POINTS ON HYPERBOLAS AT RATIONAL DISTANCE

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1 International Journal of Number Theory Vol. 8 No c World Scientific Publishing Company DOI: /S POINTS ON HYPERBOLAS AT RATIONAL DISTANCE EDRAY HERBER GOINS and KEVIN MUGO Department of Mathematics Purdue University 150 North University Street West Lafayette IN USA egoins@math.purdue.edu kmugo@math.purdue.edu Received 4 August 2011 Accepted 14 November 2011 Published 4 May 2012 Richard Guy asked for the largest set of points which can be placed in the plane so that their pairwise distances are rational numbers. In this article we consider such a set of rational points restricted to a given hyperbola. To be precise for rational numbers a b c andd such that the quantity D /2a 2 is defined and non-zero we consider rational distance sets on the conic section axy + bx + cy + d 0. We show that if the elliptic curve Y 2 X 3 D 2 X has infinitely many rational points then there are infinitely many sets consisting of four rational points on the hyperbola such that their pairwise distances are rational numbers. We also show that any rational distance set of three such points can always be extended to a rational distance set of four such points. Keywords: Rational distance sets; elliptic curves; hyperbolas. Mathematics Subject Classification 2010: 11D09 11D25 1. Introduction In Richard Guy s classic text [2] there is a discussion of the largest set S of points P t which can be placed in the plane A 2 R so that their pairwise distances P s P t are rational numbers. One can find infinitely many such points if one restricts to a line. Indeed this is the case for the set S {...P t...} consisting of those points P t x t : y t : 1 on the line L : ax + by c all in terms of the rational numbers x t ac bt a 2r r 2 +1 P s P t s t. y t bc + at b r2 1 r 2 +1 Similarly one can find infinitely many such points if one restricts to a circle. Indeed this is the case for the set S {...P t...} consisting of those points 911

2 912 E. H. Goins & K. Mugo P t x t : y t :1 on the circle S 1 :x h 2 +y k 2 r 2 all in terms of the rational numbers x t h + t4 6t 2 +1 t r 4s tst +1 y t k + 4t3 t t r P s P t s 2 +1t 2 +1 r. Moreover it can be shown that these rational distance sets are dense. The problem becomes much more interesting if one restricts to those sets S which are neither collinear nor concyclic. Richard Guy asks in problem D20 of [2] whether there exist six points P t in the plane A 2 R no three on a line L nofour on a circle S 1 all of whose mutual distances P s P t are rational; several authors have shown that infinitely many such sets S exist. In this article we ask the same question but we place the extra condition that these points are restricted to lie on a given conic section C sothatbybézout s Theorem no three will lie on a line and no five will lie on a circle. Campbell [1] showed that there exist infinitely many such sets S of four points restricted to the parabola C : y x 2. He did so by showing that a certain elliptic curve has infinitely many rational points. Motivated by this result and its proof we show the following. Theorem 1. For rational numbers a b c and d such that the quantity D ad bc/2a 2 is defined and non-zero consider the conic section C : axy + bx + cy + d 0. If the elliptic curve E D : Y 2 X 3 D 2 X has infinitely many rational points X : Y :1 then there are infinitely many rational distance sets S {P 1 P 2 P 3 P 4 } consisting of four rational points on C such that their pairwise distances P s P t are rational numbers. In Theorem 6 we show explicitly that we may choose the rational points P 1 ac + Y 1X 2 X 3 : ab a 2 X 1Y 2 Y 3 X 1 Y 2 Y 3 Y 1 X 2 X 3 P 2 P 3 P 4 ac + X 1Y 2 X 3 ac + X 1X 2 Y 3 ac + : ab a 2 Y 1X 2 Y 3 Y 1 X 2 Y 3 X 1 Y 2 X 3 : ab a 2 Y 1Y 2 X 3 Y 1 Y 2 X 3 X 1 X 2 Y 3 4D 2 Y 1 Y 2 Y 3 X 1 X 2 X 3 : ab 4D 2 a 2 X 1X 2 X 3 Y 1 Y 2 Y 3 As a consequence of eliminating the X t s and Y t s from these formulas we find that every rational distance set of three points can be extended to a rational distance set of four points..

