PYTHAGOREAN TRIPLE

A 'Pythagorean triple' consists of three positive integers ''a'', ''b'', and ''c'', such that ''a''² + ''b''² = ''c''². Such a triple is commonly written (''a'', ''b'', ''c''), and a well-known example is (3, 4, 5). If (''a'', ''b'', ''c'') is a Pythagorean triple, then so is (''ka'', ''kb'', ''kc'') for any positive integer ''k''. A 'primitive Pythagorean triple' is one in which ''a'', ''b'' and ''c'' are coprime.
The name is derived from the Pythagorean theorem, of which every Pythagorean triple is a solution. The converse is not true. For instance, the triangle with sides ''a'' = ''b'' = 1 and ''c'' = √2 is right, but (1, 1, √2) is not a Pythagorean triple because √2 is not an integer. Moreover, 1 and √2 don't have an integer common multiple because √2 is irrational.
There are 16 primitive Pythagorean triples with ''c'' ≤ 100:

( 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)	(20, 21, 29)	(28, 45, 53)	(33, 56, 65)
(36, 77, 85)	(39, 80, 89)	(48, 55, 73)	(65, 72, 97)

Contents

Generating Pythagorean triples

Properties of Pythagorean triples

Some relationships

Unit circle relationships

Half-angle relationships

A special case: the Platonic sequence

Other formulas for generating triples

Parent/child relationships

Generalizations

See also

References

External links

Generating Pythagorean triples

An effective way to generate Pythagorean triples is based on the observation that if ''m'' and ''n'' are two positive integers with ''m'' > ''n'', then
:$a\; =\; m^2\; -\; n^2,,$
:$b=\; 2mn,,$
:$c\; =\; m^2\; +\; n^2,,$
is a Pythagorean triple. It is primitive if and only if ''m'' and ''n'' are relatively prime and one of them is even (if both ''n'' and ''m'' are odd, then ''a'', ''b'', and ''c'' will be even, and so the Pythagorean triple will not be primitive). Not every Pythagorean triple can be generated in this way, but every ''primitive'' triple (possibly after exchanging ''a'' and ''b'') arises in this fashion from a unique pair of coprime numbers ''m'' > ''n''. This shows that there are infinitely many primitive Pythagorean triples. This formula was given by Euclid (c. 300 B.C.) in his book ''Elements'' and is referred to as Euclid's formula.
The relation betweeen $(m,n)$ and $(a,b)$ is that in the complex plane $a+ib$ is $m+in$ squared.
An alternate form of the Euclid formula eliminates the negative sign by making use of the relation ''m'' = ''p'' + ''q'' and ''n'' = ''p'':
:$a\; =\; q(2p\; +\; q)\; ,$
:$b\; =\; 2p(p+q)\; ,$
:$c\; =\; (p\; +\; q)^2\; +\; p^2\; ,$
See also Other formulas for generating triples below for more generating formulas.

Properties of Pythagorean triples

The properties of primitive Pythagorean triples include:

★ Exactly one of ''a'', ''b'' is odd; ''c'' is odd.

★ The area (''A'' = ''ab''/2) is an integer.

★ Exactly one of ''a'', ''b'' is divisible by 3.

★ Exactly one of ''a'', ''b'' is divisible by 4.

★ Exactly one of ''a'', ''b'', ''c'' is divisible by 5.

★ Exactly one of ''a'', ''b'', (''a'' + ''b''), (''b'' − ''a'') is divisible by 7.

★ At most one of ''a'', ''b'' is a square.

★ Every integer greater than 2 that is not congruent to 2 mod 4 is part of a primitive Pythagorean triple. Examples of integers not part of a primitive pythagorean triple: 6,10,14,18

★ Every integer greater than 2, is part of a primitive or non-primitive Pythagorean triple, for example, the integers 6,10,14, and 18 are not part of primitive triples, but are part of the non-primitive triples 6,8,10; 14,48,50 and 18,80,82.

★ There exist infinitely many Pythagorean triples whose hypotenuses are squares of natural numbers.

★ There exist infinitely many Pythagorean triples in which one of the legs is the square of a natural number.

★ There exist infinitely many Pythagorean triples in which the hypotenuse and the longer of the two legs differ by exactly one.

