Pytha

Pythagorean Triples

Three integers a, b, and c that satisfy a2 b2 = c2 are called Pythagorean Triples. There

are infinitely many such numbers and there also exists a way to generate all the triples.

Let n and m be integers, n*m. Then define(*) a = n2 - m2, b = 2nm, c = n2 m2.

The three number a, b, and c always form a Pythagorean triple. The proof is simple: (n2 -

m2)2 (2mn)2 = n4 - 2n2m2 m4 4n2m2 = n4 2n2m2 m4 = (n2 m2)2. The formulas were

known to Euclid and used by Diophantus to obtain Pythagorean triples with special

properties. However, he never raised the question whether in this way one can obtain all

possible triples.The fact is that for m and n coprime of different parities, (*) yields

coprime numbers a, b, and c. Conversely, all coprime triples can indeed be obtained in

this manner. All others are multiples of coprime triples: ka, kb, kc.As an aside, those

who mastered the arithmetic of complex numbers might have noticed that (m in)2 = (n2 -

m2) i2mn. Which probably indicates that (*) has a source in trigonometry. But the proof

below only uses simple geometry and algebra.First of all, note that if a2 b2 = c2, then

(a/c)2 (b/c)2 = 1. With x = a/c and y = b/c we get x2 y2 = 1. This is the well known

equation of the unit circle with center at the origin. Finding Pythagorean triples is

therefore equivalent to locating rational points (i.e., points (x,y) for which both x and

Pythagorean Triples

Three integers a, b, and c that satisfy a2 b2 = c2 are called Pythagorean Triples. There

are infinitely many such numbers and there also exists a way to generate all the triples.

Let n and m be integers, n*m. Then define(*) a = n2 - m2, b = 2nm, c = n2 m2.

The three number a, b, and c always form a Pythagorean triple. The proof is simple: (n2 -

m2)2 (2mn)2 = n4 - 2n2m2 m4 4n2m2 = n4 2n2m2 m4 = (n2 m2)2. The formulas were

known to Euclid and used by Diophantus to obtain Pythagorean triples with special

properties. However, he never raised the question whether in this way one can obtain all

possible triples.The fact is that for m and n coprime of different parities, (*) yields

coprime numbers a, b, and c. Conversely, all coprime triples can indeed be obtained in

this manner. All others are multiples of coprime triples: ka, kb, kc.As an aside, those

who mastered the arithmetic of complex numbers might have noticed that (m in)2 = (n2 -

m2) i2mn. Which probably indicates that (*) has a source in trigonometry. But the proof

below only uses simple geometry and algebra.First of all, note that if a2 b2 = c2, then

(a/c)2 (b/c)2 = 1. With x = a/c and y = b/c we get x2 y2 = 1. This is the well known

equation of the unit circle with center at the origin. Finding Pythagorean triples is

therefore equivalent to locating rational points (i.e., points (x,y) for which both x and

y are rational) on the unit circle. For if (p/q)2 (r/s)2 = 1, multiplication by a common

denominator leads to an identity between integers.Rational numbers approximate irrational

to any degree of accuracy. Therefore, the set of rational pairs is dense in the whole

plane. So, perhaps, one might expect that any curve should contain a lot of rational pairs

or meander wildly to avoid them. But this is not the case. The recent proof of Fermat's

Last Theorem lets us claim that the curves xN yN = 1 with N*2 contain no rational

points. But there are simpler examples. From Lindemann's theorem, we conclude that the

graph of a perfectly smooth function y = ex contains a single rational point, (0,1).

Moreover, pulling the unit circle even a little aside may change the picture drastically.

Let (xk, yk) = ( 2/k, 3/k), and consider a unit circle with center at (xk, yk). As k

grows, the point approaches the origin, but for no k, such a circle contains a rational

point.Let t be defined by(1) t = y/(x 1).