3 Points on Hyperbolas at Rational Distance 913 Corollary 2. Say that {P 1 P 2 P 3 } is a rational distance set of three points P t x t : y t :1on the hyperbola C. If we choose the point P 4 c + ay 1 + bay 2 + bay 3 + b : b + ax 1 + cax 2 + cax 3 + c : a then S {P 1 P 2 P 3 P 4 } is a rational distance set of four points on the hyperbola. This work generalizes results found by Ly Tsosie and Urresta in [3] during the Mathematical Sciences Research Institute s Undergraduate Program. 2. Main Results For the sequel we fix rational numbers a b c andd such that the quantity aad bc 0. We also fix the notation C {x : y : z P 2 axy + bxz + cyz + dz 2 0} D 2a 2 E D {X : Y : Z P 2 Y 2 Z X 3 D 2 XZ 2 }. Let S be rational distance set on C thatislets be a collection of affine rational points P t x t : y t :1onthehyperbolaC such that their pairwise distances P i P j x i x j 2 +y i y j 2 are rational numbers. We give a way to generate such a set S. Proposition 3. Fix an integer n 3. Consider a collection of nn 1/2 rational non-torsion points X ij : Y ij :1on the elliptic curve E D satisfying the nn 3/2 compatibility relations Y ij Y st Y is Y jt X ij X st X is X jt for all distinct indices i j s and t. Then the n rational points P t ac + Y ijx it X jt : ab a 2 X ijy it Y jt X ij Y it Y jt Y ij X it X jt for any distinct i j t form a rational distance S on the hyperbola C. Proof. We explain how to derive this construction. Let S {...P t...} be any rational distance set on C. Foreachpair{P i P j } of distinct points in S define the rational numbers Dx i x j X ij y i y j + x i x j 2 +y i y j. 2

4 914 E. H. Goins & K. Mugo Upon solving for y in the equation axy + bx + cy + d 0 we find the slope Xij 2 D2 y i y j 2DX ij x i x j ax i + cax j + c. If S n thereare n 2 nn 1/2 such quadratic relations among the xt sowe try and solve for the n 1 n variables xt in terms of the X ij.wehavetheidentity [ X 2 ax t + c 2 it D 2 ] [ Xjt 2 ][ ] D2 X 2 ij D 2 1 for distinct i j t. 2DX it 2DX jt 2DX ij It makes sense to define the numbers n ij X 3 ij D2 X ij X ij axi + cax j + c which must satisfy the n 2 n 1 nn 3/2 compatibility conditions n ij n st 2 axi + cax j + cax s + cax t + c n isn jt for all distinct indices i j s andt. Even though these numbers are not rational in general we still find the expression P t x t : y t :1 ac + n ij : ab a 2 n itn jt n it n jt n ij for distinct i j t. We now prove the result above. Say that we are given a collection of nn 1/2 rational non-torsion points X ij : Y ij : 1 on the elliptic curve E D satisfying the nn 3/2 compatibility relations above. Then the rational numbers n ij Y ij /X ij are nonzero so that the P t as above are on the conic section. Since we have the identity 2 ys y t P s P t x s x t 1+ n ij x s x t n ij n is n js n it n jt we see that S {...P t...} is indeed a rational distance set. X 2 st + D2 X st This result motivates study of the set E n D X ij : Y ij : Z ij P 2 nn 1 Yij 2 2 Z ij Xij 3 D2 X ij Z 2 ij X is X jt Y ij Y st X ij X st Y is Y jt ij which is a projective variety of dimension nn 1/2. The elliptic curve has involutions E D E D defined by } X : Y : Z [±1]X : Y : Z 0:0:1 Y D 2 XZ : ±D 2 YZ : X 2 X ± Y X ;

