aa r X i v : . [ m a t h . HO ] D ec A NOTE ON PYTHAGOREAN TRIPLES
ROBERTO AMATO
Abstract.
Some relations among Pythagorean triples are established. The main tool is afundamental characterization of the Pythagorean triples throught a chatetus which allowsto determine relationships with Pythagorean triples having the same chatetus raised toan integer power. Introduction
Let x , y and z be positive integers satisfying x + y = z . Such a triple ( x, y, z ) is called Pythagorean triple and if, in addition, x , y and z are co-prime, it is called primitive Pythagorean triple. Preliminarly, let us recall a recent novelformula that allows to obtain all Pythagorean triples as follows. Theorem 1.1. ( x, y, z ) is a Pythagorean triple if and only if there exists d ∈ C ( x ) suchthat (1.1) x = x, y = x d − d , z = x d + d , with x positive integer, x ≥ , and where C ( x ) = D ( x ) , if x is odd, D ( x ) ∩ P ( x ) , if x is even,with D ( x ) = (cid:8) d ∈ N such that d ≤ x and d divisor of x (cid:9) , and if x is even with x = 2 n k , n ∈ N and k ≥ odd fixed, with P ( x ) = (cid:8) d ∈ N such that d = 2 s l , with l divisor of x and s ∈ { , , . . . , n − } (cid:9) . We want to find relations between the primitive Pythagorean triple ( x, y, z ) generatedby any predeterminated x positive odd integer using (1.1) and the primitive Pythagoreantriple generated by x m with m ∈ N and m ≥
2. In this paper we take care of relations onlyfor the case in which the primitive triple ( x, y, z ) is generated whith d ∈ C ( x ) only with d = 1 and the primitive triple ( x m , y ′ , z ′ ) is generated with d m ∈ C ( x m ) only with d m = 1obtaining formulas that give us y ′ and z ′ directly from x , y , z . This is the first step toinvestigate on other relations between Pythagorean triples. Key words and phrases.
Pythagorean triples, Diophantine equations. Results
We have that the following theorem holds.
Theorem 2.1.
Let ( x, y, z ) be the primitive Pythagorean triple generated by any prede-terminated positive odd integer x ≥ using (1.1) with z − y = d = 1 and let ( x m , y ′ , z ′ ) be the primitive Pythagorean triple generated by x m , m ∈ N , m ≥ , using (1.1) with z ′ − y ′ = d m = 1 , we have the following formulas y ′ = y m − X p =1 x p , (2.1) z ′ = y m − X p =1 x p + 1 , for every m ∈ N and m ≥ .Moreover we have also (2.2) z ( − m − + m − X p =1 ( − m − − p x p = (cid:26) y ′ if m is even, z ′ if m is odd,and (2.3) z ( − m − + m − X p =1 ( − m − − p x p + ( − m − = (cid:26) z ′ if m is even, y ′ if m is odd.Proof. Let x be a positive odd integer that we consider as x = 2 n + 1, n ∈ N , so that using(1.1) with d = z − y = 1 it results the primitive Pythagorean triple(2.4) x = 2 n + 1 , y = 2 n + 2 n, z = 2 n + 2 n + 1 , while considering x m , m ∈ N , m ≥
2, using (1.1) with d m = z ′ − y ′ = 1 it results theprimitive Pythagorean triple(2.5) x m , y ′ = x m − , z ′ = x m + 12 . From the comparison between (2.4) and (2.5) we obtain y ′ = (2 n + 1) m −
