Elliptic Curves Containing Sequences of Consecutive Cubes
aa r X i v : . [ m a t h . N T ] J un ELLIPTIC CURVES CONTAINING SEQUENCES OFCONSECUTIVE CUBES
GAMZE SAVAS¸ C¸ EL˙IK AND G ¨OKHAN SOYDAN
Abstract.
Let E be an elliptic curve over Q described by y = x + Kx + L where K, L ∈ Q . A set of rational points ( x i , y i ) ∈ E ( Q ) for i = 1 , , · · · , k ,is said to be a sequence of consecutive cubes on E if the x − coordinatesof the points x i ’s for i = 1 , , · · · form consecutive cubes. In this note,we show the existence of an infinite family of elliptic curves containing alength-5-term sequence of consecutive cubes. Morever, these five rationalpoints in E ( Q ) are linearly independent and the rank r of E ( Q ) is at least5. Introduction
Let us consider a rational elliptic curve given by a Weierstrass equation y + a xy + a y = x + a x + a x + a (1.1)with a , · · · , a ∈ Q . We will say that the points ( x i , y i ) , i = 1 , · · · , k on thecurve (1.1) are in arithmetic progression of length k if the sequence x , x , · · · , x k forms an arithmetic progression (AP for short).In 1992, Lee and V´elez, [10], found infinitely many curves of type y = x + a containing k = 4-length APs. In 1999, Bremner, [3], showed that there areinfinitely many elliptic curves with k = 7 and k = 8-length APs. We shallbriefly say k -AP instead of k -length AP. Four years later, Campbell, [5], gave adifferent method to produce infinite families of elliptic curves with k = 7 and k = 8 APs. In addition, he described a method for obtaining infinite families ofquartic elliptic curves with k = 9 AP and gave an example of a quartic ellipticcurve with k = 12 AP. Two years later, Ulas [14], first described a constructionmethod for an infinite family of quartic elliptic curves on which there exists anAP with k = 10. Secondly he showed that there is an infinite family of quarticscontaining AP with k = 12. In 2006, Macleod, [11], showed that by simplifyingUlas’ approach, more general parametric solutions for APs arise with k = 10 Mathematics Subject Classification.
Primary 14G05, Secondary 11B83.
Key words and phrases.
Elliptic curves, rational points, sequences of consecutive cubes. which give a large number of examples with k = 12 and a few with k = 14.Let f ( x ) be an irreducible polynomial over Q of degree five. Consider the hy-perelliptic curve y = f ( x ). In 2009, Ulas, [15], found an infinite family of curveson which there is an AP with k = 11. In the same year, Alvarado, [1], showedthe existence of an infinite family of curves which contain APs with k = 12.Recently Dey and Maji, [7], found upper bounds for the lengths of sequencesof rational points on Mordell curves which are defined by the equations of thetype y = x + k , k ∈ Q \{ } , such that the ordinates of the points are in AP,and also when both the abscissae and ordinates of the points are seperately theterms of two APs.In 2013, Bremner and Ulas, [4], considered the sequences of rational points onelliptic curves whose x − coordinates form a “geometric progression”in Q . Theyobtained an infinite family of elliptic curves having geometric progression se-quence of length 4 and they also pointed out infinitely many elliptic curves withlength 5 geometric progression sequences can be obtained.Recently, Kamel and Sadek, [9], considered sequences of rational points on el-liptic curves given by the equation y = ax + bx + c over Q whose x − coordinatesform a sequence of consecutive squares. They showed that elliptic curves given bythe latter equation with 5 − term sequences of rational points whose x − coordinatesare elements of a sequence of consecutive squares in Q parametrized by an ellipticsurface whose rank is positive. This implies the existence of infinitely many suchelliptic curves. They also showed that these five rational points in the sequenceare linearly independent in the group of rational points of the elliptic curve theylie on. Especially, they introduced an infinite family of elliptic curves of rank ≥ x − coordinates form a sequence of “consecutive cubes ”. We considerelliptic curves given by the equation y = kx + lx + m over Q . Following thestrategy in [9], we obtain all their results which we detailed in the previousparagraph for “consecutive cubes”.2. SEQUENCES OF CONSECUTIVE CUBES
Definition 2.1.
