On a Diophantine equation of the second degree

Transcription

1 ARKIV FOR MATEMATIK Band 3 nr 33 Communicated 6 June 1956 by T. AGELL and OTTO FROST~A On a Diophantine equation of the second degree By BEr~CT STOLT 1. Introduction It is easy to solve the Diophantine equation Ax ~ + Bxy + Cy ~ + Dx + Ey +.F =0 with integral coefficients, in integers x and y when the equation represents an ellipse or a parabola in the (x, y)-plane. If the equation represents a hyperbola, the problem is much more difficult. For solving an equation of this type one may use either the theory of quadratic forms or the theory of quadratic fields. T. agell has shown ([1]-[5]) how it is possible to determine all the solutions of the Diophantine equation x ~-Dy ~= ±, (1) where D and are integers and D is not a perfect square, by quite elementary methods. The author ([6]-[8]) used these methods to the equation Consider the Diophantine equation x 2-Dy 2 =±4. (2) A u ~ +Buv + Cv 2 = ±, (3) where A, B, C and are integers and B ~ - 4A C = D is a positive integer which is not a perfect square. It is obvious that (3) can be transformed into (1) by means of linear transformations with integral coefficients. The problem of determining all the solutions of (3) in integers u and v then reduces to the problem of finding all the integral solutions x and y of (1) which satisfy certain linear congruences; see agell [4], pp However, in this way we get no general view of the solutions of (3), and it will be rather laborious to discuss the different cases which may occur. The purpose of this paper is to show how it is possible to avoid the linear transformations and congruences. In fact, for equation (3), we shall deduce inequalities analogous to those determined by agell for equation (1). We shall use the notions proposed by agell or notions analogous to them. 2s 381

2 B. STOLT, On a Diophantine equation of the second degree 2. The Diophantine equation x 2- Dy ~= 4 Consider the Diophantine equation x ~ - Dy ~ = 4, (4) where D is a positive integer which is not a perfect square. When x and y are integers satisfying this equation, the number ½(x + y V-D) is said to be an integral solution of this equation. Two solutions ½(x +y VD) and ½(x' +y' VD) are equal, if x =x" and y = y'. Among all the integral solutions of the equation there is a solution ½ (z~ + y, VD) in which x 1 and yl are the least positive integers satisfying the equation. This integral solution is called the/undamental solution. A well-known result is the following T~.ORE~. When D is a natural number which is not a per/ect square, the Diophantine equation x 2 - Dy ~ = 4 (4) has an in/inity o/integral solutions. 1/the ]undamental solution is denoted by e, every integral solution ½(x + y VD) may be written in the /orm ½(x+yVD)=+e ~, (k=o, +l, +_2, +3,...). If D ~ 5 (mod. 8), there only exist integral solutions with even x and y. 3. The classes of solutions of the Diophantine equation.4u2+ Buy + Cv ~= +. The fundamental solutions of the classes Let A and Z r be positive integers and B and C be rational integers such that B z- 4AC = D is a positive integer which is not a perfect square. Consider the Diophantine equation A u2 + Buv + Cv 2= +25. (3) It is suitable to define a solution of (3) in the following way. If u = t/a is a fractional number and v is an integer which satisfy (3), the number (2Au + By) + v V-D (5) is called a solution of (3). If (x + y~/d}/2) is an integral solution of the Diophantine equation x 2 - Dy 2 ='4, (4) the number (2Au+Bv)+vVD x+y~-d (2Aux+Bvx+Dvy)+(2Auy+Bvy+vx)V-D