Then t(x 1) = y andt2(x 1)2 = y2 = 1 - x2 = (1 x)(1 - x). We are not interested in

negative x. So let's cancel (1 x) on both sides. The result ist2(x 1) = (1 - x). Solving

for x we get(2) x = (1 - t2)/(1 t2)

From y = t(1 x) we also obtain(3) y = 2t/(1 t2)

Formula (1)-(3) show that t is rational iff both x and y are rational.There is another way

to look at the just described configuration.The configuration consists of the unit circle

centered at the origin and a straight line passing through the point (-1,0) which lies on

the circle. Unless the line is tangent to the circle, the two have a second common point.

denominator leads to an identity between integers.Rational numbers approximate irrational

to any degree of accuracy. Therefore, the set of rational pairs is dense in the whole

plane. So, perhaps, one might expect that any curve should contain a lot of rational pairs

or meander wildly to avoid them. But this is not the case. The recent proof of Fermat's

Last Theorem lets us claim that the curves xN yN = 1 with N*2 contain no rational

points. But there are simpler examples. From Lindemann's theorem, we conclude that the

graph of a perfectly smooth function y = ex contains a single rational point, (0,1).

Moreover, pulling the unit circle even a little aside may change the picture drastically.

Let (xk, yk) = ( 2/k, 3/k), and consider a unit circle with center at (xk, yk). As k

grows, the point approaches the origin, but for no k, such a circle contains a rational

point.Let t be defined by(1) t = y/(x 1).

Then t(x 1) = y andt2(x 1)2 = y2 = 1 - x2 = (1 x)(1 - x). We are not interested in

negative x. So let's cancel (1 x) on both sides. The result ist2(x 1) = (1 - x). Solving

for x we get(2) x = (1 - t2)/(1 t2)

From y = t(1 x) we also obtain(3) y = 2t/(1 t2)

Formula (1)-(3) show that t is rational iff both x and y are rational.There is another way

to look at the just described configuration.The configuration consists of the unit circle

centered at the origin and a straight line passing through the point (-1,0) which lies on

the circle. Unless the line is tangent to the circle, the two have a second common point.

In order to find this point, we have to solve simultaneously two equations: the quadratic

equation of the circle x2 y2 = 1 and the linear equation of the line. By eliminating

either x or y from the latter, and substituting the result into the former, we get a

quadratic equation in one variable with integer coefficients. One solution of this

equation is immediate - it is related to the point (-1,0), and is rational. Therefore, the

second solution of the equation is also rational and gives either x- or y-coordinate of

the second point of intersection.

(3,4,5), (5,12,13), (6,8,10), (7,24,25), (8,15,17), (9,12,15), (9,40,41), (10,24,26),

(11,60,61), (12,16,20), (12,35,37), (13,84,85), (14,48,50), (15,20,25), (15,36,39),

(15,112,113), (16,30,34), (16,63,65), (17,144,145), (18,24,30), (18,80,82), (19,180,181),

(20,21,29), (20,48,52), (20,99,101), (21,28,35), (21,72,75), (21,220,221), (22,120,122),

equation of the circle x2 y2 = 1 and the linear equation of the line. By eliminating

either x or y from the latter, and substituting the result into the former, we get a

quadratic equation in one variable with integer coefficients. One solution of this

equation is immediate - it is related to the point (-1,0), and is rational. Therefore, the

second solution of the equation is also rational and gives either x- or y-coordinate of

the second point of intersection.

(3,4,5), (5,12,13), (6,8,10), (7,24,25), (8,15,17), (9,12,15), (9,40,41), (10,24,26),

(11,60,61), (12,16,20), (12,35,37), (13,84,85), (14,48,50), (15,20,25), (15,36,39),

(15,112,113), (16,30,34), (16,63,65), (17,144,145), (18,24,30), (18,80,82), (19,180,181),

(20,21,29), (20,48,52), (20,99,101), (21,28,35), (21,72,75), (21,220,221), (22,120,122),

(23,264,265), (24,32,40), (24,45,51), (24,70,74), (24,143,145), (25,60,65), (25,312,313),

(26,168,170), (27,36,45), (27,120,123), (27,364,365), (28,45,53), (28,96,100),

(28,195,197), (29,420,421), (30,40,50), (30,72,78), (30,224,226), (31,480,481),

(32,60,68), (32,126,130), (32,255,257), (33,44,55), (33,56,65), (33,180,183),

(33,544,545), (34,288,290), (35,84,91), (35,120,125), (35,612,613), (36,48,60),

(36,77,85), (36,105,111), (36,160,164), (36,323,325), (37,684,685), (38,360,362),

(39,52,65), (39,80,89), (39,252,255), (39,760,761), (40,42,58), (40,75,85), (40,96,104),