Let E be an elliptic curve defined over a number field F by theWeierstrass equation y + a xy + a y = x + a x + a x + a , a i ∈ F. (2.1)The points ( x i , y i ) ∈ E ( F ) are said to form a sequence of “consecutive cubes”on E if there is c ∈ F such that x i = ( c + i ) , i = 1 , , · · · . LLIPTIC CURVES CONTAINING SEQUENCES OF CONSECUTIVE CUBES 3
Now we first need a result which guarantees the finiteness of the sequence ofconsecutive cubes on an elliptic curve. In 1910, Mordell conjectured that if ε is an algebraic curve over F of genus g ≥
2, then there are only finitely manyrational points on ε , i.e. the set ε ( F ) of F − rational points is finite. In 1983, thisconjecture was proved by Faltings, [8], and hence is now also known as Faltings’theorem. So, using Faltings’ theorem we give the following proposition aboutthe finiteness of consecutive cubes on an elliptic curve. Proposition 2.1.
Let E be an elliptic curve defined by (2.1) over a numberfield F . Let ( x i , y i ) ∈ E ( F ) be a sequence of consecutive cubes on E . Then thesequence ( x i , y i ) is finite.Proof. Assume without loss of generality that x i = ( c + i ) , i = 1 , , · · · , c ∈ F .This sequence leads to a sequence of rational points on the genus 5 hyperellipticcurve E ′ : y + a x y + a y = x + a x + a x + a . Thus, the points ( c + i, y ) ∈ E ′ ( F ). By using Faltings’ theorem [8], we obtainthat E ′ ( F ) is finite, so the sequence is finite. (cid:3) Next, using the above proposition, we define the length of this sequence likeAP.
Definition 2.2.
Let E be an elliptic curve over Q defined by a Weierstrassequation. Let ( x i , y i ) ∈ E ( Q ) , i = 1 , , · · · , n , be a sequence of consecutivecubes on E . Then n is called the length of the sequence.3. CONSTRUCTING ELLIPTIC CURVES CONTAINING 5- TERMSEQUENCES OF CONSECUTIVE CUBES
In this section, we investigate a family of elliptic curves given by the affineequation E : y = kx + lx + m (3.1)over Q . We will show that there exist infinitely many elliptic curves given by thelatter equation having 5 − term sequences of consecutive cubes.We consider 3 − term sequences of consecutive cubes. So, if (( c − , p ) , ( c , q ),and (( c + 1) , r ) lie in E ( Q ), where c ∈ Q , then these rational points form a3 − term sequence of consecutive cubes. Using these points, we obtain p = k ( c − + l ( c − + m,q = kc + lc + m,r = k ( c + 1) + l ( c + 1) + m. Hence, solving this system gives the following
GAMZE SAVAS¸ C¸ EL˙IK AND G ¨OKHAN SOYDAN k = [(3 c + 3 c + 1) p + ( − c − q + (3 c − c + 1) r ] / c (27 c + 54 c + c + 2) ,l = [ − (9 c + 36 c + 84 c + 126 c + 126 c + 84 c + 36 c + 9 c + 1) p + (18 c + 168 c + 252 c + 72 c + 2) q − (9 c − c + 84 c − c + 126 c − c + 36 c − c + 1) r ] / c − c + 1)(9 c + 9 c + 24 c + 21 c + 13 c + 6 c + 2) c,m = [(6 c + 33 c + 83 c + 126 c + 126 c + 84 c + 36 c + 9 c + c ) p + ( − c − c + 72 c − c + 4 c + 12) q + (6 c − c + 83 c − c + 126 c − c + 36 c − c + c ) r ] / c − c + 1)(9 c + 9 c + 24 c + 21 c + 13 c + 6 c + 2) . (3.2)Then, we obtain the following result: Remark 1.
