Editing Pythagorean triple (section)

==Elementary properties of primitive Pythagorean triples==

===General properties===

The properties of a primitive Pythagorean triple {{math|(''a'', ''b'', ''c'')}} with {{math|''a'' < ''b'' < ''c''}} (without specifying which of {{math|''a''}} or {{math|''b''}} is even and which is odd) include:
* <math>\tfrac{(c-a)(c-b)}{2}</math> is always a perfect square.<ref>{{citation|title=The Pythagorean Theorem: The Story of Its Power and Beauty|first=Alfred S.|last=Posamentier|author-link=Alfred S. Posamentier|publisher=Prometheus Books|year=2010|isbn=9781616141813|page=[https://archive.org/details/pythagoreantheor0000posa/page/156 156]|url=https://archive.org/details/pythagoreantheor0000posa/page/156}}.</ref> As it is only a necessary condition but not a sufficient one, it can be used in checking if a given triple of numbers is ''not'' a Pythagorean triple.  For example, the triples {{math|{6, 12, 18}{{null}}}} and {{math|{1, 8, 9}{{null}}}} each pass the test that {{math|(''c'' − ''a'')(''c'' − ''b'')/2}} is a perfect square, but neither is a Pythagorean triple.
*When a triple of numbers {{math|''a''}}, {{math|''b''}} and {{math|''c''}} forms a primitive Pythagorean triple, then {{math|(''c'' minus the even leg)}} and one-half of {{math|(''c'' minus the odd leg)}} are both perfect squares; however this is not a sufficient condition, as the numbers {{math|{1, 8, 9}{{null}}}} pass the perfect squares test but are not a Pythagorean triple since {{math|1{{sup|2}} + 8{{sup|2}} ≠ 9{{sup|2}}}}.
*At most one of {{math|''a''}}, {{math|''b''}}, {{math|''c''}} is a square.<ref>For the nonexistence of solutions where {{math|''a''}} and {{math|''b''}} are both square, originally proved by Fermat, see {{citation|title=Elementary Number Theory with Applications|first=Thomas|last=Koshy|publisher=Academic Press|year=2002|isbn=9780124211711|page=545|url=https://books.google.com/books?id=-9pg-4Pa19IC&pg=PA545}}. For the other case, in which {{math|''c''}} is one of the squares, see {{citation|title=Numbers and Geometry|series=[[Undergraduate Texts in Mathematics]]|first=John|last=Stillwell|author-link=John Stillwell|publisher=Springer|year=1998|isbn=9780387982892|page=133|url=https://books.google.com/books?id=4elkHwVS0eUC&pg=PA133}}.</ref>
*The area of a Pythagorean triangle cannot be the square<ref name="Carmichael"/>{{rp|p. 17}} or twice the square<ref name="Carmichael"/>{{rp|p. 21}} of an integer.
*Exactly one of {{math|''a''}}, {{math|''b''}} is [[divisible]] by 2 (is [[even number|even]]), and the hypotenuse {{math|''c''}} is always odd.<ref name=Sierpinski6>{{harvnb|Sierpiński|2003|pp=4–6}}</ref>
*Exactly one of {{math|''a''}}, {{math|''b''}} is divisible by 3, but never {{math|''c''}}.<ref>{{citation |title=Proceedings of the Southeastern Conference on Combinatorics, Graph Theory, and Computing, Volume 20 |publisher=Utilitas Mathematica Pub |year=1990 |isbn=9780919628700 |page=141 |url=https://books.google.com/books?id=G3U_AQAAIAAJ}}</ref><ref name=Sierpinski />{{rp|23–25}}
*Exactly one of {{math|''a''}}, {{math|''b''}} is divisible by 4,<ref name=Sierpinski/> but never {{math|''c''}} (because {{math|''c''}} is never even).
*Exactly one of {{math|''a''}}, {{math|''b''}}, {{math|''c''}} is divisible by 5.<ref name=Sierpinski/>
*The largest number that always divides ''abc'' is 60.<ref name=MacHale/>
*Any odd number of the form {{math|2''m''+1}}, where {{math|''m''}} is an integer and {{math|''m''>1}}, can be the odd leg of a primitive Pythagorean triple. See [[Pythagorean triple#Almost-isosceles Pythagorean triples|almost-isosceles primitive Pythagorean triples]] section below. However, only even numbers divisible by 4 can be the even leg of a primitive Pythagorean triple. This is because [[Pythagorean triple#Generating a triple|Euclid's formula]] for the even leg given above is {{math|2''mn''}} and one of {{math|''m''}} or {{math|''n''}} must be even.