(40,198,202), (40,399,401), (41,840,841), (42,56,70), (42,144,150), (42,440,442),

(43,924,925), (44,117,125), (44,240,244), (44,483,485), (45,60,75), (45,108,117),

(45,200,205), (45,336,339), (46,528,530), (48,55,73), (48,64,80), (48,90,102),

(48,140,148), (48,189,195), (48,286,290), (48,575,577), (49,168,175), (50,120,130),

(50,624,626), (51,68,85), (51,140,149), (51,432,435), (52,165,173), (52,336,340),

(52,675,677), (54,72,90), (54,240,246), (54,728,730), (55,132,143), (55,300,305),

(56,90,106), (56,105,119), (56,192,200), (56,390,394), (56,783,785), (57,76,95),

(57,176,185), (57,540,543), (58,840,842), (60,63,87), (60,80,100), (60,91,109),

(60,144,156), (60,175,185), (60,221,229), (60,297,303), (60,448,452), (60,899,901),

(62,960,962), (63,84,105), (63,216,225), (63,280,287), (63,660,663), (64,120,136),

(64,252,260), (64,510,514), (65,72,97), (65,156,169), (65,420,425), (66,88,110),

(66,112,130), (66,360,366), (68,285,293), (68,576,580), (69,92,115), (69,260,269),

(69,792,795), (70,168,182), (70,240,250), (72,96,120), (72,135,153), (72,154,170),

(72,210,222), (72,320,328), (72,429,435), (72,646,650), (75,100,125), (75,180,195),

(75,308,317), (75,560,565), (75,936,939), (76,357,365), (76,720,724), (77,264,275),

(26,168,170), (27,36,45), (27,120,123), (27,364,365), (28,45,53), (28,96,100),

(28,195,197), (29,420,421), (30,40,50), (30,72,78), (30,224,226), (31,480,481),

(32,60,68), (32,126,130), (32,255,257), (33,44,55), (33,56,65), (33,180,183),

(33,544,545), (34,288,290), (35,84,91), (35,120,125), (35,612,613), (36,48,60),

(36,77,85), (36,105,111), (36,160,164), (36,323,325), (37,684,685), (38,360,362),

(39,52,65), (39,80,89), (39,252,255), (39,760,761), (40,42,58), (40,75,85), (40,96,104),

(40,198,202), (40,399,401), (41,840,841), (42,56,70), (42,144,150), (42,440,442),

(43,924,925), (44,117,125), (44,240,244), (44,483,485), (45,60,75), (45,108,117),

(45,200,205), (45,336,339), (46,528,530), (48,55,73), (48,64,80), (48,90,102),

(48,140,148), (48,189,195), (48,286,290), (48,575,577), (49,168,175), (50,120,130),

(50,624,626), (51,68,85), (51,140,149), (51,432,435), (52,165,173), (52,336,340),

(52,675,677), (54,72,90), (54,240,246), (54,728,730), (55,132,143), (55,300,305),

(56,90,106), (56,105,119), (56,192,200), (56,390,394), (56,783,785), (57,76,95),

(57,176,185), (57,540,543), (58,840,842), (60,63,87), (60,80,100), (60,91,109),

(60,144,156), (60,175,185), (60,221,229), (60,297,303), (60,448,452), (60,899,901),

(62,960,962), (63,84,105), (63,216,225), (63,280,287), (63,660,663), (64,120,136),

(64,252,260), (64,510,514), (65,72,97), (65,156,169), (65,420,425), (66,88,110),

(66,112,130), (66,360,366), (68,285,293), (68,576,580), (69,92,115), (69,260,269),

(69,792,795), (70,168,182), (70,240,250), (72,96,120), (72,135,153), (72,154,170),

(72,210,222), (72,320,328), (72,429,435), (72,646,650), (75,100,125), (75,180,195),

(75,308,317), (75,560,565), (75,936,939), (76,357,365), (76,720,724), (77,264,275),

(77,420,427), (78,104,130), (78,160,178), (78,504,510), (80,84,116), (80,150,170),

(80,192,208), (80,315,325), (80,396,404), (80,798,802), (81,108,135), (81,360,369),

(84,112,140), (84,135,159), (84,187,205), (84,245,259), (84,288,300), (84,437,445),