As above, for given p, q, r ∈ Q ( c ), we know the existence of k, l, m ∈ Q ( c ) such that the ordered pairs (( c − , p ) , ( c , q ) and (( c + 1) , r ) are threerational points lying on (3.1).Now, secondly assuming (( c + 2) , s ) is a rational point on (3.1), we obtain a4 − term sequence of consecutive cubes on (3.1). Putting the values k, l, m and(( c + 2) , s ) in (3.1), one can find s = [(84 + 2109 c + 626 c + 243 c + 27 c + 1026 c + 2646 c + 4536 c + 5292 c + 4159 c ) p + ( − c − c − c − c − c − c − c − c − − c ) q + (702 c + 1134 c + 138 c − c + 243 c + 81 c + 252 c + 1008 c + 84 − c ) r ] / (3 c − c + 1)(3 c + 1)(1 + 3 c + 3 c )( c + 2) c. (3.3)Thus, we need to find the elements p, q, r and s in Q ( c ) which satisfy theequation (3.3). Now let’s explain how to find the general solution ( p, q, r, s ) forequation (3.3).Consider the quadratic surface S : a x + a y + a z + a t = 0over Q and the line aQ + bQ = ( a + bu : a + bv : a + bw : a )connecting the rational points Q = (1 : 1 : 1 : 1) and Q = ( u : v : w : 0)lying on S in three-dimensional projective space P . The intersection of S and LLIPTIC CURVES CONTAINING SEQUENCES OF CONSECUTIVE CUBES 5 aQ + bQ yields the quadratic equation( a + a + a + a ) a + ( a u + a v + a w ) b + (2 a u + 2 a v + 2 a w ) ab = 0 . Using Q and Q lying on S , one solves this quadratic equation and obtainsformulae for solutions ( x, y, z, t ). Since ( p, q, r, s ) = (1 , , ,
1) is a solution forequation (3.3), applying the above procedure to (3.3) gives the general solution( p, q, r, s ) with the following parametrization: p = ( c + 1) (cid:0) c + 3 c + 1 (cid:1) (cid:0) c + 6 c + 4 (cid:1) (cid:0) c + 9 c + 7 (cid:1) (cid:0) c + 2 c + 3 (cid:1) u + 3 (cid:0) c + c + 1 (cid:1) (cid:0) c + 1 (cid:1) (cid:0) c + 9 c + 7 (cid:1) (cid:0) c + 6 c + 18 c + 8 (cid:1) v − (cid:0) c + c + 1 (cid:1) (cid:0) c − c + 1 (cid:1) (cid:0) c + 6 c + 4 (cid:1) (cid:0) c + 3 c + 15 c + 7 (cid:1) w − (cid:0) c + c + 1 (cid:1) (cid:0) c + 1 (cid:1) (cid:0) c + 9 c + 7 (cid:1) (cid:0) c + 6 c + 18 c + 8 (cid:1) vu + 6 (cid:0) c + c + 1 (cid:1) (cid:0) c − c + 1 (cid:1) (cid:0) c + 6 c + 4 (cid:1) (cid:0) c + 3 c + 15 c + 7 (cid:1) wu,q = − ( c + 1) (cid:0) c + 3 c + 1 (cid:1) (cid:0) c + 6 c + 4 (cid:1) (cid:0) c + 9 c + 7 (cid:1) (cid:0) c + 2 c + 3 (cid:1) u − (cid:0) c + c + 1 (cid:1) (cid:0) c + 1 (cid:1) (cid:0) c + 9 c + 7 (cid:1) (cid:0) c + 6 c + 18 c + 8 (cid:1) v − (cid:0) c + c + 1 (cid:1) (cid:0) c − c + 1 (cid:1) (cid:0) c + 6 c + 4 (cid:1) (cid:0) c + 3 c + 15 c + 7 (cid:1) w c + 1) (cid:0) c + 3 c + 1 (cid:1) (cid:0) c + 6 c + 4 (cid:1) (cid:0) c + 9 c + 7 (cid:1) (cid:0) c + 2 c + 3 (cid:1) vu + 6 (cid:0) c + c + 1 (cid:1) (cid:0) c − c + 1 (cid:1) (cid:0) c + 6 c + 