Normal (2D) Pythagoras Theorem Numbers

Pythagoras theorem applies to a triangle with lengths a, b and c as shown in diagram.It indicates that

*the square of the hypotenuse is equal to the sum of the squares of the other two sides*

If you look at the diagram, this means that a

^{2}+ b

^{2}= c

^{2}

The table below indicates some of the "integer" values that obey the rule and form right-angled triangles.

3 | 4 | 5 | and all multiples e.g. | 6 | 8 | 10 |

5 | 12 | 13 | and all multiples e.g. | 10 | 24 | 26 |

7 | 24 | 25 | and all multiples e.g. | 14 | 48 | 50 |

8 | 15 | 17 | and all multiples e.g. | 16 | 30 | 34 |

9 | 40 | 41 | and all multiples e.g. | 18 | 80 | 82 |

12 | 35 | 37 | and all multiples e.g. | 24 | 70 | 74 |

20 | 21 | 29 | and all multiples e.g. | 40 | 42 | 58 |

Extended (3D) Pythagoras Theorem Numbers

Pythagoras theorem can be extended for trianglular prisms with lengths a, b c, and diagonal d as shown in diagram.It indicates that

*the square of the diagonal is equal to the sum of the squares of the three main edges*

If you look at the diagram a

^{2}+ b

^{2}+ c

^{2}= d

^{2}

The table below indicates some of the "integer" values that obey the rule and form right-angled prisms.

1 | 2 | 2 | 3 |

1 | 4 | 8 | 9 |

2 | 4 | 4 | 6 |

2 | 3 | 6 | 7 |

2 | 6 | 9 | 11 |

3 | 6 | 6 | 9 |

4 | 4 | 7 | 9 |

4 | 8 | 8 | 12 |

5 | 10 | 10 | 15 |

6 | 6 | 8 | 11 |

Sum of 3 Cubes

There are very complex mathematical theories as to whether any set of integers exist which could
fulfil the equation ......
a^{3}+ b

^{3}= c

^{3}

However, integer solutions for extended version of the cubic equation ...... a

^{3}+ b

^{3}+ c

^{3}= d

^{3}...... include

1 | 6 | 8 | 9 |

2 | 12 | 16 | 18 |

3 | 4 | 5 | 6 |

3 | 10 | 18 | 19 |

4 | 3 | 5 | 6 |

5 | 3 | 4 | 6 |

6 | 1 | 8 | 9 |

6 | 8 | 10 | 12 |

7 | 14 | 17 | 20 |

8 | 1 | 6 | 9 |

8 | 6 | 10 | 12 |

9 | 12 | 15 | 18 |

10 | 3 | 18 | 19 |

10 | 8 | 6 | 12 |

12 | 2 | 16 | 18 |

12 | 15 | 9 | 18 |

12 | 16 | 20 | 24 |

14 | 7 | 17 | 20 |

15 | 9 | 12 | 18 |

16 | 2 | 12 | 18 |

16 | 12 | 20 | 24 |

17 | 7 | 14 | 20 |

18 | 3 | 10 | 19 |

20 | 12 | 16 | 24 |

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