5 Points on Hyperbolas at Rational Distance 915 so the projective variety has involutions E n D E n D as well. This corresponds to the involution n ij ±n ij in the formulas above. We give some examples of this projective variety. Case n 3.Then n 2 n D 1 0sothatE 3 E D E D E D is that projective variety of dimension 3 which is the product of elliptic curves. E D 3 E D E D E D E D E D E D X 23 : Y 23 : Z 23 X 13 : Y 13 : Z 13 X 12 : Y 12 : Z 12 X 1 : Y 1 : Z 1 X 1 : Y 1 : Z 1 X 2 : Y 2 : Z 2 X 2 : Y 2 : Z 2 X 3 : Y 3 : Z 3 X 3 : Y 3 : Z 3 In other words the formulae in Proposition 3 uniquely determine P 1 P 2 andp 3. We use this to generate rational distance sets S on the hyperbola C. Corollary 4. If X t : Y t :1are three rational points on E D which are not points of finite order then S {P 1 P 2 P 3 } is a rational distance set on C : axy + bx + cy + d 0when we choose P 1 P 2 P 3 ac + Y 1X 2 X 3 Y 1 X 2 X 3 : ab a 2 X 1Y 2 Y 3 X 1 Y 2 Y 3 ac + X 1Y 2 X 3 : ab a 2 Y 1X 2 Y 3 Y 1 X 2 Y 3 X 1 Y 2 X 3 ac + X 1X 2 Y 3 : ab a 2 Y 1Y 2 X 3. Y 1 Y 2 X 3 X 1 X 2 Y 3 In particular when E D has positive rank there are infinitely many rational distance sets S of three points on the hyperbola C.

6 916 E. H. Goins & K. Mugo Proof. The desired rational points P 1 ac + Y 23X 12 X 13 : ab a 2 X 23Y 12 Y 13 X 23 Y 12 Y 13 Y 23 X 12 X 13 P 2 ac + Y 13X 12 X 23 : ab a 2 X 13Y 12 Y 23 X 13 Y 12 Y 23 Y 13 X 12 X 23 P 3 ac + Y 12X 13 X 23 : ab a 2 X 12Y 13 Y 23 X 12 Y 13 Y 23 Y 12 X 13 X 23 come about by relabeling the three points on E D as X 1 : Y 1 : Z 1 X 23 : Y 23 : Z 23 X 2 : Y 2 : Z 2 X 13 : Y 13 : Z 13 X 3 : Y 3 : Z 3 X 12 : Y 12 : Z 12. The result follows directly from Proposition 3. As a specific example consider the conic section C : xy +120.ThenD 6 and the elliptic curve E 6 has the three rational points 12 : 36 : 1 50 : 35 : 8 and : : 12167; which correspond to n 1 3 n 2 7/10 and n /851 respectively. We find the affine rational points P : : P : : P : :1 P 1 P P 1 P P 2 P lying on the hyperbola so that S {P 1 P 2 P 3 } is a rational distance set on C. A plot of this configuration can be found in Fig. 1. Case n 4.Then n 2 n 1 2sothatn12 n 34 n 13 n 24 n 14 n 23 are the two compatibility conditions which define the projective variety E D 4 of dimension 6. We have a result that is similar to E D 3 E D E D E D. Proposition 5. E D 4 is the 3-fold fiber product of the two-dimensional variety { } E D u : v : wt P 2 P 1 v 4 w 3 uu + D 2 w u + D 2 T 4 wu 2 D 4 T 4 w 2 2 corresponding to the projection E D P 1 which sends u : v : wt T.

7 Points on Hyperbolas at Rational Distance Fig. 1. Rational distance set on xy Proof. We will explain the diagram E D 4 E D E D E D E D E D P 1 E D P u1 : v 1 : w 1 T u1 : v 1 : w 1 T u2 : v 2 : w 2 T u3 : v 3 : w 3 T u2 : v 2 : w 2 T T u3 : v 3 : w 3 T