4 (cid:1) (cid:0) c + 3 c + 15 c + 7 (cid:1) wv,r = − ( c + 1) (cid:0) c + 3 c + 1 (cid:1) (cid:0) c + 6 c + 4 (cid:1) (cid:0) c + 9 c + 7 (cid:1) (cid:0) c + 2 c + 3 (cid:1) u + 3 (cid:0) c + c + 1 (cid:1) (cid:0) c + 1 (cid:1) (cid:0) c + 9 c + 7 (cid:1) (cid:0) c + 6 c + 18 c + 8 (cid:1) v + 3 (cid:0) c + c + 1 (cid:1) (cid:0) c − c + 1 (cid:1) (cid:0) c + 6 c + 4 (cid:1) (cid:0) c + 3 c + 15 c + 7 (cid:1) w c + 1) (cid:0) c + 3 c + 1 (cid:1) (cid:0) c + 6 c + 4 (cid:1) (cid:0) c + 9 c + 7 (cid:1) (cid:0) c + 2 c + 3 (cid:1) wu − (cid:0) c + c + 1 (cid:1) (cid:0) c + 1 (cid:1) (cid:0) c + 9 c + 7 (cid:1) (cid:0) c + 6 c + 18 c + 8 (cid:1) wv,s = − ( c + 1) (cid:0) c + 3 c + 1 (cid:1) (cid:0) c + 6 c + 4 (cid:1) (cid:0) c + 9 c + 7 (cid:1) (cid:0) c + 2 c + 3 (cid:1) u + 3 (cid:0) c + c + 1 (cid:1) (cid:0) c + 1 (cid:1) (cid:0) c + 9 c + 7 (cid:1) (cid:0) c + 6 c + 18 c + 8 (cid:1) v − (cid:0) c + c + 1 (cid:1) (cid:0) c − c + 1 (cid:1) (cid:0) c + 6 c + 4 (cid:1) (cid:0) c + 3 c + 15 c + 7 (cid:1) w . (3.4)See [12] for details about finding parametric rational solutions of a homoge-nous polynomial of degree 2 in several variables. Remark 2.
The above argument shows that given p, q, r, s ∈ Q ( c, u, v, w ), thereexist k, l, m ∈ Q ( c ) such that the ordered pairs (( c − , p ) , ( c , q ) , (( c + 1) , r ) , (( c + 2) , s ) are four rational points on (3.1). GAMZE SAVAS¸ C¸ EL˙IK AND G ¨OKHAN SOYDAN
Next we consider the case when (( c − , t ) ∈ E ( Q ). In this case, there existsa 5 − term sequence of consecutive cubes on (3.1). Then one obtains t = Ku + Lu + M u + N u + P (3.5)with K = [( c + 1) (cid:0) c + 3 c + 1 (cid:1) (cid:0) c + 6 c + 4 (cid:1) (cid:0) c + 9 c + 7 (cid:1) (cid:0) c + 2 c + 3 (cid:1) ] ,L = −
24 ( c + 1) (cid:0) c + 2 c + 3 (cid:1) (cid:0) c + 3 c + 1 (cid:1) (cid:0) c + 4 (cid:1) (cid:0) c + 6 c + 4 (cid:1) (3 c + 9 c + 7) (cid:0) c − c + 1 (cid:1) (cid:0) c + 3 c + 27 c + 1 (cid:1) v + 32 c ( c + 1) (cid:0) c + 2 c + 3 (cid:1) (3 c + 3 c + 1) (cid:0) c + 4 (cid:1) (cid:0) c + 6 c + 4 (cid:1) (cid:0) c + 9 c + 7 (cid:1) (cid:0) c + 8 (cid:1) (3 c − c + 4) w,M = 6 (1215 c + 4131 c + 24543 c + 49383 c + 134460 c + 152118 c + 263619 c + 229491 c + 297153 c + 223859 c + 208374 c + 123018 c + 52116 c + 16504 c + 480) (cid:0) c + 9 c + 7 (cid:1) v − (cid:0) c + c + 1 (cid:1) (cid:0) c + 4 (cid:1)(cid:0) c + 6 c + 4 (cid:1) (cid:0) c + 9 c + 7 (cid:1) (189 c + 2340 c − c + 7383 c − c + 7365 c + 740 c − c + 1552 c + 21) wv + 2 (2835 c − c + 37881 c − c + 156276 c − c + 395718 c − c + 374198 c − c − c − c − c + 2688 c + 441) (cid:0) c + 6 c + 4 (cid:1) w ,N = −