★ There exist infinitely many Pythagorean triples in which the hypotenuse and the longer of the two legs differ by exactly two.

★ There exist infinitely many Pythagorean triples in which the hypotenuse and the longer of the two legs differ by exactly three.

★ For each natural number ''n'', there exist ''n'' Pythagorean triples with different hypotenuses and the same area.

★ For each natural number ''n'', there exist at least ''n'' different Pythagorean triples with the same leg ''a'', where ''a'' is some natural number

★ For each natural number ''n'', there exist at least ''n'' different triangles with the same hypotenuse.

★ In every Pythagorean triple, the radius of the incircle and the radii of the three excircles are natural numbers.

★ There is no Pythagorean triple in which the hypotenuse and one leg are the legs of another Pythagorean triple.

Some relationships

If ''a''² + ''b''2 = ''c''² is a primitive Pythagorean triple, where ''a'' is odd, then
:
:
:
:
where each fraction is reduced to lowest terms and ''m'' > ''n''.
It can also be shown that
:$b(m^2-n^2)\; =\; a(2mn),$
:$(m/n)b\; -\; a\; =\; c\; ,$
:$(n/m)b\; +\; a\; =\; c\; ,$
Additional relationships among the sides:
:$c\; -\; b\; =\; (m\; -\; n)^2\; ,$
:$c\; +\; b\; =\; (m\; +\; n)^2\; ,$
:$a^2\; =\; c^2\; -\; b^2\; =\; (c\; -\; b)(c\; +\; b),$
:$c\; -\; a\; =\; (m^2\; +\; n^2)\; -\; (m^2\; -\; n^2)\; =\; 2n^2,$
:$c\; =\; a\; +\; (m^2\; +\; n^2)\; -\; (m^2\; -\; n^2)\; =\; a\; +\; 2n^2,$
:$a\; =\; c\; -\; (m^2\; +\; n^2)\; -\; (m^2\; -\; n^2)\; =\; c-\; 2n^2\; ,$

The radius, ''r'', of the inscribed circle can be found by:
:$r\; =\; ab/(a+b+c)\; ,$
where
:$a\; =\; 2r\; +1\; ,$
:$b\; =\; 2r(r+1),$
:$c\; =\; 2r^2\; +\; 2r\; +\; 1,$
:$r\; =\; n\; (m-n)\; ,$
:
The unknown sides of a triple can be calculated directly from the radius of the incircle, ''r'', and the value of a single known side, ''a''.
:''k'' = ''a'' − 2''r''
:''b'' = 2''r'' + (2 ''r''²/''k'')
:''c'' = ''b''+ ''k'' = 2''r'' + (2''r''² /''k'') + ''k''
The solution to the 'Incircles' problem shows that, for any circle whose radius is a whole number k, we are guaranteed at least one right angled triangle containing this circle as its inscribed circle where the lengths of the sides of the triangle are a primitive Pythagorean triple:
:''a''=2''k''(''k''+1)
:''b''=2''k''+1
:''c''=2''k''²+2''k''+1
The perimeter ''P'' and area ''L'' of a primitive Pythagorean triple triangle are
:''P'' = ''a'' + ''b'' + ''c'' = 2''m''(''m'' + ''n'')
:''L'' = ''ab''/2 = ''mn''(''m''² − ''n''²)
The shortest side will be ''a'' if one of the following conditions is met:
:
:
:
:
:
:$a^2\; =\; c^2\; -\; b^2\; =\; (c\; -\; b)(c\; +\; b),$
:$c\; -\; a\; =\; (m^2\; +\; n^2)\; -\; (m^2\; -\; n^2)\; =\; 2n^2,$
:$c\; =\; a\; +\; (m^2\; +\; n^2)\; -\; (m^2\; -\; n^2)\; =\; a\; +\; 2n^2,$
:$a\; =\; c\; -\; (m^2\; +\; n^2)\; -\; (m^2\; -\; n^2)\; =\; c-\; 2n^2\; ,$
More relationships among the sides:
:$c^4=(a^2-b^2)^2+(2ab)^2,$
:$a^4=(c^2+b^2)^2-(2cb)^2,$
:$b^4=(c^2+a^2)^2-(2ca)^2,$
:$a^2b^2=(c^2+ab)^2-(ca+cb)^2=(c^2-ab)^2-(ca-cb)^2,$
see: http://www.geocities.com/fredlb37/node8.html

