International Journal of Pure an Applie Mathematics Volume 35 No. 3 007, 365-374 ON PYTHAGOREAN QUADRUPLES Eray Goins 1, Alain Togbé 1 Department of Mathematics Purue University 150 North University Street, West Lafayette, IN 47907, USA e-mail: egoins@purue.eu Department of Mathematics Purue University North Central 1401S, U.S. 41, Westville, IN 46391, USA e-mail: atogbe@purue.eu Abstract: We consier the multplicative properties of integer quaruples (a,b,c,) satisfying a +b +c as a generalization of Pythagorean triples. In the proces, we present a group structure on the rational points on the unit sphere minus the poles an iscuss a factorization result. AMS Subject Classification: 11E5, 11E0 Key Wors: Pythagorean triples, ternary quaratic forms, sums of squares 1. Introuction We are all familiar with the statement given a right triangle with legs of length a an b an hypotenuse of length c, then a + b c ; this is known as the Pythagorean Theorem. There are many such examples integral right triangles: 3 + 4 5, 5 + 1 13, 8 + 15 17. (1) Such integral triples (a, b, c) are calle Pythagorean triples. Given such a triple, there exist relatively prime integers m an n such that a c m n m + n an b c mn m + n. () Receive: January 8, 007 c 007, Acaemic Publications Lt. Corresponence author
366 E. Goins, A. Togbé To see why, let (a,b,c) be a Pythagorean triple, an choose integers m an n such that b/(c + a) n/m. Then b/(c a) (c + a)/b, so that ( a c + a c c a )/( c + a + c a ) ( m b b b b n n )/ ( m m n m) + n (3) which simplifies to () (see for example [3], pp. 151-154). The goal of this paper is to stuy a generalization to four variables, the so-calle Pythagorean quaruples, where we consier integers a, b, c, such that a + b + c. There are many such examples: 1 + + 3, + 5 + 14 + 10 + 11 15, 1 + 6 + 18 6 + 6 + 17 6 + 10 + 15 19. (4) Such quaruples have a parametrization similar to the well-known triples, as outline in the following result. Theorem 1. For each Pythagorean quaruple (a,b,c,) with 0, there exist relatively prime integers m, n, p such that a mp p + m + n, b np p + m + n, c p m n p + m + n. (5) There exist many similar formulas which give exact values for a, b, c, an but unfortunately there is not one parametrization for all Pythagorean quaruples (see for example [6], p. 1464). The result above circumvents this problem by consiering ratios. It is easy to show that the Pythagorean triples are close uner multiplication. That is, given two such triples one generates a thir through the operation (a 1, b 1, c 1 ) (a, b, c ) (a 1 a b 1 b, a 1 b + a b 1, c 1 c ). (6) This operation inuces an associative, commutative multiplicative structure on the Pythagorean triples; see for example [3], p. 116. In this paper, we consier the associative, commutative operation (a 1, b 1, c 1, 1 ) (a, b, c, ) (a 1 a b 1 b, a 1 b + a b 1, c 1 + c 1, c 1 c + 1 ) (7) an attempt to ask questions of factorization.
ON PYTHAGOREAN QUADRUPLES 367 Our main result may be state as follows. Let P (a,b,c,) be a Pythagorean ( quaruple, an enote n as the greatest common ivisor of a, b, c, ; then P 0 a n, b n, c n, ) n is also a Pythagorean quaruple, an we say that h(p) /n is the height of P. Define the conjugacy class [P] as the collection of all scalar multiples of P 0, where we allow sign changes an permutations a b, a c, b c as well. This set correspons to the ientity a + b + c. We will prove the following result. Theorem. Let P be a Pythagorean quaruple with h(p) > 3. Then there exist Pythagorean quaruples P 1, P with h(p 1 ), h(p ) < h(p) such that [P] [P 1 P ]. As an example, consier the ientity 3 +4 5. Viewe as a Pythagorean quaruple, we have the factorization [(0, 3, 4, 5)] [(0, 8, 6, 10)] [(,, 1, 3) (,, 1, 3)]. (8) In a sense, the ientity 3 +4 5 is generate by the ientity 1 + + 3. The paper is organize as follows. First, we use stereographic projection to efine a group structure on the points on the unit sphere. Next, we iscuss factorization of the rational points by introucing the notions of height an irreucibility. Finally, we apply these results to Pythagorean quaruples. Our proofs ultimately rely on the composition (a, b, c, ) ( a, b, c ) a + ib c +, (9) where the multiplicative properties of the complex numbers inuce the multiplicative properties of the Pythagorean quaruples.. Multiplication on the Unit Sphere Ultimately we wish to stuy Pythagorean quaruples (a,b,c,), but we note that through the map (a, b, c, ) ( a, b, c ), where ( a ) + ( b ) ( c ) + 1, (10) it suffices to consier rational points on the unit sphere. To this en, we begin with a iscussion of the real points.