3 - uv') ARKIV FSR MATEMATIK. Bd 3 nr 33 is also a solution of (3). This solution is said to be associated with the solution [(2A u + By) + ~/D]/2. The set of all solutions associated with each other forms a class o/ solutions of (3). If u and v are two integers satisfying (3), the number [(2Au + By) + v~-d]]2 is called an integral solution of (3). It is possible to decide whether the two given solutions [(2Au + By) + vg-d]/2 and [(2Au' + By') + v'v-d]/2 belong to the same class or not. In fact, it is easily seen that the necessary and sufficient condition for these two solutions to be associated with each other is that the two numbers [2Auu' + B(uv' + u'v) + 2Cvv']/ and (vu" - / be integers. If u = t/a is a fractional number and v is an integer which satisfy (3), - u and - v also satisfy (3). It is apparent that the two solutions _ [(2Au +By)+v /D]/2 belong to the same class. Let u and v be two integers satisfying (3). Then there exists a fractional number u' = - (Au + Bv)/A such that the numbers u' and v satisfy (3). So do the numbers - u' = (Au Bv)/A and - v. It is easily seen that the solutions belong to the same class. _~ (2Au' + By) +v ~D - 2 =~ 2 (2Au + By) -v V-D Suppose that K is the class consisting of the solutions [(2 A u i + By,) + v~ VD]/2, i = 1,2,3,... Then it is evident that the solutions [(2Au, + Bvt) -v~ ~/D]/2, i = 1, 2,3..., also constitute a class, which may be denoted by K'. The classes K and K' are said to be conjugates of each other. Conjugate classes are in general distinct but may sometimes coincide. Among all the solutions [(2Au + By) + v~/d]2 in a given class K we now choose a solution [(2Au 1 By1) + vlvd]/2 in the following way: Let v 1 be the least nonnegative value of v which occurs in K. If K and K' do not coincide, then the number u 1 is also uniquely determined; for the solution (2Au~ BVl) Vl~/D --(2Au~ Bv,) vlv-l) 2 2 belongs to the conjugate class K'. If K and K' coincide, we get a uniquely determined u 1 by prescribing u 1 ~_u'l. The solution [(2Au 1 BVl) v I ~/D]/2 defined in this way is said to be the/undamental solution o/the class. The case v 1 = 0 can only occur when the classes coincide. We prove THEOREM 1. Suppose that [(2Au + By)+ v V-D]I2 is the/undamental solution o/ the class K o/ the Diophantine equation A u 2 Buy + Cv 2 =, (7) where A and are positive integers and B and C rational integers, and further D = B ~ - 383

4 B. stolt, On a Diophantine equation of the second degree 4A C is a positive integer which is not a per/ect square. I] (xx + Yl ~/D)/2 isthe/undamental solution o/equation (4), we have the inequality 0_<-v-_< (xl-2). (8) Proo]. If equality (8) is true for a class K, it is also true for the conjugate class K'. Thus we can suppose that (2A u + By) is positive. We easily get Consider the solution (2Au + By) +v V-D xl-y, 2 2 VD (2Au + Bv)x 1- Dvy 1 + (vx 1 - (2Au + Bv)yl)VD 4 which belongs to the same class as [(2Au + By) + v V~)]/2. Since [(2Au + By) + v ~/D]/2 is the fundamental solution, and since by (9) [(2Au + Bv)x 1 -DVyl]/4 is positive, we must have (2Au + Bv)x 1- Dry1 >= 2Au + Bv 4 2 (10) From this inequality it follows that and finally This proves inequality (8). (2Au + By) 2 (x 1-2) 2 => Dv2y~ D v 2 A >= x1_2- THEOREM 2. Suppose that [(2Au + By)+v l/d]/2 is the lundamental solution o/ the class K o/the Diophantine equation Au ~ + Buy + Cv z = -, (11) where A and are positive integers and B and C rational integers, and/urther D = B ~ - 4A C is a positive integer which is not a per/ect square. If (x 1 + Yl ~/D)/2 is the/undamental solution o/equation (4), we have the inequality 384 O< v~_ ~ (x~ + 2). (12)