*The hypotenuse {{math|''c''}} (which is always odd) is the sum of two squares. This requires all of its prime factors to be [[Pythagorean prime|primes of the form {{math|4''n'' + 1}}]].<ref>{{citation|title=Roots to Research: A Vertical Development of Mathematical Problems|first=Judith D.|last=Sally|author-link=Judith D. Sally|publisher=American Mathematical Society|year=2007|isbn=9780821872673|pages=74–75|url=https://books.google.com/books?id=nHxBw-WlECUC&pg=PA74}}.</ref> Therefore, c is of the form {{math|4''n'' + 1}}. A sequence of possible hypotenuse numbers for a primitive Pythagorean triple can be found at {{OEIS|A008846|}}.
*The area {{math|1=(''K'' = ''ab''/2)}} is a [[congruent number]]<ref>This follows immediately from the fact that ''ab'' is divisible by twelve, together with the definition of congruent numbers as the areas of rational-sided right triangles. See e.g. {{citation|title=Introduction to Elliptic Curves and Modular Forms|volume=97|series=Graduate Texts in Mathematics|first=Neal|last=Koblitz|publisher=Springer|year=1993|isbn=9780387979663|page=3|url=https://books.google.com/books?id=99v9XcOjhO4C&pg=PA3}}.</ref> divisible by 6.
*In every Pythagorean triangle, the radius of the [[incircle]] and the radii of the three [[incircle|excircles]] are positive integers. Specifically, for a primitive triple the radius of the incircle is {{math|1=''r'' = ''n''(''m'' &minus; ''n'')}}, and the radii of the excircles opposite the sides {{math|''m''{{sup|2}} &minus; ''n{{sup|2}}''}}, ''2mn'', and the hypotenuse {{math|''m''{{sup|2}} + ''n''{{sup|2}}}} are respectively {{math|''m''(''m'' &minus; ''n'')}}, {{math|''n''(''m'' + ''n'')}}, and {{math|''m''(''m'' + ''n'')}}.<ref>{{citation|title=A Survey of Classical and Modern Geometries: With Computer Activities|first=Arthur|last=Baragar|publisher=Prentice Hall|year=2001|isbn=9780130143181|at=Exercise 15.3, p. 301}}</ref>
*As for any right triangle, the converse of [[Thales' theorem]] says that the diameter of the [[circumcircle]] equals the hypotenuse; hence for primitive triples the circumdiameter is {{math|''m''{{sup|2}} + ''n''{{sup|2}}}}, and the circumradius is half of this and thus is rational but non-integer (since {{math|''m''}} and {{math|''n''}} have opposite parity).
*When the area of a Pythagorean triangle is multiplied by the [[curvature]]s of its incircle and 3 excircles, the result is four positive integers {{math|''w'' > ''x'' > ''y'' > ''z''}}, respectively. Integers {{math|−''w'', ''x'', ''y'', ''z''}} satisfy [[Descartes' theorem|Descartes's Circle Equation]].<ref name=Bernhart>{{citation |last1=Bernhart |first1=Frank R. |last2=Price |first2=H. Lee |title=Heron's formula, Descartes circles, and Pythagorean triangles |year=2005 |arxiv=math/0701624}}</ref> Equivalently, the radius of the [[Isoperimetric point#Isoperimetric points and Soddy circles|outer Soddy circle]] of any right triangle is equal to its semiperimeter. The outer Soddy center is located at {{math|''D''}}, where {{math|''ACBD''}} is a rectangle, {{math|''ACB''}} the right triangle and {{math|''AB''}} its hypotenuse.<ref name=Bernhart/>{{rp|p. 6}}
*Only two sides of a primitive Pythagorean triple can be simultaneously prime because by [[Pythagorean triple#Generating a triple|Euclid's formula]] for generating a primitive Pythagorean triple, one of the legs must be composite and even.<ref>{{cite OEIS|A237518|Least primes that together with prime(n) forms a Heronian triangle|mode=cs2}}</ref> However, only one side can be an integer of perfect power <math>p \ge 2</math> because if two sides were integers of perfect powers with equal exponent <math>p</math> it would contradict the fact that there are no integer solutions to the [[Beal conjecture#Partial results|Diophantine equation]] <math>x^{2p} \pm y^{2p}=z^2</math>, with <math>x</math>, <math>y</math> and <math>z</math> being pairwise coprime.<ref name=DM>H. Darmon and L. Merel. Winding quotients and some variants of Fermat’s Last Theorem, J. Reine Angew. Math. 490 (1997), 81–100.</ref>