8 918 E. H. Goins & K. Mugo A rational point on the projective variety E D 4 is in the form X1 : Y 1 : Z 1 X 2 : Y 2 : Z 2 X 3 : Y 3 : Z 3 P X 1 : Y 1 : Z 1 X 2 : Y 2 : Z 2 X 3 : Y 3 : Z 3 where X t : Y t : Z t andx t : Y t : Z t are six rational points on the elliptic curve E D thatis Y1 2 Z 1 X1 3 D 2 X 1 Z1 2 Y 1 2 Z 1 X 13 D 2 X 1Z 12 Y2 2 Z 2 X2 3 D2 X 2 Z2 2 Y 2 2 Z 2 X 23 D 2 X 2 Z 22 Y3 2 Z 3 X3 3 D2 X 3 Z3 2 Y 3 2 Z 3 X 33 D 2 X 3 Z 2 3 and we have the two compatibility relations Y 1 Y 1 Y 2 Y 2 Y 3 Y 3. X 1 X 2 X 3 X 1 X 2 GivensucharationalpointP on E D 4 define the three rational points u t : v t : w t T on E D in terms of u t 4D 3 Xt 2 X ty t Z tx t 2 D 2 Zt 2 X t2 D 2 Z t2 v t 8D 5 Xt 2 X t2 Z t [D 2 Zt 2 X t + DZ t 2 Xt 2 X t DZ t 2 ] w t Y t Xt 2 D 2 Zt 2 2 X t DZ tx t + DZ t 3 T 2D X t X t Y t Y t where T is independent of t. Hence the expression X 3 Q u 1 : v 1 : w 1 u 2 : v 2 : w 2 u 3 : v 3 : w 3 T represents a point on the 3-fold fiber product of E D with respect to the projection E D P 1 which sends u : v : wt T. Conversely given such a rational point Q on the 3-fold fiber product define the rational points six rational points X t : Y t : Z t andx t : Y t : Z t on the elliptic curve E D in terms of X t Tv 2 t w t Y t v t u 2 t D 4 T 4 w 2 t Z t T 3 u t + D 2 w t u 2 t D 4 T 4 w 2 t X t Dw2 t u2 t D4 T 4 wt 2 * [2Dvt 2 w2 t u2 t D4 T 4 wt 2 u2 t +2D2 u t w t + D 4 T 4 wt 2 ] Y t 2D2 v t w 3 t [2Dv2 t w2 t u2 t D4 T 4 w 2 t u2 t +2D2 u t w t + D 4 T 4 w 2 t ] Z t w 2 t u 2 t D 4 T 4 w 2 t 3.

9 Points on Hyperbolas at Rational Distance 919 It is easy to verify that the two compatibility relations hold: Y t X t u2 t D4 T 4 w 2 t Tv t w t Y t X t 2Dv t w t u 2 t D4 T 4 w 2 t Y t t X t X t Y 2D T so that we have a rational point P on the projective variety E D 4. While the projective variety E D 4 is somewhat cumbersome the surface E D gives us some insight on how to generate a family of rational distance sets S consisting of four rational points P t on the hyperbola C. Theorem 6. Continue notation as in Proposition 5. 1 There exists a non-trivial maps such that the composition E D E D P 1 is a constant map. 2 If X t : Y t :1are three rational points on E D which are not points of finite order then S {P 1 P 2 P 3 P 4 } is a rational distance set on C : axy + bx + cy + d 0when we choose P 1 P 2 P 3 P 4 ac + Y 1X 2 X 3 ac + X 1Y 2 X 3 ac + X 1X 2 Y 3 ac + Y 1 X 2 X 3 X 1 Y 2 Y 3 : ab a 2 X 1Y 2 Y 3 : ab a 2 Y 1X 2 Y 3 Y 1 X 2 Y 3 X 1 Y 2 X 3 : ab a 2 Y 1Y 2 X 3 Y 1 Y 2 X 3 X 1 X 2 Y 3 4D 2 Y 1 Y 2 Y 3 X 1 X 2 X 3 : ab 4D 2 a 2 X 1X 2 X 3 Y 1 Y 2 Y 3 3 When E D has positive rank there are infinitely many rational distance sets S of four points on the hyperbola C.. Proof. Define the map E D E D by X : Y : Z X 2 Z :X 2 + D 2 Z 2 Y : Z 3 1. Clearly this is a non-trivial map whose composition with the projection E D P 1 which sends u : v : wt T is a constant map. Say that X t : Y t : 1 are three rational points on E D which are not points of finite order. Then we have three points u t : v t : z t 1 on the surface E D which corresponds to a rational point on the 3-fold fiber product. Using the expressions