(84,585,591), (84,880,884), (85,132,157), (85,204,221), (85,720,725), (87,116,145),

(87,416,425), (88,105,137), (88,165,187), (88,234,250), (88,480,488), (88,966,970),

(90,120,150), (90,216,234), (90,400,410), (90,672,678), (91,312,325), (91,588,595),

(92,525,533), (93,124,155), (93,476,485), (95,168,193), (95,228,247), (95,900,905),

(96,110,146), (96,128,160), (96,180,204), (96,247,265), (96,280,296), (96,378,390),

(96,572,580), (96,765,771), (98,336,350), (99,132,165), (99,168,195), (99,440,451),

(99,540,549), (100,105,145), (100,240,260), (100,495,505), (100,621,629), (102,136,170),

(102,280,298), (102,864,870), (104,153,185), (104,195,221), (104,330,346), (104,672,680),

(105,140,175), (105,208,233), (105,252,273), (105,360,375), (105,608,617), (105,784,791),

(108,144,180), (108,231,255), (108,315,333), (108,480,492), (108,725,733), (108,969,975),

(110,264,286), (110,600,610), (111,148,185), (111,680,689), (112,180,212), (112,210,238),

(112,384,400), (112,441,455), (112,780,788), (114,152,190), (114,352,370), (115,252,277),

(115,276,299), (116,837,845), (117,156,195), (117,240,267), (117,520,533), (117,756,765),

(119,120,169), (119,408,425), (120,126,174), (120,160,200), (120,182,218), (120,209,241),

(120,225,255), (120,288,312), (120,350,370), (120,391,409), (120,442,458), (120,594,606),

(120,715,725), (120,896,904), (121,660,671), (123,164,205), (123,836,845), (124,957,965),

(125,300,325), (126,168,210), (126,432,450), (126,560,574), (128,240,272), (128,504,520),

(129,172,215), (129,920,929), (130,144,194), (130,312,338), (130,840,850), (132,176,220),

(132,224,260), (132,351,375), (132,385,407), (132,475,493), (132,720,732), (133,156,205),

(80,192,208), (80,315,325), (80,396,404), (80,798,802), (81,108,135), (81,360,369),

(84,112,140), (84,135,159), (84,187,205), (84,245,259), (84,288,300), (84,437,445),

(84,585,591), (84,880,884), (85,132,157), (85,204,221), (85,720,725), (87,116,145),

(87,416,425), (88,105,137), (88,165,187), (88,234,250), (88,480,488), (88,966,970),

(90,120,150), (90,216,234), (90,400,410), (90,672,678), (91,312,325), (91,588,595),

(92,525,533), (93,124,155), (93,476,485), (95,168,193), (95,228,247), (95,900,905),

(96,110,146), (96,128,160), (96,180,204), (96,247,265), (96,280,296), (96,378,390),

(96,572,580), (96,765,771), (98,336,350), (99,132,165), (99,168,195), (99,440,451),

(99,540,549), (100,105,145), (100,240,260), (100,495,505), (100,621,629), (102,136,170),

(102,280,298), (102,864,870), (104,153,185), (104,195,221), (104,330,346), (104,672,680),

(105,140,175), (105,208,233), (105,252,273), (105,360,375), (105,608,617), (105,784,791),

(108,144,180), (108,231,255), (108,315,333), (108,480,492), (108,725,733), (108,969,975),

(110,264,286), (110,600,610), (111,148,185), (111,680,689), (112,180,212), (112,210,238),

(112,384,400), (112,441,455), (112,780,788), (114,152,190), (114,352,370), (115,252,277),

(115,276,299), (116,837,845), (117,156,195), (117,240,267), (117,520,533), (117,756,765),

(119,120,169), (119,408,425), (120,126,174), (120,160,200), (120,182,218), (120,209,241),

(120,225,255), (120,288,312), (120,350,370), (120,391,409), (120,442,458), (120,594,606),

(120,715,725), (120,896,904), (121,660,671), (123,164,205), (123,836,845), (124,957,965),

(125,300,325), (126,168,210), (126,432,450), (126,560,574), (128,240,272), (128,504,520),

(129,172,215), (129,920,929), (130,144,194), (130,312,338), (130,840,850), (132,176,220),

(132,224,260), (132,351,375), (132,385,407), (132,475,493), (132,720,732), (133,156,205),