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This is Problem 9 on Project Euler:

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

a² + b² = c²

For example, 3² + 4² = 9 + 16 = 25 = 5².

There exists exactly one Pythagorean triplet for which a + b + c = 1000.
Find the product abc.

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

m² – n²
2mn
m² + n²

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:

(m² – n²) + (2mn) + (m² + n²) = 1000
This can be simplified to 2m² + 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 = m² – n² = (20 * 20) – (5 * 5) = 375

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

c = m² + n² = (20 * 20) + (5 * 5) = 425

a + b + c = 1000 and ² + b² = c², 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

This entry was posted on Wednesday, August 11th, 2010 at 9:56 am and is filed under Mathematics, Programming. You can follow any responses to this entry through the RSS 2.0 feed. Both comments and pings are currently closed.