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

Pythagoras a b c triangle

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):

pythagoras squares: a^2+b^2=c^2

Note:

Pythagorean Triples

A famous example of a Pythagorean Triples:

3,4,5 Triangle
The 3,4,5 Triangle
32 + 42 = 52
9 + 16 = 25

Two more examples:

5,12,13 Triangle   9,40,41 Triangle
5, 12, 13   9, 40, 41
52 + 122 = 132   92 + 402 = 412
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 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)
...   ... 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)

odd square numbers

And also every odd number can be expressed as a difference of the squares of two consecutive numbers. See Squares and Odd Numbers, or 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:

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

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):

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

Example: m=1 and n=2

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.

But the list only has the first set (a,b,c) which is a Pythagorean Triple (called primitive Pythagorean Triples), so the multiples of (a,b,c), such as (2a,2b,2c), (3a,3b,3c), etc are not in the list.

Example: (3,4,5) is a Pythagorean Triple. (6,8,10) is also a Pythagorean Triple, but is not shown as it is just (3,4,5) times 2.

(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)

by ganesh