This is Problem 9 on Project Euler:

A Pythagorean triplet is a set of three natural numbers, a < b < c, for which,

a^{2}+b^{2}=c^{2}For example, 3

^{2}+ 4^{2}= 9 + 16 = 25 = 5^{2}.There exists exactly one Pythagorean triplet for which

a+b+c= 1000.

Find the productabc.

Pythagorean triplets can be generated by using the following formulas, where m and n are integers and m is greater than n:

m^{2} – n^{2}

2mn

m^{2} + n^{2}

So for the purposes of this problem we are looking for values for m and n where these three formulas add up to 1000, that is:

(m^{2} – n^{2}) + (2mn) + (m^{2} + n^{2}) = 1000

This can be simplified to 2m^{2} + 2mn = 1000

Dividing by 2 and factoring by m we have m(m+n) = 500

Therefore m must be a factor of 500. Calculating m, m + n and n we get the following table:

m | m + n | n | notes |

1 | 500 | 499 | Reject, because n is greater than m |

2 | 250 | 248 | Reject, because n is greater than m |

5 | 100 | 95 | Reject, because n is greater than m |

10 | 50 | 40 | Reject, because n is greater than m |

20 | 25 | 5 | A solution |

25 | 20 | -5 | Reject, because n cannot be negative |

50 | 10 | -40 | Reject, because n cannot be negative |

100 | 5 | -95 | Reject, because n cannot be negative |

250 | 2 | -248 | Reject, because n cannot be negative |

500 | 1 | -499 | Reject, because n cannot be negative |

Therefore the only solution is where m = 20 and n = 5. Plugging these values into our original expressions we have:

a = m^{2} – n^{2} = (20 * 20) – (5 * 5) = 375

b = 2mn = 2 * 20 * 5 = 200

c = m^{2} + n^{2} = (20 * 20) + (5 * 5) = 425

a + b + c = 1000 and ^{2} + `b`^{2} = `c`^{2}, so the answer to the problem is a * b * c.

We have not had to use a computer to solve this problem, but a brute-force approach in Delphi could be this:

procedure TfrmMain.Problem9; var a, b, c : integer; begin for a := 1 to 998 do begin for b := a + 1 to 999 do begin c := 1000 - (a + b); if (a * a) + (b * b) = (c * c) then begin ShowMessage(IntToStr(a * b * c)); exit; end; end; end; end;

This code just generates sets of numbers where a + b + c = 1000 and then checks the set for being a Pythagorean triple. If it is, the product is displayed. There are no optimizations in this code, which could be improved considerably.

Tags: Mathematics, Programming, Project Euler

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