Pythagorean Triples - Advanced
(You may like to read Pythagoras' Theorem
and Introduction to Pythagorean Triples first)
A "Pythagorean Triple" is a set of positive integers a, b and c that fits the rule:
a^{2} + b^{2} = c^{2}
Triangles
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
A famous example of a Pythagorean Triples:
The 3,4,5 Triangle |
3^{2} + 4^{2} = 5^{2} |
9 + 16 = 25 |
Two more examples:
5, 12, 13 | 9, 40, 41 | |
5^{2} + 12^{2} = 13^{2} | 9^{2} + 40^{2} = 41^{2} | |
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, 5):
Let n be any integer greater than 1, then 3n, 4n and 5n are also 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) | |
4 | (12,16,20) | |
... | ... etc ... |
So we can make infinitely many triples just using the (3,4,5) triple.
Primitive Triples
In the case above (3,4,5) is a primitive triple,
But all its multiples, such as (6,8,10) etc, are not.
Primitive triples have this property: a, b and c share no common factors.
Example: (3,4,5)
3, 4 and 5 share no common factors, so (3,4,5) is a primitive triple
Example: (6,8,10)
6, 8 and 10 share a common factor of 2, so (6,8,10) is not a primitive triple
Euclid's Proof of 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 (one after the other) whole numbers is always an odd number.
Examples:
- 2^{2} − 1^{2} = 4 − 1 = 3 (an odd number),
- 3^{2} − 2^{2} = 9 − 4 = 5 (an odd number),
- 4^{2} − 3^{2} = 16 − 9 = 7 (an odd number),
- etc
Can you see how subtracting squares make odd numbers in this picture?
See Squares and Odd Numbers, or have a look at this table as an example:
n | n^{2} | n^{2} minus previous n^{2} |
---|---|---|
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 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 is even too. Similarly when one of a and b is odd and the other is even, c has 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 = m^{2} − n^{2}
- b = 2mn
- c = m^{2} + n^{2}
Then a, b and c form a Pythagorean Triple. This is known as "Euclid's formula".
Example: m=2 and n=1
- a = 2^{2} − 1^{2} = 4 − 1 = 3
- b = 2 × 2 × 1 = 4
- c = 2^{2} + 1^{2} = 4 + 1 = 5
And we get the first Pythagorean Triple (3,4,5).
Similarly, when m=3 and n=2 we get the next Pythagorean Triple (5,12,13).
This method creates all primitive triples, but we may need to swap a and b to see:
Example: when m=4 and n=1 we get (15,8,17), which is also (8,15,17)
It also creates some non-primitive triples (that are multiples of primitive triples):
Example: when m=3 and n=1 we get (8,6,10), which is also (6,8,10) by swapping.
But (6,8,10) is just (3,4,5) times 2
List of the First Few
Here is a list of all primitive Pythagorean Triples for a, b, and c less than 1000.
The list has only primitive triples, so (3,4,5) is there, but (6,8,10) etc are not
(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) |