368 E. Goins, A. Togbé Recall that the unit sphere S (R) {(x 1, x, x 3 ) R 3 x 1 + x + x 3 1} is isomorphic with the extene complex numbers uner the stereographic projection map (x 1, x, x 3 ) (x 1 + ix )/(1 + x 3 ) (see [], p. 8-9 for etails, our formulas are ifferent in that we map the north pole (0,0,1) 0 an the south pole (0, 0, 1) ). The multiplicative structure of the complex numbers inuces a corresponing structure on the unit sphere. Theorem 3. Let k be a fiel containe in R, an enote the poleless unit sphere by G(k) S (k) {(0,0, ±1)} { (x 1, x, x 3 ) k 3 x 1 + x + x 3 1, x 3 ±1 } (11) an efine the operation : G(k) G(k) G(k) as (x 1, x, x 3 ) (y 1, y, y 3 ) ( x1 y 1 x y, x 1 y + x y 1, ) x 3 + y 3. (1) This makes G(k) into a commutative group, where the ientity is O (1,0,0) an the inverse of a point P (x 1, x, x 3 ) is [ 1]P (x 1, x, x 3 ). The reaer shoul keep in min that although these formulas may seem a bit o, the unerlying structure is closely tie to that of the complex numbers. This construction is equivalent to G(R) C with stereographic projection the group isomorphism. Proof. Given two points (x 1,x,x 3 ), (y 1,y,y 3 ) G(k) we efine the expression (x 1,x,x 3 ) (y 1,y,y 3 ) (z 1,z,z 3 ) base on the prouct x 1 + ix 1 + x 3 y1 + iy 1 + y 3 z 1 + iz 1 + z 3, where (z 1, z, z 3 ) G(k). (13) Noting that z 1 + iz 1 + z 3 z 1 + z (1 + z 3 ) 1 z 3 (1 + z 3 ) 1 z 3 1 + z 3, (14) we may take norms of both sies to ai in solving for z 3 : 1 x 3 1 + x 3 1 y 3 1 + y 3 1 z 3 1 + z 3 z 3 x 3 + y 3 (15)
ON PYTHAGOREAN QUADRUPLES 369 (showing in particular that z 3 ±1), while consiering real an imaginary parts to ai in solving for z 1 an z : z 1 x 1 y 1 x y an z x 1 y + x y 1. (16) The statements about associativity an commutativity follow from wellknown multiplicative properties of the complex numbers. The point (1, 0, 0) 1 uner stereographic projection, an (x 1, x, x 3 ) x 1 ix 1 x 3 x 1 + x x 1 + ix 1 + x 3 1 x 3 ( ) x1 + ix 1 (17) 1 + x 3 thereby verifying the statements about the ientity an the inverse. 3. Factorization on the Rational Unit Sphere We now focus on the case k Q. We give some examples of the operation efine in the previous section. Both ( 0, 4 5, 3 ( 5) an 11, 6 11, 9 11) are such rational points on the unit sphere. We fin that ( 0, 4 5, 3 ) ( 5 3, 3, 1 ) ( 3 3, 3, 1 ), 3 ( ) ( 11, 6 11, 9 11 3, 1 3, ) ( 3 3, 3, 1 ) (18). 3 This motivates the concept of factorization but first we nee a way to measure when one point is larger than another. Definition 1. Write x ( a, b, c ) G(Q) with a, b, c, integers. We efine the height of x as the integer h( x) gc(a,b,c,). (19) Write x x 1 x with x 1, x G(Q). We say that x is reucible if h( x 1 ), h( x ) < h( x); an irreucible if no such x 1, x exist. As G(Q) is a group uner, we can always fin x such that x x 1 x given x an x 1. The importance of reucibility is in bouning the heights of x 1 an x.