5 AIIKIY FOR MATEMATIK. Bd 3 nr 33 Proo[. If equality (12) is true for a class K, it is also true for the conjugate class K'. Thus we can suppose (2A u + By) >= O. We clearly have x~v~ Y~ ~-I[(2Au+Bv)2+4A]>D Y~(2Au+Bv)~ Thus xlv - yl(2au + By) >0. (13) Consider the solution (2Au+Bv)+vV-D xl-y~ (2Au+Bv)x~-Dvyx+(vx~-(2Au+Bv)y~) which belongs to the same class as [(2Au + Bv)+v ~/D]/2. Since [(2Au + By)+ v I/D]/2 is the fundamental solution of the class, and since, by (13), [xlv - Yl (2Au -4- By)]]4 is positive, we must have xlv -Yl (2Au + Bv) >v. 4 =2 (14) From this inequality it follows that (zl - 2)2v 2 > 2 = yl (2A u + Bv) 2 = y~ (Dv 2-4A ) or v 2 <-~ (x I +2). This proves inequality (12). As only integral solutions are of interest, we prove TrIEOR~M 3. I/one o/the solutions o/ the class K is an integral solution, every solution of K is integral. Proo[. Let K be a class and [(2Au + By) + v V-D]]2 be an integral solution of it. If there existed a solution [(2A u~ + Bye) + v~ ~/D]/2 belonging to K, where u 1 = t/a were no integer, we would have an integral solution (x + y ~/D) / 2 of (4) such that would hold. Hence (2Au+Bv)+vVD x+yl/d (2Aul +BVl)+V 1VD v (x + By) u(x - By) Cry, v 1 + Auy. Ul Both u 1 and v 1 are integers, for if B is even, D is divisible by 4 and x is divisible by 2. If B is odd, D is odd. In that case both x and y are either even or odd. This proves the theorem. 385

6 n. STOLT, On a Diophantine equation of the second degree If the solutions of a class K are integral, K is called a class o/integral solutions. If the classes K and K' are conjugates of each other, it may happen that K but not K' is a class of integral solutions. From the preceding theorems we deduce at once TH~ORV.M 4. 1/ A and are positive integers and B and C rational integers, and /urther D = B z - 4A C is a positive integer which is not a per/ect square, the Diophantine equations (7) and (11) have a ]inite number o/classes o/integral solutions. The ]unda. mental solutions o/all the classes can be ]ound after a/inite number o/trials by means o/the inequalities in Theorems 1 and 2. [(2Au 1-4- By1). v 1 I/D]/2 is the/undamental solution o/the class K, we obtain all the solutions [(2Au By) v ~/D]/2 by the /ormula (2Au+ Bv)+vVD (2Aul + Bvl)+v~VL) x+yvd where (x + y V-D) [2 runs through all the solutions o/(4), including +_ 1. The Diophantine equations (7) and (11) have no integral solutions at all when they have no integral solutions satis/ying inequality (8), or (12) respectively. We next prove THEOREM 5. The necessary and su[/icient condition [or the solutions o/the Diophantine equation (2A u -4- By) + v (-D (2Au 1 + By1) "4"v I/D 2 2 Au ~ + Buv +Cv ~ = + (3) to belong to the same class is that UVl -- Ul v be an integer. Proo]. We already know that a necessary and sufficient condition is that 2Auu 1 + B (uv 1 + UlV ) -4-2Cvv 1 uv 1 - UlV be integers. Thus it is sufficient to show that 2Auu l + B(uv l +ulv)+ 2Cvv I is an integer when (uv 1 - UxV )/ is an integer. Multiplying the equations 386