Unit circle relationships

An arbitrary rational slope, ''t'' on the unit circle can be written ''t'' = ''n''/''m'' where ''m'' and ''n'' are integers and ''m'' > ''n''. Other unit circle relationships are shown below:

:$cos\; heta\; =\; \{m^2-n^2\; over\; m^2+n^2\}\; =\; \{1-t^2\; over\; 1+t^2\}=\; \{a\; over\; c\}$
:$sin\; heta\; =\; \{2mn\; over\; m^2+n^2\}\; =\; \{2t\; over\; 1+t^2\}\; =\; \{b\; over\; c\}$
:$an\; heta\; =\; \{2mn\; over\; m^2-n^2\}\; =\; \{2t\; over\; 1-t^2\}\; =\; \{b\; over\; a\}$
:$x^2\; +\; y^2\; =\; 1\; ,$

Half-angle relationships

:$anleft(\{\; heta\; over\; 2\}$
ight) = {n over m},
:
ight) = {m-n over m+n}.

A special case: the Platonic sequence

The case ''n'' = 1 of the more general construction of Pythagorean triples has been known for a long time. Proclus, in his commentary to the 47th Proposition of the first book of Euclid's Elements, describes it as follows:
In equation form, this becomes:
''a'' is odd (Pythagoras, c. 540 BC):
:$mbox\{side\; \}a\; :\; mbox\{side\; \}b\; =\; \{a^2\; -\; 1\; over\; 2\}\; :\; mbox\{side\; \}c\; =\; \{a^2\; +\; 1\; over\; 2\}.$
''a'' is even (Plato, c. 380 BC):
:$mbox\{side\; \}a\; :\; mbox\{side\; \}b\; =\; left(\{a\; over\; 2\}$
ight)^2 - 1 : mbox{side }c = left({a over 2}
ight)^2 + 1
It can be shown that all Pythagorean triples are derivatives of the basic Platonic sequence (x,y,z) = t, (t^2-1)/2 and (t^2+1)/2 by allowing ''a'' to take non-integer rational values. If t is replaced with the rational fraction m/n in the sequence, the 'standard' triple generator 2mn, m^2-n^2 and m^2+n^2 results. It follows that every triple has a corresponding rational t value which can be used to generate a similar (ie equiangular) triangle with rational sides in the same proportion as the original. For example, the Platonic equivalent of (6,8,10) is (3/2; 2, 5/2). The Platonic sequence itself can be derived by following the steps for 'splitting the square' described in Diophantus II.VIII.