12 = (cid:2) (2 n + 1) − (cid:3) h (2 n + 1) m − + (2 n + 1) m − + . . . + 1 i = (cid:0) n + 4 n (cid:1) m − X p =1 (2 n + 1) p = (2 n + 2 n ) m − X p =1 (2 n + 1) p = y m − X p =1 x p , that is the first of (2.1), and because d m = z ′ − y ′ = 1 we obtain also z ′ = y m − X p =1 x p + 1 , NOTE ON PYTHAGOREAN TRIPLES 3 that is the second of (2.1).Moreover, if m is odd, using (2.4) and (2.5) we obtain z ′ = (2 n + 1) m + 12 = (cid:2) (2 n + 1) + 1 (cid:3) h (2 n + 1) m − − (2 n + 1) m − + . . . − (2 n + 1) + 1 i = (2 n + 2 n + 1) m − X p =1 ( − m − − p (2 n + 1) p = z m − X p =1 ( − m − − p x p , that is the second case of (2.2), and because d m = z ′ − y ′ = 1 we obtain also y ′ = z m − X p =1 ( − m − − p x p − , that is the second case of (2.3).At last, if m is even, we proof that we obtain(2.6) y ′ = z − m − X p =1 ( − m − − p x p , that is the first case of (2.2) and because d m = z ′ − y ′ = 1 we obtain also z ′ = z − m − − p X p =1 ( − m − − p x p + 1 , that is the first case of (2.3).In order to prove that (2.6) holds we can write it using (2.4) and (2.5) in the following way(2.7) (2 n + 1) m −
12 = (2 n + 2 n + 1) − m − X p =1 ( − m − − p (2 n + 1) p , and we prove that (2.7) is an identity. In fact(2 n + 1) m − n + 4 n + 2) − m − X p =1 ( − m − − p (2 n + 1) p , (2 n + 1) m − (cid:2) (2 n + 1) + 1) (cid:3) − m − X p =1 ( − m − − p (2 n + 1) p , (2 n + 1) m = − (2 n + 1) + m − X p =1 ( − m − − p (2 n + 1) p +1) + m − X p =1 ( − m − − p (2 n + 1) p , R. AMATO (2 n + 1) m = − (2 n + 1) + h ( − m − (2 n + 1) + ( − m − (2 n + 1) + ( − m − (2 n + 1) + . . . − (2 n + 1) m − + (2 n + 1) m i + h ( − m − (2 n + 1) + ( − m − (2 n + 1) + ( − m − (2 n + 1) + ( − m − (2 n + 1) + . . . − (2 n + 1) m − + (2 n + 1) m − i , (2.8)and because m is even, after simplifying (2.8) we get(2 n − m = (2 n − m , so we prove that (2.7) is an identity and, therefore, (2.6) holds. Consequently, formulas(2.1), (2.2) and (2.3) have thus been proved.Obviously, because z − y = d = 1, then we can obtain also other relations between ( x, y, z )and ( x m , y ′ , z ′ ), for example (2.1) are equivalent to y ′ = z + y m − X p =1 x p − ,z ′ = z + y m − X p =1 x p . By similar way we can obtain other relations from (2.2) and (2.3). (cid:3)
To prove the accuracy of formulas (2.1), (2.2) and (2.3) we consider the following example.
Example 2.1.
We give the following table that can be extended for each primitive triples x , y , z , and x s , y ′ , z ′ with x − y = 1 and x ′ − y ′ = 1.Usung (2.1) we obtain x = 3 y = 4 z = 5 x = 3 y ′ = 4(1 + 3 ) = 40 z ′ = 41 x = 3 y ′ = 4(1 + 3 + 3 ) = 364 z ′ = 365 x = 3 y ′ = 4(1 + 3 + 3 + 3 ) = 3280 z ′ = 3281 x = 3 y ′ = 4(1 + 3 + 3 + 3 + 3 ) = 29524 z ′ = 29525 x = 3 y ′ = 4(1 + 3 + 3 + 3 + 3 + 3 ) = 265720 z ′ = 265721... ... ...While, using (2.2) and (2.3) we obtain x = 3 y = 4 z = 5 x = 3 y ′ = 5( − ) = 40 z ′ = 41 x = 3 z ′ = 5(1 − + 3 ) = 365 y ′ = 364 x = 3 y ′ = 5( − − + 3 ) = 3280 z ′ = 3281 x = 3 z ′ = 5(1 − + 3 − + 3 ) = 29525 y ′ = 29524 x = 3 y ′ = 5( − − +3 − +3 ) = 265720 z ′ = 265721... ... ... NOTE ON PYTHAGOREAN TRIPLES 5
References [1] R. Amato,
A characterization of pythagorean triples , JP Journal of Algebra, Number Theory andApplications (2017), 221–230[2] W. Sierpinski, Elementary theory of numbers , PWN-Polish Scientific Publishers, 1988.(R. Amato)
Department of Engineering, University of Messina, 98166 Messina - (Italy)
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