72 ( c + c + 1)(3 c + 4)( c − c + 1)(3 c + 1)(3 c + 6 c + 18 c + 8)(3 c + 3 c + 27 c + 1)(3 c + 9 c + 7) v + 48 ( c + c + 1)(3 c + 4)(3 c + 6 c + 4)(3 c + 9 c + 7)(27 c + 450 c − c + 2019 c − c + 2325 c − c − c + 16 c + 21) v w + 24 ( c + c + 1)(3 c + 4)(3 c + 6 c + 4)(3 c + 9 c + 7)(135 c + 1440 c − c + 3345 c − c + 2715 c + 3100 c + 2550 c + 1520 c − vw − c ( c + c + 1)(3 c + 4)(3 c − c + 1)(3 c − c + 4)(3 c + 3 c + 15 c + 7)( c + 8)(3 c + 6 c + 4) w , LLIPTIC CURVES CONTAINING SEQUENCES OF CONSECUTIVE CUBES 7 P = [3( c + c + 1) (cid:0) c − c + 1 (cid:1) (cid:0) c + 6 c + 4 (cid:1) (3 c + 3 c + 15 c + 7)] w + 72 (cid:0) c − c + 1 (cid:1) (cid:0) c + 4 (cid:1) (cid:0) c + 6 c + 4 (cid:1) (cid:0) c − c + 7 (cid:1) (3 c + 3 c + 15 c + 7) (cid:0) c − c + 27 c − (cid:1) (cid:0) c + c + 1 (cid:1) w v −
18 (6561 c + 2187 c + 91125 c + 42039 c + 287712 c − c − c + 6399 c + 316035 c + 232191 c + 1581642 c + 299082 c + 294228 c + 248472 c − c + c + 1) w v + 72 (cid:0) c + 1 (cid:1) (cid:0) c + 4 (cid:1) (cid:0) c + 9 c + 7 (cid:1) (cid:0) c − c + 7 (cid:1) (3 c − c + 27 c − (cid:0) c + 6 c + 18 c + 8 (cid:1) (cid:0) c + c + 1 (cid:1) wv + [3 (cid:0) c + c + 1 (cid:1) (cid:0) c + 1 (cid:1)(cid:0) c + 9 c + 7 (cid:1) (cid:0) c + 6 c + 18 c + 8 (cid:1) ] v . We see that the above expressions are homogeneous in v and w . So, we mayassume that w = 1. Now we consider the curve H : Y = KX + LX + M X + N X + P (3.6)over Q ( c, v ). This equation of the form (3.6) is birationally equivalent to anelliptic curve χ defined by the form χ : V = U − IU − J (3.7)where I = 12 KP − LN + M (3.8)and J = 72 KM P + 9
LM N − KN − L P − M . (3.9)The discriminant ∆( χ ) of χ is given by (4 I − J ) /
27, and the specialization of χ is singular only if ∆( χ ) = 0. Furthermore, the point R = (cid:18) L − KM K , L + 8 K N − KLM K / (cid:19) (3.10)lies in χ ( Q ( c, v )) since K is a square (see [6, chapter 3, pp. 89-91] for details).Now we are ready to give the following result. Theorem 3.1.
The curve (3.6) is birationally equivalent over Q ( c, v ) to anelliptic curve χ with rank χ ( Q ( c, v )) > .Proof. Write the homogenous form of the curve (3.6). Then one gets Y = KX + LX Z + M X Z + N XZ + P Z with a rational point T = ( X : Y : Z ) = (1 : ( c + 1)(3 c + 3 c + 1)(3 c + 6 c + 4)(3 c + 9 c + 7)( c + 2 c + 3) : 0).According to the above procedure, the curve (3.6) is birationally equivalent to(3.7) and taking c = 3, v = , using (3.8)-(3.10), one gets the specialization˜R =( , ) of the point R on the special-ized elliptic curve GAMZE SAVAS¸ C¸ EL˙IK AND G ¨OKHAN SOYDAN ψ : Y = X − X + 30476125037279414454071839383853830234262941440938082304735091890625 . By MAGMA, [2], we see that the point ˜R is a point of infinite order on ψ . Thus,by the Silverman specialization theorem, [13, Theo. 11.4], the point R is ofinfinite order on χ . (cid:3) Corollary 3.1.1.
