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# Time of Birth

We are looking for solutions to the equation $x^2 = y^2 + z^2$ where the woman was born in the year $y$ (between $0$ A.D. and $1997$), she lived $z$ years (at least $1$ year and not more than $120$ years) and she died in the year $x$ where $x < 1997$ . There are twenty possible solutions. If she was born in $0$ A.D. there are ten possible solutions. The remaining solutions are:

This can never happen in the future (taking $x^2 > 1997$). Even if the woman had a longer life there are still no solutions in the past for a lifespan of $121$ years, but there are four solutions for a lifespan of $144$ years. In the next $2000$ years (assuming a lifespan of no more than $400$ years) the only solutions are:

The sets of numbers $x$, $y$ and $z$ are Pythagorean triples.

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We are looking for solutions to the equation $x^2 = y^2 + z^2$ where the woman was born in the year $y$ (between $0$ A.D. and $1997$), she lived $z$ years (at least $1$ year and not more than $120$ years) and she died in the year $x$ where $x < 1997$ . There are twenty possible solutions. If she was born in $0$ A.D. there are ten possible solutions. The remaining solutions are:

Born | Lived | Died |

$y^2$ | $z^2$ | $x^2$ |

$9$ | $16$ | $25$ |

$16$ | $9$ | $25$ |

$36$ | $64$ | $100$ |

$64$ | $36$ | $100$ |

$144$ | $25$ | $169$ |

$144$ | $81$ | $225$ |

$225$ | $64$ | $289$ |

$576$ | $49$ | $625$ |

$576$ | $100$ | $676$ |

$1600$ | $81$ | $1681$ |

This can never happen in the future (taking $x^2 > 1997$). Even if the woman had a longer life there are still no solutions in the past for a lifespan of $121$ years, but there are four solutions for a lifespan of $144$ years. In the next $2000$ years (assuming a lifespan of no more than $400$ years) the only solutions are:

Born | Lived | Died |

$y^2$ | $z^2$ | $x^2$ |

$2304$ | $196$ | $2500$ |

$2304$ | $400$ | $2704$ |

$3600$ | $121$ | $3721$ |

$3969$ | $256$ | $4225$ |

The sets of numbers $x$, $y$ and $z$ are Pythagorean triples.

How many four digit square numbers are composed of even numerals? What four digit square numbers can be reversed and become the square of another number?

Square numbers can be represented as the sum of consecutive odd numbers. What is the sum of 1 + 3 + ..... + 149 + 151 + 153?