Math is Fun

Math is Fun Home PageHOME

A-Z Listing




Top-rated teacher Web sites rated by teacher usage
Edunet site of the week January 2001
Maths Is Fun is ICRA Registered

Copyright © 2006
MathsIsFun.com

Pythagorean Triples - Advanced

Pythagoras' Theorem states that the sum of the areas of the two squares formed along the two small sides of a right angled triangle equals the area of the square formed along the longest.

Let a, b, and c be the three sides of a right angled triangle. (A right angled triangle is a triangle in which any one of the angles is equal to 90 degrees.) The longest side of the right angled triangle is called the 'hypotenuse'.

Pythagoras theorem is written in the form of an equation is:

a2 + b2 = c2

where c is the hypotenuse while a and b are the other sides of the triangle.

If a, b, and c are positive integers, they are together called Pythagorean Triples.

The smallest such Pythagorean Triple is 3, 4 and 5. It can be seen that 32 + 42 = 52 (9+16=25).

Here are some examples:

pythagorean triple pythagorean triple
3,4,5 Triangle 5,12,13 triangle 9,40,41 Triangle
32 + 42 = 52 52 + 122 = 132 92 + 402 = 412

 

Endless

The set of Pythagorean Triples is endless.

It is easy to prove this with the help of the first Pythagorean triple, (3, 4, and 5):

Let n be any integer greater than 1: 3n, 4n and 5n would also be a set of Pythagorean triple. This is true because:

(3n)2 + (4n)2 = (5n)2

n (3n, 4n, 5n)
2 (6,8,10)
3 (9,12,15)
... ... etc ...

So, you can make infinite triples just using the (3,4,5) triple.

Euclid's Proof that there are Infinitely Many Pythagorean Triples

However, Euclid used a different reasoning to prove the set of Pythagorean triples is unending.

The proof was based on the fact that the difference of the squares of any two consecutive numbers is always an odd number.

For example, 22 - 12 = 4-1 = 3, 152 - 142 = 225-196 = 29.

And also every odd number can be expressed as a difference of the squares of two consecutive numbers. Have a look at this table as an example:

n n2 Difference
1 1  
2 4 4-1 = 3
3 9 9-4 = 5
4 16 16-9 = 7
5 25 25-16 = 9
... ... ...

And there are an infinite number of odd numbers.

Since there are infinite number of odd numbers, and a part of them are perfect squares, there are an infinite number of odd squares (Since a fraction of infinity is also infinity). Therefore, there are infinite Pythagorean triples.

Properties

It can be observed that the Pythagorean triple consists of:

  • all even numbers, or
  • two odd numbers and an even number.

The Pythagorean triple can never be made up of all odd numbers or two even numbers and one odd number. This is true because:

  • (i) The square of an odd number is an odd number and the square of an even number is an even number.
  • (ii) The sum of two even numbers is an even number and the sum of an odd number and an even number is in odd number.

Therefore, if one of a and b is odd and the other is even, c would have to be odd. Similarly, if both a and b are even, c would be an even number too!

Constructing Pythagorean Triples

It is easy to construct sets of Pythagorean triples. If m and n are any two natural numbers,
Let a = n2 - m2, b = 2nm, c = n2 + m2. Then, a, b, and c form a Pythagorean triple.

For example, let m=1 and n=2.
a = 22-12 = 4-1 = 3
b = 2 × 2 × 1 = 4
c = 22+12 = 5
Thus, we obtain the first Pythagorean triple (3,4,5).

Similarly, when m=2 and n=3 we get the next Pythagorean triple (5,12,13).

List of the First Few

Here is a list of all Pythagorean triples where a, b, and c are less than 1,000.

The list only contains the first set (a,b,c) which is a Pythagorean triple (primitive Pythagorean triples). The multiples of (a,b,c), (ie. (na,nb,nc)), which also form a Pythagorean triple are not given in the list. For example, it has already been seen that (3,4,5) is a Pythagorean triple and so is (6,8,10). However, (6,8,10) is obtained by multiplying (3,4,5) by 2. Hence only (3,4,5) would be shown

(3,4,5) (5,12,13) (7,24,25) (8,15,17) (9,40,41)
(11,60,61) (12,35,37) (13,84,85) (15,112,113) (16,63,65)
(17,144,145) (19,180,181) (20,21,29) (20,99,101) (21,220,221)
(23,264,265) (24,143,145) (25,312,313) (27,364,365) (28,45,53)
(28,195,197) (29,420,421) (31,480,481) (32,255,257) (33,56,65)
(33,544,545) (35,612,613) (36,77,85) (36,323,325) (37,684,685)
(39,80,89) (39,760,761) (40,399,401) (41,840,841) (43,924,925)
(44,117,125) (44,483,485) (48,55,73) (48,575,577) (51,140,149)
(52,165,173) (52,675,677) (56,783,785) (57,176,185) (60,91,109)
(60,221,229) (60,899,901) (65,72,97) (68,285,293) (69,260,269)
(75,308,317) (76,357,365) (84,187,205) (84,437,445) (85,132,157)
(87,416,425) (88,105,137) (92,525,533) (93,476,485) (95,168,193)
(96,247,265) (100,621,629) (104,153,185) (105,208,233) (105,608,617)
(108,725,733) (111,680,689) (115,252,277) (116,837,845) (119,120,169)
(120,209,241) (120,391,409) (123,836,845) (124,957,965) (129,920,929)
(132,475,493) (133,156,205) (135,352,377) (136,273,305) (140,171,221)
(145,408,433) (152,345,377) (155,468,493) (156,667,685) (160,231,281)
(161,240,289) (165,532,557) (168,425,457) (168,775,793) (175,288,337)
(180,299,349) (184,513,545) (185,672,697) (189,340,389) (195,748,773)
(200,609,641) (203,396,445) (204,253,325) (205,828,853) (207,224,305)
(215,912,937) (216,713,745) (217,456,505) (220,459,509) (225,272,353)
(228,325,397) (231,520,569) (232,825,857) (240,551,601) (248,945,977)
(252,275,373) (259,660,709) (260,651,701) (261,380,461) (273,736,785)
(276,493,565) (279,440,521) (280,351,449) (280,759,809) (287,816,865)
(297,304,425) (300,589,661) (301,900,949) (308,435,533) (315,572,653)
(319,360,481) (333,644,725) (336,377,505) (336,527,625) (341,420,541)
(348,805,877) (364,627,725) (368,465,593) (369,800,881) (372,925,997)
(385,552,673) (387,884,965) (396,403,565) (400,561,689) (407,624,745)
(420,851,949) (429,460,629) (429,700,821) (432,665,793) (451,780,901)
(455,528,697) (464,777,905) (468,595,757) (473,864,985) (481,600,769)
(504,703,865) (533,756,925) (540,629,829) (555,572,797) (580,741,941)
(615,728,953) (616,663,905) (696,697,985)