*There are no Pythagorean triangles in which the hypotenuse and one leg are the legs of another Pythagorean triangle; this is one of the equivalent forms of [[Fermat's right triangle theorem]].<ref name="Carmichael">{{citation
 | last = Carmichael | first = Robert D. | author-link = Robert Daniel Carmichael
 | publisher = John Wiley & Sons
 | title = Diophantine Analysis
 | url = https://archive.org/details/diophantineanaly00carm
 | year = 1915}}</ref>{{rp|p. 14}}
*Each primitive Pythagorean triangle has a ratio of area, {{math|''K''}}, to squared [[semiperimeter]], {{math|''s''}}, that is unique to itself and is given by<ref>{{citation |last1=Rosenberg |first1=Steven |last2=Spillane |first2=Michael |last3=Wulf |first3=Daniel B. |title=Heron triangles and moduli spaces |journal=Mathematics Teacher |volume=101 |pages=656–663 |date=May 2008 |doi=10.5951/MT.101.9.0656 |url=http://www.nctm.org/publications/article.aspx?id=19484}}</ref>
:: <math>\frac{K}{s^2} = \frac{n(m-n)}{m(m+n)} = 1-\frac{c}{s}.</math>
*No primitive Pythagorean triangle has an integer altitude from the hypotenuse; that is, every primitive Pythagorean triangle is indecomposable.<ref name=Yiu/>
*The set of all primitive Pythagorean triples forms a rooted [[ternary tree]] in a natural way; see [[Tree of primitive Pythagorean triples]].
*Neither of the [[acute angle]]s of a Pythagorean triangle can be a [[rational number]] of [[Degree (angle)|degrees]].<ref>{{mathworld|title=Rational Triangle|id=RationalTriangle|mode=cs2}}</ref>  (This follows from [[Niven's theorem]].)

===Special cases===

In addition, special Pythagorean triples with certain additional properties can be guaranteed to exist:
*Every integer greater than 2 that is not [[singly and doubly even|congruent to 2 mod 4]] (in other words, every integer greater than 2 which is ''not'' of the form {{math|4''k'' + 2}}) is part of a primitive Pythagorean triple. (If the integer has the form {{math|4''k''}}, one may take {{math|1=''n'' = 1}} and {{math|1=''m'' = 2''k''}} in Euclid's formula; if the integer is {{math|2''k'' + 1}}, one may take {{math|1=''n'' = ''k''}} and  {{math|1=''m'' = ''k'' + 1}}.)
*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 {{math|(6, 8, 10)}}, {{math|(14, 48, 50)}} and {{math|(18, 80, 82)}}.
*There exist infinitely many Pythagorean triples in which the hypotenuse and the longest leg differ by exactly one. Such triples are necessarily primitive and have the form {{math|(2''n'' + 1, 2''n''{{sup|2}} + 2''n'', 2''n''{{sup|2}} + 2''n'' +1)}}. This results from Euclid's formula by remarking that the condition implies that the triple is primitive and must verify {{math|1=(''m''{{sup|2}} + ''n''{{sup|2}}) - 2''mn'' = 1}}. This implies {{math|1=(''m'' – ''n''){{sup|2}} = 1}}, and thus {{math|1=''m'' = ''n'' + 1}}. The above form of the triples results thus of substituting {{math|''m''}} for {{math|''n'' + 1}} in Euclid's formula. 
*There exist infinitely many primitive Pythagorean triples in which the hypotenuse and the longest leg differ by exactly two. They are all primitive, and are obtained by putting {{math|1=''n'' = 1}} in Euclid's formula. More generally, for every integer {{math|''k'' > 0}}, there exist infinitely many primitive Pythagorean triples in which the hypotenuse and the odd leg differ by {{math|2''k''{{sup|2}}}}. They are obtained by putting {{math|1=''n'' = ''k''}} in Euclid's formula.
*There exist infinitely many Pythagorean triples in which the two legs differ by exactly one.  For example, 20{{sup|2}} + 21{{sup|2}} = 29{{sup|2}}; these are generated by Euclid's formula when <math>\tfrac{m-n}{n}</math> is a [[Generalized continued fraction|convergent]] to <math>\sqrt2.</math>
*For each positive integer {{math|''k''}}, there exist {{math|''k''}} Pythagorean triples with different hypotenuses and the same area.