7 AI{KIV FOR M/LTEMATIK, Bd 3 nr 33 Au 2 + Buy + Cv 2 = +_, Au~ + Bulv 1 + Cv~ = +_ (14) we get (2Auu 1 + B(uvl + UlV ) + 2Cvvl) ~ - D(uv 1 - ulv) 2 = 4 2. (15) It is apparent from (15) that 2Auu 1 + B(uvl + ulv)+ 2Cvv I is divisible by when uv 1 - UlV is divisible by. 4. Quasi-ambiguous classes Let K be a class of solutions of the Diophantine equation Au ~ + Buy +Cv ~= ±. (3) Further let [(2Au + By) + v V-D]/2 be a solution of K. If the number (2Au+Bv)v (16) is an integer, K is called a quasi-ambiguous class. If u and v are integers, K is a quasiambiguous class of integral solutions. We prove T~o~.~ 6. 17/the conjugate classes K and K' o/(3) coincide, the resulting class is quasi-ambiguous. Proof. Let K and K' be a pair of conjugate classes of (3), and let [(2Au By) ~/-D]/2 and [(2Au' + By) v l/d]/2 be one solution of every class. As the classes coincide, according to Theorem 5 the number (u -u')v/ is an integer. As (2Au + By) = - (2Au' + By), we get By = -A(u-u'). Hence we get -(2Au'+Bv)v=A(u'u')v (2AU+Bv)v Clearly this number is an integer. This proves the theorem. It is apparent from Example 4 that two conjugate classes K and K' may be quas i- ambiguous without coinciding. THEOREM 7. Let K and K' be two conjugate classes of integral solutions. I/ their ]undamental solutions are [(2Au + Bv) v ~/D]/2 and [(2Au' + Bv) v ]/D]/2, respectively, and if we have v = O, the classes K and K' coincide. Let K be a class o/ integral solutions, i/it has a solnt!on [(2Au + By)+ v ~D]2, where u = 0, and if C divides B, the class is quasi-ambiguous. Proof. Let K and K' be two conjugate classes of integral solutions, and let [(2A U + By) + v VD]/2 and [(2Au' + By) +v ~/D]/2 be their fundamental solutions. The necessary and sufficient condition for the two classes to coincide it that the number 387

8 B, STOLT, On a Diophantine equation of the second degree (U --?~') V be an integer. H we have v = O, the condition is clearly fulfilled. Let K be a class of integral solutions, and let [(2A u + Bv) + v VD] ]2 be a solution of it, where u = 0. If we put the values 0 and v into the Diophantine equation we get ± = Cv ~. Au ~ + Buy + Cv ~ = +_, (3) The condition for the class K to be quasi-ambiguous is that the number be an integer. Putting u = 0 and _+ = Cv 2 into (16) we get (2Au+Bv)v 2v (16) This proves the theorem. (2Au + Bv)v B Icl TtIEOREM 8. A necessary condition/or the Diophantine equation A u ~ + Buy + Cv 2 = +_ (3) to have quasi-ambiguous classes is = st ~, where s is a square-/ree integer that divides dad. Proo/. Let Au ~ + Buy =Cv ~ = _+ (3) be a Diophantine equation, and suppose that u = w/a is a fractional number and v an integer which satisfy (3). Then the Diophantine equation (2Au + By) 2- Dv ~ = +_4A (18) is solvable in integers (2Au + By) and v. Suppose that we have = st 2, where s is square-free. Further suppose that (3) has a quasi-ambiguous class, and let the number [(2Au + Bv) + v V-D]/2 be a solution of it. Then the number (2A u + Bv) v (19) s t 2 is an integer. Let P be a prime that divides t. If p divides v, it follows from (18) that it also divides (2Au + By). If p and v are coprime, p~ divides (2Au + Bv) as is easily seen from (19). But then it follows from (18) that p~ divides D. ow let p be a prime that divides s. If p divides v, it also divides (2Au + Bv). But then p2 divides 4A and thus p divides 4A. If p and v are coprime, p divides (2Au + By) and thus p divides D. This proves the theorem. 388