370 E. Goins, A. Togbé Proposition 1. Write x ( a, b, ) c G(Q) for integers a, b, c, an. (a) h( x) is o. (b) If h( x) 3 then x is irreucible. (c) If h( x) > 3 an c+ gc(a,b,c,) is a multiple of 4 then x is reucible. Given an o integer m+1, we can always fin a point x G(Q) such that h( x). This is equivalent to expressing 8 m +m +1 a +b +c as the sum of three integral squares, a result which was known to Legenre, see [3], p. 17 an [4], article 91. Proof. (a) Assume to the contrary, that x ( a, b, c ) G(Q) has even height for relatively prime integers a, b, c, an. Then a + b + c, where now is a multiple of 4. It is well-known that this happens only when a, b, c are even as well, which contraicts the assumption that a, b, c, an are relatively prime. Hence, h( x) must be o. (b) Assume that x is reucible with h( x) 3. From the efinitions, if h( x) 1, then x is irreucible; an from (a) the value h( x) is not possible; so h( x) 3. By our assumption, there exist x 1, x G(Q), each of height h( x 1 ) h( x ) 1 such that x x 1 x. But the only points of height 1 are in the form (±1, 0, 0), (0, ±1, 0), or (0, 0, ±1), (0) an so the prouct x 1 x is of the same form. These points have height 1, an so x must have height 1 as well. Again, this is a contraiction. (c) Choose x ( a, b, ) c G(Q) in terms of relatively prime integers a, b, c, an. Assume that h( x) > 3 an c + 4n for some integer n. Since (a + b) ab + c ab + 4n( c), (1) the sum a + b m is even as well. Define the rational points ( x 1 3, 3, 1 ) ( m an x 3 n, m a n, 3n ) n in G(Q). () It is easy to see that x x 1 x. The first point has height h( x 1 ) 3 < h( x). Since c < we have the inequality h( x ) n 3 c 3 + c 4 < h( x). (3) 4 Hence x is reucible.
ON PYTHAGOREAN QUADRUPLES 371 c+ gc(a,b,c,) The conition that be a multiple of 4 is a bit strong. In fact, there are irreucible points x G(Q), where h( x) > 3; take for example x ( 7, 3 7, 6 7 ). To remey this, we consier instea a conjugacy class which can always be factore. Definition. Let x (x 1, x, x 3 ) G(Q). Denote the conjugacy class of x as the set [ x] {( ±x σ(1), ±x σ(), ±x σ(3) ) σ Sym(3) } G(Q). (4) We say that [ x] is reucible if there is a representative x 0 [ x] such that x 0 is reucible; an irreucible if no such x 0 exists. For instance, while x ( 7, 3 7, 6 7 ) is irreucible as a point, [ x] [ ( 7, 3 7, 6 7 )] is reucible as a conjugacy class: [( 7, 3 7, 6 )] [( 7 7, 6 )] [( 7, 3 7 3, 1 ) ( 3, 3 3, 3, 1 )]. (5) 3 This motivates the main result of this section. Theorem 4. Let x G(Q). (a) h( x) is inepenent of choice of representative x 0 [ x]. (b) h( x) 3 if an only if [ x] is irreucible. (c) h( x) > 3 if an only if [ x] is reucible. Proof. For the following, write x ( a, b, ) c in terms of relatively prime integers a, b, c, an, so that h( x). (a) [ x] simply consists of permutations an sign changes of the coorinates, an each representative x 0 [ x] has the same enominator. Hence h( x 0 ) as well. (b) ( ) If h( x) 3 then h( x 0 ) 3 for each representative x 0 [ x]. Each x 0 is irreucible by Proposition 1, so [ x] is irreucible. ( ) Now assume that h( x) > 3. By Proposition 1, m + 1 is o, an not all of a, b, c, are even. We may choose a representative x 0 ( a, b, c ) with c n + 1 o as well. Note that c + (m + n + 1) an c + (m + n) 4n, (6) so choose the sign of c so that c + is a multiple of 4; this can be one because either m+n or m+n+1 is even. Then by Proposition 1 again, x 0 is reucible, so by efinition [ x] is reucible. (c) This is the contrapositive of (b).
