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Pythagorean Triples
pythagorean
triples | Pythagorean quadruples | Fermat's
last theorem
More on Pythagorean Triples
A Pythagorean triple is a set of three numbers a, b and c such that:
a2 + b2 = c2
In other words, three natural numbers that satisfy Pythagoras' theorem.
Pythagorean triples consists of either all even
numbers, or two odd numbers and an even number, where one of a or b will
be even.
Back in 300 BC a Greek guy named Euclid came up with a way of constructing
a set of Pythagorean triples. Take m and n, which can be any two whole
numbers. Then a, b, and c of a Pythagorean triple are:
a = n2 - m2
b = 2nm
c = n2 + m2
(Yes, a is the difference of two squares, while c is the sum of those two squares.)
There are 16 Pythagorean triples where all the digits are less than
100. Note the pattern where the first of the triple is odd.
| c = b + 1 |
c = b + 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) |
(16, 63, 65) |
Others: (20, 21, 29) (28, 45, 53) (33, 56,
65)
(36, 77, 85) (39, 80, 89) (48, 55, 73) (65, 72, 97) |
Here are all the Pythagorean triples where a, b, and c are less than
1,000 (multiples are not included), with the same c=b+1 pattern
moved to the front of the rows.
(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) [This is the last in the format c=b+1 with b and c
less than 1,000.]
(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)
Pythagorean Quadruples
By the way, there are also Pythagorean quadruples.
a2 + b2 + c2 = d2
32 + 82 + 362 = 372
Some small Pythagorean quadruples:
(1,2,2,3) (2,3,6,7) (1,4,8,9) (4,4,7,9)
(2,6,9,11) (6,6,7,11) (3,4,12,13) (2,5,14,15) (2, 10, 11, 15) (1,12,12,17)
(8,9,12,17) (1,6,18,19) (6,6,17,19) (6,10,15,19) (4,5,20,21) (4,8,19,21)
(4,13,16,21) (8,11,16,21) (3,6,22,23) (3,14,18,23) (6,13,18,23) (9,
12, 20, 25) (12, 15, 16, 25) (2,7,26,27) (2,10,25,27) (2,14,23,27)
(7,14,22,27) (10,10,23,27) (3,16,24,29) (11,12,24,29) (12,16,21,29)
Fermat's Last Theorem
One of the famous puzzles in mathematics is known as Fermat's
Last Theorem. The theorem itself basically states that you can't do a3 + b3 = c3 for
any whole numbers, or for any higher powers (eg, a4 + b4 =
c4, etc), either.
If an integer n is greater than 2, then the equation
an + bn = cn
has no solutions in non-zero integers a, b, and c.
It was long thought to be true, but how could it be proved?
In 1637 a mathematician
named Pierre de Fermat left a note (in Latin) in the margin of a book
saying "I have a truly marvellous proof of this proposition which
this margin is too narrow to contain."
All Fermat's other theorems
were (eventually) proved or disproved, but this one held on, and defied
proof for hundreds of years. Fermat did prove the case for
n=4, which reduced the problem to proving the theorem for prime numbers.
Wikipedia:
Over the next two centuries (1637–1839), the conjecture was proven
for only the primes 3, 5, and 7, although Sophie Germain proved a special
case for all primes less than 100. In the mid-19th century, Ernst Kummer
proved the theorem for a large (probably infinite) class of primes known
as regular primes. Building on Kummer's work and using sophisticated
computer studies, other mathematicians were able to prove the conjecture
for all odd primes up to four million.
But the general proof for all n still hadn't been found. And so
Fermat's Last Theorem remained unproved for 358 years, when English mathematician
Andrew Wiles finally published a proof in 1995.
Most mathematicians doubt that Fermat really
had a proof, but the problem became the most famous unsolved – and
now most famous solved – problem in mathematics. It is thought to have
generated more false proofs than any other theorem in mathematics.
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