10 920 E. H. Goins & K. Mugo in Eq. * we find six rational points on E D given by X 14 : Y 14 : Z 14 X 1 : Y 1 : Z 1 DX1 2 D2 Z1 2 :2D2 Y 1 Z 1 : X 1 + DZ 1 2 X 24 : Y 24 : Z 24 X 2 : Y 2 : Z 2 DX2 2 D 2 Z2:2D 2 2 Y 2 Z 2 : X 2 + DZ 2 2 X 34 : Y 34 : Z 34 X 3 : Y 3 : Z 3 DX1 2 D2 Z1 2 :2D2 Y 1 Z 1 : X 1 + DZ 1 2 with the first three as labeled in Corollary 4. We have chosen our maps so that Y ij /X ij Y st /X st 2D for all distinct i j s andt sousingtheformulasin Proposition 3 we have the four rational points as above with the last being P 4 ac + Y 12X 14 X 24 : ab a 2 X 12Y 14 Y 24 X 12 Y 14 Y 24 Y 12 X 14 X 24 ac + Y 3X 1 X 2 X 3 Y 1 Y 2 ac + : ab a 2 X 3Y 1 Y 2 Y 3 X 1 : a 2 X 2 4D 2 Y 1 Y 2 Y 3 X 1 X 2 X 3 : ab 4D 2 a 2 X 1X 2 X 3 Y 1 Y 2 Y 3. Using Corollary 4 we find precisely what is listed above. As a consequence we find that we can always extend a rational distance set S of three points on a hyperbola to a set of four points. Corollary 2. Say that {P 1 P 2 P 3 } is a rational distance set of three points P t x t : y t :1on the hyperbola C : axy + bx + cy + d 0. If we choose the point P 4 c + ay 1 + bay 2 + bay 3 + b : b + ax 1 + cax 2 + cax 3 + c : a then S {P 1 P 2 P 3 P 4 } is a rational distance set of four points on the hyperbola. Proof. Following the proof of Proposition 3 we can express the points P t in the form P 1 ac + Y 1X 2 X 3 : ab a 2 X 1Y 2 Y 3 X 1 Y 2 Y 3 Y 1 X 2 X 3 P 2 ac + X 1Y 2 X 3 : ab a 2 Y 1X 2 Y 3 Y 1 X 2 Y 3 X 1 Y 2 X 3

11 P 3 Points on Hyperbolas at Rational Distance 921 ac + X 1X 2 Y 3 : ab a 2 Y 1Y 2 X 3 Y 1 Y 2 X 3 X 1 X 2 Y 3 wherewehavechosenthenot necessarily rational numbers Y 1 X 1 Y 2 X 2 Y 3 X 3 a a a Following Theorem 6 we choose P 4 ac + which is seen to be a rational point on C. 1 ax2 + cax 3 + c ay2 + bay 3 + b a 1 ax1 + cax 3 + c ay1 + bay 3 + b a 1 ax1 + cax 2 + c ay1 + bay 2 + b. a Y 1 Y 2 Y 3 4D 2 : ab 4D 2 a 2 X 1X 2 X 3 X 1 X 2 X 3 Y 1 Y 2 Y 3 As a specific example consider again the conic section C : xy We have seen that D 6 and the elliptic curve E 6 has the three rational points 12 : 36 : 1 50 : 35 : 8 and : : These correspond to the rational points 144 : 6480 : : : and : : respectively on the surface E D. We find the Fig. 2. Rational distance set on xy +120.

12 922 E. H. Goins & K. Mugo affine rational points P : : P : : P : : P : :1 P 1 P P 1 P P 2 P P 1 P P 2 P P 3 P lying on the hyperbola so that S {P 1 P 2 P 3 P 4 } is a rational distance set on C. A plot of this configuration can be found in Fig. 2. References [1] G. Campbell Points on y x 2 at rational distance Math. Comp electronic. [2] R. K. Guy Unsolved Problems in Number Theory Problem Books in Mathematics Vol. 1 Springer 2004 pp. xviii+437. [3] M. Ly S. Tsosie and L. P. Urresta Rational distance sets on conic sections MSRI-UP J