Pythagorean Triples - Advanced

(You may like to read about Pythagoras' Theorem
or an Introduction to Pythagorean Triples first)

 

A "Pythagorean Triple" is a set of positive integers, a, b and c that fits the rule:

a2 + b2 = c2

Triangles

Pythagoras

And when we make a triangle with sides a, b and c it will be a right angled triangle (see Pythagoras' Theorem for more details):

Note:

  • c is the longest side of the triangle, called the "hypotenuse"
  • a and b are the other two sides

Pythagorean Triples

Examples of Pythagorean Triples are:

3, 4, 5 5, 12, 13 9, 40, 41
pythagorean triple pythagorean triple
32 + 42 = 52 52 + 122 = 132 92 + 402 = 412
9 + 16 = 25 25 + 144 = 169 (try it yourself)

Endless

The set of Pythagorean Triples is endless.

We can prove this with the help of the first Pythagorean Triple (3, 4, and 5):

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

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

Examples:

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

So we can make infinitely many triples just using the (3,4,5) triple.

Euclid's Proof that there are Infinitely Many Pythagorean Triples

But 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.

Examples:

22 - 12 = 4-1 = 3 (an odd number),

152 - 142 = 225-196 = 29 (an odd number)

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 the perfect squares form a subset of the odd numbers, and a fraction of infinity is also infinity, it follows that there must also be an infinite number of odd squares. So there are an infinite number of Pythagorean Triples.

Properties

An interesting fact: a Pythagorean Triple always consists of:

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

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

  • The square of an odd number is an odd number and the square of an even number is an even number.
  • 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.

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

Constructing Pythagorean Triples

It is easy to construct sets of Pythagorean Triples.

When m and n are any two positive integers (m < n):

  • a = n2 - m2
  • b = 2nm
  • c = n2 + m2

Then a, b, and c form a Pythagorean Triple.

Example: 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)