249, ... (Sloane's A033315) which occur at , 5, 10, 13, 29, 46, 53, 61, 109, 181, ... (Sloane's A033316). Values of the incrementally largest minimal are 2, 4, 6, 180, 1820, 3588, 9100, 226153980, ... (Sloane's A033319), which occur at , 5, 10, 13, 29, 46, 53, 61, ... (Sloane's A033320).

**References**

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.

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

Dörrie, H. *100 Great Problems of Elementary Mathematics: Their History and Solutions.* New York: Dover, 1965.

Lagarias, J. C. ``On the Computational Complexity of Determining the Solvability or Unsolvability of the Equation
.''
*Trans. Amer. Math. Soc.* **260**, 485-508, 1980.

Sloane, N. J. A. Sequences A033313, A033314, A033315, A033316, A033317, A033318, A033319, and A033320 in ``An On-Line Version of the Encyclopedia of Integer Sequences.'' http://www.research.att.com/~njas/sequences/eisonline.html.

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.*
Lupton, AZ: Erhus University Press, 1996.

Stillwell, J. C. *Mathematics and Its History.* New York: Springer-Verlag, 1989.

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

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1999-05-26