Other formulas for generating triples

'I, II: ' Pythagoras' and Plato's formulas have been described above. The methods below appear in various sources, often without attribution as to their origin.
'III.' Given an integer ''n'', the triple can be generated by the following two procedures:
:$a=\; 2n\; +\; 1\; ,:,\; b=2n(n\; +\; 1)\; ,:,\; c\; =\; 2n(n\; +\; 1)\; +\; 1$
Example: When ''n'' = 2 the triple produced is 5, 12, and 13
(This formula is actually the same as method I, substituting ''m'' with 2''n'' + 1.)
Alternatively, one can generate triples from even integers using the following formulas:
:$a=\; 2m\; ,:,\; b=m^2\; -\; 1\; ,:,\; c\; =\; m^2\; +\; 1$
Example: When ''m'' = 4 the triple produced is 8, 15, and 17
(This formula is another specific case of method I, substituting ''n'' with 1).
'IV.' Given the integers ''n'' and ''x'',
:$a=\; 2x^2\; +\; 2nx\; ,:,\; b=\; 2nx\; +\; n^2\; ,:,\; c=2x^2\; +\; 2nx\; +\; n^2$
Example: For ''n'' = 3 and ''x'' = 5, ''a'' = 80, ''b'' = 39, ''c'' = 89.
(This formula is actually the same as method I, substituting ''m'' and ''n'' with ''n''+''x'' and ''x''.)
'V.' Triples can be calculated using this formula: $2xy\; =\; z^2$, x,y,z > 0 where the following relations hold:
''x'' = ''c'' − ''b'', ''y'' = ''c'' − ''a'', ''z'' = ''a'' + ''b'' − ''c'' and ''a'' = ''x'' + ''z'', ''b'' = ''y'' + ''z'', ''c'' = ''x'' + ''y'' + ''z'' and ''r'' = ''z''/2 , where
''x'', ''y'', and ''z'' are the three sides of the triple and ''r'' is the radius of the inscribed circle.
Pythagorean triples can then be generated by choosing any '''even''' integer ''z''.
''x'' and ''y'' are any two factors of $z^2/2$.
Example: Choose ''z'' = 6. Then $z^2/2\; =18.$
The three factor-pairs of 18 are: (18, 1), (2, 9), and (6, 3). All three factor pairs will produce triples using the above equations.
''z'' = 6, ''x'' = 18, ''y'' = 1 produces the triple ''a'' = 18 + 6 = 24, ''b'' = 1 + 6 = 7, ''c'' = 18 + 1 + 6 = 25.
''z'' = 6, ''x'' = 2, ''y'' = 9 produces the triple ''a'' = 2 + 6 = 8, ''b'' = 9 + 6 = 15, ''c'' = 2 + 9 + 6 = 17.
''z'' = 6, ''x'' = 6, ''y'' = 3 produces the triple ''a'' = 6 + 6 = 12, ''b'' = 3 + 6 = 9, ''c'' = 6 + 3 + 6 = 15.
'VI.' An infinity by infinity matrix ''M'' of Pythagorean triples (PNTs), which has some particularly desirable properties can be generated by taking:
: $a(r,k)\; =\; 4rk\; +\; 2k(k-1),$
: $b(r,\; k)\; =\; 4r(r+k-1)\; -\; 2k\; +\; 1,$
: $c(r,k)\; =\; 4r(r+k-1)\; +\; 2k(k-1)\; +\; 1,$
where ''r'' is the row number and ''k'' is the column number. Note that ''a'' is always doubly even, while ''b'' and ''c'' are always odd. Not more than the first ''k'' rows in column ''k'' will have ''a > b.'' Each row is a family of PNTs with the hypotenuse ''c'' of each PNT in row ''r'' exceeding the even side a by the square of the rth odd number. The Pythagorean formula for generating PNTs (section I, above) with ''a'' and ''b'' reversed to make ''a'' the even side, and ''m'' being any natural number:
: $k\; =\; m,$
: $b\; =\; 2k+1,$
: $a\; =\; (b^2-1)/2,$
: $c\; =\; a+1\; =\; (b^2+1)/2,$
yields the first row (r = 1) of ''M'', and the Platonic formula (section II, above) using a = 4m instead of 2m, to eliminate derivative PNTs:
: $r\; =\; m,$
: $a\; =\; 4r,$
: $b\; =\; 4r^2-1,$
: $c\; =\; b+2\; =\; 4r^2+1,$
yields the first column (k = 1) of ''M''.
Each column is a family of PNTs with the hypotenuse of each PNT in column ''k'' exceeding the odd side ''b'' by twice the square of ''k''. For example ''M''(6,4) = {120, 209, 241} 241 − 120 = 121, the square of the sixth odd number (11), and 241 − 209 = 32, which is twice the square of 4.
Below is a small portion of the matrix. The PNTs of row 1 are all relatively prime (primitive), but every other row contains derivative (not relatively prime) PNTs Iff the column number is a power of 2, the PNTs in that column are all primitive. For every odd prime factor p of the column number, the middle row of each group of p rows (r = (p+1)/2 + np, where n >= 0) will contain a PNT which is derivative. In the table below these are indicated by angle brackets. If j is 2 or a factor of k, then M(r, jk) is derivative if and only if M(r, k) is derivative. Fewer than 20% of the PNTs in M are derivative.
column-> 1 2 3 4 5
row a b c a b c a b c a b c a b c
1 4 3 5 12 5 13 24 7 25 40 9 41 60 11 61
2 8 15 17 20 21 29 <36 27 45> 56 33 65 80 39 89
3 12 35 37 28 45 53 48 55 73 72 65 97 <100 75 125>
4 16 63 65 36 77 85 60 91 109 88 105 137 120 119 169
5 20 99 101 44 117 125 <72 135 153> 104 153 185 140 171 221
6 24 143 145 52 165 173 84 187 205 120 209 241 160 231 281
The ''a'''s of each column ''k'' are an arithmetic sequence with difference ''4''k, and the ''b'''s of each row r are an arithmetic sequence with difference 4''r''-2. The ''a's,'' ''b's,'' and ''c's'' of any row or column are each monotonically increasing.
If the two legs of a PNT differ by 1, the longer leg and the hypotenuse form the coordinates of a larger PNT in ''M'' the legs of which differ by 1. ''M''(1,1) = {4, 3, 5}. ''M''(4,5) = {120, 119, 169}.
''M''(120,169) = {137904, 137903, 195025}, etc. Thus, a Pythagorean triangle can be found, the acute angles of which are arbitrarily close to 45 degrees. As Martin (1875) describes, each such triple has the form
:$(2P\_\{n\}P\_\{n+1\},\; P\_\{n+1\}^2\; -\; P\_\{n\}^2,\; P\_\{n+1\}^2\; +\; P\_\{n\}^2).$
where $P\_i$ are the Pell numbers.
'VII.' Generalized Fibonacci Series:
A pythagorean triple can be generated by using any two arbitrary integers, a and b using the following procedures:
a. select any two integers a and b
b. define c = a+b
c. define d = b+c
The integers a,b,c,d are a generalized Fibonacci series. The sides of the triple are computed as follows:
side 1 = $2bc$
side 2 = $ad$
hypotenuse = $b^2\; +\; c^2$
example let a = 69 and b = 75, then c = 69+75 =144 and d= 75+144=219
side 1 = $2cdot\; 75cdot\; 144=21600$
side 2 = $69cdot\; 219\; =\; 15111$
hypotenuse = $75^2\; +\; 144^2\; =\; 26361$
$21600^2\; +\; 15111^2\; =\; 26361^2$
'VIII.' Progression of Whole and Fractional Numbers:
Take a progression of whole and fractional numbers:
1 1/3, 2 2/5 , 3 3/7 , 4 4/9 etc.
The properties of this progression are:
a) the whole numbers are those of the common series and have unity as their common difference b) the numerators of the fractions, annexed to the whole numbers, are also the natural numbers. 3) the denominators of the fractions are the odd numbers, 3,5,7, etc.
To calculate a pythagorean triple:
select any term of this progression and reduce it to an improper fraction. For example, take the term 3 3/7. The improper fraction is 24/7. The numbers 7 and 24 are the sides, a and b, of a right triangle. The hypotenuse is one greater than the largest side.
1 1/3 yields the 3,4,5 triple; 2 2/5 gives 5,12,13 ; 3 3/7 yields gives 7,24, 25 ;
4 4/9 gives 9,40,41 and so forth.

