Editing Pythagorean triple (section)

===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>