Let a nontrivial sequence of consecutive rational cubes be ( c − , ( c − , c , ( c + 1) , ( c + 2) . Then there are infinitely many ellipticcurves of the form E j : y = k j x + l j x + m j , = j ∈ Z , such that ( c + i ) , i = − , − , , , , is the x − coordinate of a rational point on E j . Furthermore, thesefive rational points are linearly independent.Proof. Set c = c , v = v and w = 1 in (3.4). Then one gets the elliptic curve χ c ,v , : t = Ku + Lu + M u + N u + P (3.11) K, L, M, N, P ∈ Q , and according to Theorem 3.1 its rank is positive. Then wewill find a point R = ( u, t ) of infinite order in χ c ,v , ( Q ). Set jR = ( u j , t j ) with0 = j ∈ Z to be the j − th multiple of the point R in χ c ,v ( Q ).Now, substituting c = c , v = v , w = 1 and u = u j into the formulae for p, q, r, s ∈ Q ( c, u, v, w ) in (3.2), one gets the rational numbers p j , q j , r j , s j re-spectively. Then substituting p j , q j , r j , s j into formulae for k, l, m ∈ Q ( c, p, q, r )in (3.2) one obtains the rational numbers k j , l j , m j respectively.Hence we constructed an infinite family of elliptic curves E j : y = k j x + l j x + m j with 0 = j ∈ Z . This infinite family E j of elliptic curves has the point(( c − , p j ) , ( c , q j ) , (( c + 1) , r j ) , (( c + 2) , s j ) , (( c − , t j ) ∈ E j ( Q ). Thismeans that we get an infinite family of elliptic curves with a 5 − term sequence ofrational points whose x − coordinates consist of a sequence of consecutive cubesin Q .Now, we will show that the points (( c − , p j ) , ( c , q j ) , (( c + 1) , r j ) , (( c +2) , s j ) , (( c − , t j ) ∈ E j ( Q ) are linearly independent. To do this we first needto find a point ( u, t ) lying in (3.11).Consider the equation (3.11). Taking c = 3 , v = , w = 1, we obtain thecurve t = 63404527588416 u − u + 19793578415844699648550525 u + 478172417894196583574016303077775625 . (3.12) LLIPTIC CURVES CONTAINING SEQUENCES OF CONSECUTIVE CUBES 9
Completing the square on the RHS of (3.12), we obtain a point of infinite order( u, t ) = ( 6054777653 , ∈ χ , , ( Q ) . So, using the specialization c = 3, v = , w = 1, u = we obtain aspecialized elliptic curve E : y = 101931764760453272870450501152925375 x + 1706408630108593668606722657955417125 x +501846962320384035129646905616917886230000625with the following set of consecutive cubes in E ( Q ) =(1 , , (2 , , (3 , , (4 , , (5 , . By MAGMA, we see that these rational points are linearly independent.By the Silverman Specialization Theorem, we can say that the points(( c − , p j ) , ( c , q j ) , (( c + 1) , r j ) , (( c + 2) , s j ) , (( c − , t j ) are linearly inde-pendent in E j over Q ( c, v, u j ). This completes the proof of the corollary. (cid:3) Remark 3.
The previous corollary implies the existence of an infinite family ofelliptic curves whose rank r ≥ − term sequence of consecutive cubes, weassume that the point (( c + 3) , z ) is a rational point on (3.1). So, the followingrelation is satisfied z = K ′ u + L ′ u + M ′ u + N ′ u + P ′ with K ′ , L ′ , M ′ , N ′ , P ′ ∈ Q ( c, v ). So we give the following remark. Remark 4.
The existence of a 6 − term sequence of consecutive cubes on theelliptic curve depends on the existence of a rational point ( u, t, z ) on the algebraiccurve defined by the following intersection C : t = Ku + Lu + M u + N u + P, z = K ′ u + L ′ u + M ′ u + N ′ u + P ′ . The genus of the curve C is 5. Thus , Falting’s Theorem says that for given c ∈ Q ,there are only finitely many elliptic curves over Q defined by y = kx + lx + m where ( c + j ) , j = − , − , , , , x − coordinates of a 6 − term sequenceof consecutive cubes. Acknowledgements.
We would like to thank the referee for carefully readingour manuscript and for giving such constructive comments which substantiallyhelped improving the presentation of the paper. And also we thank ProfessorMohammad Sadek for his useful discussions. The second author was supportedby the Research Fund of Uluda˘g University under Project No: F-2016/9.
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E-mail address : [email protected]; [email protected] G¨okhan Soydan , Department of Mathematics, Uluda˘g University, 16059 Bursa,Turkey
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