Parent/child relationships

All primitive Pythagorean triples can be generated from the 3-4-5 triangle by using the 3 linear transformations T1, T2, T3 below, where ''a'' ,''b'', ''c'' are sides of a triple:
new side a new side b new side c
T1: a - 2b + 2c 2a - b + 2c 2a - 2b + 3c
T2: a + 2b + 2c 2a + b + 2c 2a + 2b + 3c
T3: -a + 2b + 2c -2a + b + 2c -2a + 2b + 3c
If one begins with 3, 4, 5 then all other primitive triples will eventually be produced. In other words, every primitive triple will be a “parent” to 3 additional primitive triples.
example: Let ''a'' = 3, ''b'' = 4, ''c'' = 5.
new side a new side b new side c
3 - (2×4) + (2×5) = 5 (2×3) - 4 + (2×5) = 12 (2×3) - (2×4) + (3×5) = 13
3 + (2×4) + (2×5) = 21 (2×3) + 4 + (2×5) = 20 (2×3) + (2×4) + (3×5) = 29
-3 + (2×4) + (2×5) = 15 -(2×3) + 4 + (2×5) = 8 -(2×3) + (2×4) + (3×5) = 17
The linear transformations T1, T2, and T3 have a geometric interpretation in the language of quadratic forms. They are closely related to (but are not equal to) reflections generating the orthogonal group of x² + y² - z² over the integers.
For further discussion of parent-child relationships in triples, see: http://mathworld.wolfram.com/PythagoreanTriple.html and
“The Modular Tree of Pythagoras”, Robert Alperin, Department of Mathematics and Computer Science, San Jose State University, San Jose California) http://www.math.sjsu.edu/~alperin/pt.pdf and http://www.faust.fr.bw.schule.de/mhb/pythagen.htm

