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单词 PrimeFactorsOfPythagoreanHypotenuses
释义

prime factors of Pythagorean hypotenuses


The possible hypotenusesMathworldPlanetmath of thePythagorean trianglesMathworldPlanetmath (http://planetmath.org/PythagoreanTriangle) form theinfiniteMathworldPlanetmath sequenceMathworldPlanetmath

5, 10, 13, 15, 17, 20, 25, 26, 29, 30, 34, 35, 37, 39, 40, 41, 45,

the mark of which ishttp://oeis.org/search?q=a009003&languagePlanetmathPlanetmath=english&go=SearchA009003 in the corpus of the integersequences of http://oeis.org/OEIS.  This sequence has the subsequence A002144

5, 13, 17, 29, 37, 41, 53, 61, 73, 89, 97, 101, 109, 113, 137,

of the odd Pythagorean primes.

Generally, the hypotenuse c of a Pythagorean triangle(Pythagorean tripleMathworldPlanetmath) may be characterised by being thecontraharmonic mean

c=u2+v2u+v

of some two different integers u and v (as has been shownin the parent entry), but also by the

Theorem.  A positive integer c is the length of thehypotenuse of a Pythagorean triangle if and only if at leastone of the prime factorsMathworldPlanetmath of c is of the form 4n+1.

Lemma 1.  All prime factors of the hypotenuse c ina primitive Pythagorean triple are of the form 4n+1.

This can be proved here by making the antithesis that thereexists a prime 4n-1 dividing c.  Then also

4n-1c2=a2+b2=(a+ib)(a-ib)

where a and b are the catheti in the triple.  But 4n-1is prime also in the ring [i] of the Gaussianintegers, whence it must divide at least one of the factorsa+ib and a-ib.  Apparently, that would imply that4n-1 divides both a and b.  This means that thetriple (a,b,c) were not primitive, whencethe antithesis is wrong and the lemma true. 

Also the converseMathworldPlanetmath is true in the following form:

Lemma 2.  If all prime factors of a positive integerc are of the form 4n+1, then c is the hypotenuse ina Pythagorean triple.  (Especially, any prime 4n+1 isfound as the hypotenuse in a primitive Pythagorean triple.)

Proof.  For proving this, one can start from Fermat’stheorem, by which the prime numbersMathworldPlanetmath of such form are sums oftwo squares (see thehttp://en.wikipedia.org/wiki/Proofs_of_Fermat's_theorem_on_sums_of_two_squaresTheorem on sums of two squares by Fermat). Since the sums of two squares form a set closed undermultiplicationPlanetmathPlanetmath, now also the product c is a sum of twosquares, and similarly is c2, i.e. c is the hypotenusein a Pythagorean triple. 

Proof of the Theorem.  Suppose that c is thehypotenuse of a Pythagorean triple (a,b,c); dividing thetriple members by their greatest common factor we get aprimitive triple (a,b,c) where  cc.  ByLemma 1, the prime factors of c, being also prime factors ofc, are of the form 4n+1.

On the contrary, let’s suppose that a prime factor p of c=pd  is of the form 4n+1.  Then Lemma 2 guaranteesa Pythagorean triple (r,s,p), whence also (rd,sd,c) isPythagorean and c thus a hypotenuse. 

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更新时间:2025/5/4 5:56:05