37 E. Goins, A. Togbé 4. Factorization of Pythagorean Quaruples We now consier the composition of maps ( a (a, b, c, ), b, c ) a + ib c + (7) to gain more information about the multiplicative nature of Pythagorean quaruples. For the sake of completeness, we show how to parametrize all such quaruples. Theorem 5. For each Pythagorean quaruple (a,b,c,) with 0, there exist relatively prime integers m, n, p such that a mp p + m + n, b np p + m + n, c p m n p + m + n. (8) In [1], one can see that some Pythagorean quaruples are in the form a αβ + γ δ, b αγ β δ, c α β γ + δ, α + β + γ + δ ; (9) for α, β, γ, an δ integers. Our formulas are relate by setting m αβ + γ δ, n αγ β δ, p α + δ. (30) Compare the above formulas to those in [5] an [6]. Proof. There exist relatively prime integers m, n, p such that p (a + ib) (c + )(m + in). (31) Consiering real an imaginary parts an using the fact that a + b + c we fin that a as esire. mp p + m + n, b np p + m + n, c p m n p + m + n (3) Now that we know how to factor points on the rational unit sphere, we iscuss factorizations of Pythagorean quaruples. We begin by iscussing conjugacy classes.
ON PYTHAGOREAN QUADRUPLES 373 Definition 3. Let P (a, b, c, ) be a nontrivial Pythagorean quaruple i.e. integers such that a + b + c but c. Denote the conjugacy class of P as the set [P] { (a 0, b 0, c 0, 0 ) Z 4 0 0, ( a0, b 0, c ) } 0 [ x], (33) 0 0 0 where x ( a, b, c ). For any representative P0 [P] efine the height h(p 0 ) as the height of a representative P 0 [P]. We say that [P] is reucible (irreucible, respectively) if [ x] is reucible (irreucible, respectively). We present a more intuitive way to view this efinition of conjugacy class. If P (a, b, c, ) is a Pythagorean quruple, set ( a P 0 n, b n, c n, ), where n gc(a,b,c,). (34) n Then [P] is the collection of all scalar multiples of P 0, where we allow sign changes an permutations a b, a c, b c as well. For example, the Pythagorean quaruples with height either 3, 5, or 7 generate the classes [(1,,, 3)], [(0, 3, 4, 5)], an [(, 3, 6, 7)], respectively. This fact correspons to the ientities 1 + + 3, 3 + 4 5, an + 3 + 6 7, (35) respectively. We may now state the main result of the paper. Theorem 6. Let P (a, b, c, ) be a nontrivial Pythagorean quaruple, an efine the operation as (a 1, b 1, c 1, 1 ) (a, b, c, ) (a 1 a b 1 b, a 1 b + a b 1, c 1 + c 1, c 1 c + 1 ). (36) (a) This makes such quaruples into a commutative monoi, with ientity O (1,0,0,1). (b) [P] is irreucible if an only if h(p) 3. (c) [P] is reucible if an only if h(p) > 3. The operation may also be realize through the map efine by a b 0 0 ϕ : Z 4 Mat 4 (Z), (a, b, c, ) b a 0 0 0 0 c. (37) 0 0 c
374 E. Goins, A. Togbé Then ϕ(p 1 P ) ϕ(p 1 ) ϕ(p ) is the prouct of the matrices, ϕ(o) is the ientity matrix, an the nontrivial Pythagorean ( ) quaruples ( correspon ) to the a b c submonoi of the image efine by et et 0. b a c Proof. The results follow irectly from Theorems 3 an 4 via the mapping (a,b,c,) ( a, b, c ). Acknowlegments The secon author is grateful Purue University North Central for the support. References [1] Robert D. Carmichael, The Theory of Numbers an Diophantine Analysis, Dover Publications Inc., New York (1959). [] John B. Conway, Functions of one Complex Variable, Springer-Verlag, New York, Secon Eition (1978). [3] H. Davenport, The Higher Arithmetic, Cambrige University Press, Cambrige, seventh eition (1999); An Introuction to the Theory of Numbers, Chapter VIII by J.H. Davenport. [4] Carl Frierich Gauss, Disquisitiones Arithmeticae, Springer-Verlag, New York (1986); Translate an with a preface by Arthur A. Clarke, Revise by William C. Waterhouse, Cornelius Greither an A. W. Grootenorst an with a preface by Waterhouse. [5] L.J. Morell, Diophantine Equations, Acaemic Press, Lonon (1969). [6] Eric W. Weisstein, CRC Concise Encyclopeia of Mathematics, CRC Press, Boca Raton, FL (1999).