Generalizations

A set of four positive integers ''a'', ''b'', ''c'' and ''d'' such that ''a''² + ''b''²+ ''c''² = ''d''² is called a Pythagorean quadruple.
A generalization of the concept of Pythagorean triples is the search for triples of positive integers ''a'', ''b'', and ''c'', such that ''a''^''n'' + ''b''^''n'' = ''c''^''n'', for some ''n'' strictly greater than 2. Pierre de Fermat in 1637 claimed that no such triple exists, a claim that came to be known as Fermat's last theorem, though it was far from the last theorem Fermat discovered. The first proof was given by Andrew Wiles in 1994.
There exist sets of four positive integers ''a'', ''b'', ''c'' and ''d'' such that ''a''³ + ''b''³+ ''c''³ = ''d''³
The smallest such set is {3, 4, 5, 6}.

See also

★ Heronian triangle

★ Pythagorean prime

★ Modular arithmetic

★ Trigonometric identity

★ Tangent half-angle formula

★ Plimpton 322

References

★ Thomas L. Heath, ''The Thirteen Books of Euclid's Elements Vol. 1 (Books I and II)'', Dover Publications; 2nd edition (June 1, 1956) ISBN 0-486-60088-2

★ Waclaw Sierpinski, ''Pythagorean Triangles'', Dover Publications, 2003. ISBN 0-486-43278-5

★ Rational right angled triangles nearly isosceles, Martin, Artemas, , , The Analyst, 1875

External links

★ http://mathworld.wolfram.com/PythagoreanTriple.html has an extensive discussion of Pythagorean triples.

★ http://nrich.maths.org/mathsf/journalf/jul01/inspire1 discusses the radius of the incircle

★ http://www.math.clemson.edu/~rsimms/neat/math/pyth/ provides a Javascript calculator for the (''m''² − ''n''², 2''mn'', ''m''² + ''n''²) formula, and shows how to derive the formula.

★ http://www.faust.fr.bw.schule.de/mhb/pythagen.htm a JavaScript calculator which illustrates the 3-fold tree structure of the set of all primitive Pythagorean triples.

★ Pythagorean Triples at cut-the-knot

★ The Trinary Tree(s) underlying Primitive Pythagorean Triples at cut-the-knot

★ Fermat's Last Theorem Blog Covers topics in the history of Fermat's Last Theorem from Pythagorean triples to Wiles' proof.

★ http://www.geocities.com/fredlb37/node2.html http://www.geocities.com/fredlb37/node1.html

★ http://math.nmsu.edu/~history/book/euclidpt.pdf

★ http://nrich.maths.org/askedNRICH/edited/1276.html Shows how to generate new triples by multiplication of two triples.

★ http://mathforum.org/dr.math/faq/faq.pythag.triples.html

★ http://www.mcs.surrey.ac.uk/Personal/R.Knott/Pythag/pythag.html Links to several on-line Triple calculators; discussion of the incircle formula

★ http://www.math.rutgers.edu/~erowland/pythagoreantriples.html

★ http://nrich.maths.org/public/viewer.php?obj_id=1332&part=index&refpage=monthindex.php

★ http://depts.gallaudet.edu/mathcs/papers/baseslopegenpt.htm contains a method for calculating the unknown sides of a triple when given only a single side.

★ http://www.faust.fr.bw.schule.de/mhb/pythagen.htm Description of how to transform a triple into 3 new triples.

★ Pythagorean Triplets

★ Sloane's Online Encyclopedia of Integer Sequences Contains over 160 lists associated with pythagorean triples

★ 'A Pythagorean Triple Generator and Sequencer' Displays results in table form for ease-of-use.

★ http://www.mcs.surrey.ac.uk/Personal/R.Knott/Pythag/pythag.html

★ http://people.wcsu.edu/sandifere/Academics/2007Spring/Mat342/PythagTrip02.pdf