## A Collection of Problems on Hyperbolas and Special Polygonal Numbers: Problems on HyperbolasThis book contains a reasonable collection of problems on hyperbolas represented by binary quadratic Diophantine equations. From the integer solutions of each of the above equations, the relations among special polygonalnumbers are obtained. The formal prerequisites for the material are minimal. It is hoped that these problems may create an interest in the hearts of researchers and lovers of mathematics who approach it with pure love for its beauty. There is no wonder that binary quadratic Diophantine equations in connection with polygonal numbers are beautiful and tricky enough to keep a mathematician occupied for entire life. |

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### Contents

1 | |

Hyperbola Represented By Negative Pell Equation 1936 | 19 |

Hyperbola represented By 3750 | 37 |

Hyperbola represented By 5162 | 51 |

### Common terms and phrases

11 Problem and(x Applying Brahmagupta lemma Consider the hyperbola consider the pellian consider the positive defined by JC Diophantine equations distinct integer solutions DxOxn Dyny equation X2 examples are given examples P1 n+1 f g n+1 F ſ following Table given by JC given by y hyperbola given HYPERBOLA REPRESENTED irrational roots occur least positive solution Let n,}and ſm Let n,n Let n,n}and m,n m1+s n,n}and m s+1 n+1 F Nasty number Note that t5.n Note that tº Numerical examples P1 O-ºn Observations occur in pairs P1 n+1 Whil P1 n+1 Wnt1 pellian equation perfect square Problem Pl n 11 Pl n+1 positive integer solution positive integers defined positive pell equation r1 F represent the integer S+1 2 Note sequence of integer sequences of positive ſm s+1 smallest positive integer sº Applying Brahmagupta solution of 1.14 Solution T.I. solution x,y solutions º solutions to 1.7 solving