*For each positive integer {{math|''k''}}, there exist at least {{math|''k''}} different primitive Pythagorean triples with the same leg {{math|''a''}}, where {{math|''a''}} is some positive integer (the length of the even leg is 2''mn'', and it suffices to choose {{math|''a''}} with many factorizations, for example {{math|1=''a'' = 4''b''}}, where {{math|''b''}} is a product of {{math|''k''}} different odd primes; this produces at least {{math|2<sup>''k''</sup>}} different primitive triples).<ref name=Sierpinski/>{{rp|30}}
*For each positive integer {{math|''k''}}, there exist at least {{math|''k''}} different Pythagorean triples with the same hypotenuse.<ref name=Sierpinski/>{{rp|31}}
*If {{math|1=''c'' = ''p{{sup|e}}''}} is a [[prime power]], there exists a primitive Pythagorean triple {{math|1=''a''{{sup|2}} + ''b''{{sup|2}} = ''c''{{sup|2}}}} if and only if  the prime {{math|''p''}} has the form {{math|1=4''n'' + 1}}; this triple is unique [[up to]] the exchange of ''a'' and ''b''. 
*More generally, a positive integer {{mvar|c}} is the hypotenuse of a primitive Pythagorean triple if and only if each [[prime factor]] of {{mvar|c}} is [[modular arithmetic#Congruence|congruent]] to {{math|1}} modulo {{math|4}}; that is, each prime factor has the form {{math|4''n'' + 1}}. In this case, the number of primitive Pythagorean triples {{math|(''a'', ''b'', ''c'')}} with {{math|''a'' < ''b''}} is {{math|2{{sup|''k''−1}}}}, where {{mvar|k}} is the number of distinct prime factors of {{mvar|c}}.<ref>{{citation
 | last = Yekutieli | first = Amnon
 | arxiv = 2101.12166
 | doi = 10.1080/00029890.2023.2176114
 | issue = 4
 | journal = [[The American Mathematical Monthly]]
 | mr = 4567419
 | pages = 321–334
 | title = Pythagorean triples, complex numbers, abelian groups and prime numbers
 | volume = 130
 | year = 2023}}</ref> 
*There exist infinitely many Pythagorean triples with square numbers for both the hypotenuse {{math|''c''}} and the sum of the legs {{math|''a'' + ''b''}}. According to Fermat, the '''smallest''' such triple<ref>{{citation |author-link=Clifford A. Pickover |last=Pickover |first=Clifford A. |chapter=Pythagorean Theorem and Triangles |chapter-url=https://books.google.com/books?id=JrslMKTgSZwC&pg=PA40 |title=The Math Book |publisher=Sterling |year=2009 |isbn=978-1402757969 |page=40 |url=https://books.google.com/books?id=JrslMKTgSZwC}}</ref> has sides {{math|1=''a'' = 4,565,486,027,761}}; {{math|1=''b'' = 1,061,652,293,520}}; and {{math|1=''c'' = 4,687,298,610,289}}. Here {{math|1=''a'' + ''b'' = 2,372,159{{sup|2}}}} and {{math|1=''c'' = 2,165,017{{sup|2}}}}. This is generated by Euclid's formula with parameter values {{math|1=''m'' = 2,150,905}} and {{math|1=''n'' = 246,792}}.
*There exist non-primitive [[Integer triangle#Pythagorean triangles with integer altitude from the hypotenuse|Pythagorean triangles with integer altitude from the hypotenuse]].<ref>{{citation
 | last = Voles | first = Roger
 | date = July 1999
 | doi = 10.2307/3619056
 | issue = 497
 | journal = [[The Mathematical Gazette]]
 | jstor = 3619056
 | pages = 269–271
 | title = 83.27 Integer solutions of <math>a^{-2}+b^{-2}=d^{-2}</math>
 | volume = 83| s2cid = 123267065
 }}</ref><ref name="Richinik">{{citation
 | last = Richinick | first = Jennifer
 | date = July 2008
 | doi = 10.1017/s0025557200183275
 | issue = 524
 | journal = [[The Mathematical Gazette]]
 | jstor = 27821792
 | pages = 313–316
 | title = 92.48 The upside-down Pythagorean theorem
 | volume = 92| s2cid = 125989951
 }}</ref> Such Pythagorean triangles are known as '''decomposable''' since they can be split along this altitude into two separate and smaller Pythagorean triangles.<ref name=Yiu>{{citation |first=Paul |last=Yiu |title=Heron triangles which cannot be decomposed into two integer right triangles |url=http://math.fau.edu/yiu/Southern080216.pdf |year=2008 |publisher=41st Meeting of Florida Section of Mathematical Association of America |page=17 }}</ref>