# Pytha Essay

This essay has a total of 3356 words and 10 pages.

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

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History of Math Mathematics, study of relationships among quantities, magnitudes, and properties and of logical operations by which unknown quantities, magnitudes, and properties may be deduced. In the past, mathematics was regarded as the science of quantity, whether of magnitudes, as in geometry, or of numbers, as in arithmetic, or of the generalization of these two fields, as in algebra. Toward the middle of the 19th century, however, mathematics came to be regarded increasingly as the scienc
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Pi The History of Pi A little known verse in the bible reads “And he made a molten sea, ten cubits from the one brim to the other; it was round all about, and his height was five cubits; and a line of thirty cubits did compass it about(I Kings 7, 23).” This passage from the bible demonstrates the ancient nature of the irrational number pi. Pi in fact is mentioned in a number of verses throughout the bible. In II Chronicles 4,2, in the passage describing the building of the great temple of Solomo
• Pytha
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 Dioph
• Sir Isaac Newton
Sir Isaac Newton Sir Isaac Newton Through his early life experiences and with the knowledge left by his predecessors, Sir Isaac Newton was able to develop calculus, natural forces, and optics. From birth to early childhood, Isaac Newton overcame many personal, social, and mental hardships. It is through these experiences that helped create the person society knows him as in this day and age. The beginning of these obstacles started at birth for Newton. Isaac was born premature on Christmas Day 1
• Abstract 4
Abstract 4 On Tuesday the 1st of February in the year two thousand the Physics 37 class performed an experiment on projectile motion. First the class divided it’s self in to groups of three or more members. The class was then thought how to us the equipment and the devices. The equipment and devices consisted of a ballistic pendulum apparatus, plain white and carbon paper, spirit level, short support rod, one and two meter sticks, large cardboard and a metric ruler. Other equipment and devices c
• Recently, the media has spent an increasing amount
afd Recently, the media has spent an increasing amount of broadcast time on new technology. The focus of high-tech media has been aimed at the flurry of advances concerning artificial intelligence (AI). What is artificial intelligence and what is the media talking about? Are these technologies beneficial to our society or mere novelties among business and marketing professionals? Medical facilities, police departments, and manufacturing plants have all been changed by AI but how? These questions
• Agfa
agfa Recently, the media has spent an increasing amount of broadcast time on new technology. The focus of high-tech media has been aimed at the flurry of advances concerning artificial intelligence (AI). What is artificial intelligence and what is the media talking about? Are these technologies beneficial to our society or mere novelties among business and marketing professionals? Medical facilities, police departments, and manufacturing plants have all been changed by AI but how? These question
• Recently, the media has spent an increasing amount
AI Recently, the media has spent an increasing amount of broadcast time on new technology. The focus of high-tech media has been aimed at the flurry of advances concerning artificial intelligence (AI). What is artificial intelligence and what is the media talking about? Are these technologies beneficial to our society or mere novelties among business and marketing professionals? Medical facilities, police departments, and manufacturing plants have all been changed by AI but how? These questions