9 ARKIV FOR MATEMATIK. Bd 3 nr umerical examples Finally, we give some examples which illustrate the preceding theorems. Example u s + 29uv + v z = 31, D = 5. The fundamental solution of the equation x2-5y~=4 is (3 +V5-)/2. For the fundamental solutions, according to inequality (8), we get 0 -< v _< 35. We find the fundamental solutions [(2Au~+Bv~)+vil/D]/2, i=1,2...,8, where vl. 2=1, u1=6/19, u~=-5/ll; Vz.~=14, us=-ll/19, u4 =-15/11; vs. 6 = 23, u~ = - 249/209, u6 = - 2; v7.8 = 35, u7 = - 2, us = - 597/209. Thus the equation has only two classes of integral solutions. Example 2. u ~+3uv+v ~= -5, D=5. For the fundamental solutions, according to inequality (12), we get 0 _< v ~ 1. We find the fundamental solution [(2Au + By) +v ~/-D]/2, where v = 1, u = 1. The equation is also satisfied by the numbers v = 1, u = -4. However, according to Theorem 5 there is only one class, and this class is quasi-ambiguous, according to Theorem 6. 'Example 3. 3u2+7uv+3v 2= -13, D=13. The fundamental solution of the equation x ~ -13y 2 =4 is ( V]-3)/2. For the fundamental solutions, according to inequality (12), we get 0 _< v _< 6. We find the fundamental solutions [(2A u~ + B vi) + v, I/-D] ] 2, i = 1,2, where vl~ =5, u1=11/3, us= -8. Thus the equation has only one class of integral solutions, and this class is quasi-ambiguous. Example 4. 2u 2+5uv+v ~=16=24,D=17. The fundamental solution of the equation x 2-17y ~ = 4 is ( ~Zl-7)/2. For the fundamental solutions, according to inequality (8), we get 0 -< v _< 10. We find the fundamental solutions [(2Au~ + 33v,) + v~ l/d]/2, i = 1,2,...,6, where Vl. 2 =2, Ul =2, u2 = -6; v3.4 =4, u3 =0, u4 = - 10; v5.6 =7, u~ = - 1, u6 = - 33/2. According to Theorem 7 the solutions [(2Aui + Bvj) + vj I/D]/2, where ~ = 3 or 4, belong to quazi-ambiguous classes. Example 5. u s + 5uv + 2v 2 = 32 = 25, D =17. For the fundamental solutions, according to inequality (8), we get 0 _< v _ 10. We find the fundamental solutions [(2Au i + Bvi) + v, V-D]/2, i = 1,2,..., 6, where v~.2 =2, ul =2, u2 = - 12; vz. 4 =4, ua =0, u4 = - 20; v~.6 =7, u5 = - 2, u6 = According to Theorem 7 the solution [(2A u 3 + Bva) + v a [/3]/2, where u a = 0, belongs to a class which is not quasi-ambiguous. Example 6. 3u ~' + 14uv + 4v 2 = 259 = 7.37, D = 148. The fundamental solution of the equation x 2-148y ~ = 4 is ( ~/~8)[2. For the fundamental solutions, according to inequality (8), we get 0 -< v _< 27. We find the fundamental solutions [(2A u s + Bvi) + v~ I/D]/2, i = 1,2, where vl. 2 = 4, u I = 3, u S = -65/3. According to Theorem 8 the only class of integral solutions is not quasi-ambiguous. 28t 389

10 Bo SToLT, On a Diophantine equation of the second degree REFERECES 1. T. ~A(]ELL, En element~er metoc[e tl! a bestemme gitterpunktene ph en hyperbel. orsk Matem. Tidsskr. 26 (1944), ElementAr talteori. Uppsala 1950, t~oer die I)arstellung ganzer Zahlen durch eine indefinite bin~re quadratische FOrm. Arehiv der Mathematik 2 (1950), Introduction t~ umber Theory. Uppsala 1951, Bemerkung tiber clie diophantische 'Gleichung u s -Dv s = C. Archiv der Mathematik 3 (1952), B. STOLT, On the Diophantine equation u s -Dv s = 4. Arkiv fsr matematik 2 (1951), On the Diophantine equation u s -Dv s = ~-41V, Part II. Arkiv fsr matematik 2 (1952), On the Diophantine equation tt 2 -Dv ~ = + 4_, Part III. Arkiv ffr matematik 3 (1954), Tryckt den 18 januari Uppsala Almqvist & Wiksells Boktryckeri AB