aa r X i v : . [ m a t h . N T ] S e p SEQUENCES ASSOCIATED TO ELLIPTIC CURVES
BET ¨UL GEZER
Abstract.
Let E be an elliptic curve defined over a field K (with char ( K ) =2) given by a Weierstrass equation and let P = ( x, y ) ∈ E ( K ) be a point.Then for each n ≥ γ ∈ K ∗ we can write the x - and y -coordinatesof the point [ n ] P as[ n ] P = (cid:18) φ n ( P ) ψ n ( P ) , ω n ( P ) ψ n ( P ) (cid:19) = (cid:18) γ G n ( P ) F n ( P ) , γ H n ( P ) F n ( P ) (cid:19) where φ n , ψ n , ω n ∈ K [ x, y ], gcd( φ n , ψ n ) = 1 and F n ( P ) = γ − n ψ n ( P ) , G n ( P ) = γ − n φ n ( P ) , H n ( P ) = γ − n ω n ( P )are suitably normalized division polynomials of E . In this work we show thecoefficients of the elliptic curve E can be defined in terms of the sequences ofvalues ( G n ( P )) n ≥ and ( H n ( P )) n ≥ of the suitably normalized division poly-nomials of E evaluated at a point P ∈ E ( K ). Then we give the general termsof the sequences ( G n ( P )) n ≥ and ( H n ( P )) n ≥ associated to Tate normal formof an elliptic curve. As an application of this we determine square and cubeterms in these sequences. Introduction
Let E denote an elliptic curve defined over a field K given by a Weierstrassequation(1.1) E : y + a xy + a y = x + a x + a x + a .For background on elliptic curves, see [29] and [30]. Let E ( K ) be the group of K -rational points on E , let O denote the point at infinity, the identity for the group K -rational points. Let K ( E ) denote the function field of E over K . Then z = − x/y ∈ K ( E ) is a uniformizer at O and the invariant differential ω = dx/ (2 y + a x + a )has an expansion as a formal Laurent series in a formal neighborhood of O suchthat ω ( z ) = (1 + a z + ( a + a ) z + · · · ) dz .This series has coefficients in Z [ a , a , a , a , a ], and the uniformizer z and thedifferential ω at O satisfy ( ω/dz )( O ) = 1. Let n ≥ n ]( z ) ∈ K [[ z ]] be the power series defining the multiplication-by- n map on the formal groupof E . The n - division polynomial F n (normalized relative to the uniformizer z ) isthe unique function F n ∈ K ( E ) with divisor [ n ] − ( O ) − n ( O ) such that z n F n [ n ]( z ) ! ( O ) = 1 Date : 25. 09. 2019.2010
Mathematics Subject Classification.
Key words and phrases.
Elliptic curves, rational points on elliptic curves, division polynomials,elliptic divisibility sequences, squares, cubes. as defined in [31, Definition 1], see also [18] for details.If E is an elliptic curve over C , then E has a complex uniformization Φ : C /L → E ( C ), with a lattice L ⊂ C . The classical n - division polynomial ψ n of an ellipticcurve C /L can be expressed in terms of the Weierstrass σ -function: ψ n ( z ) = ψ n ( z, L ) = σ ( nz, L ) σ ( z, L ) n for all n ≥ σ ( z, L ) is the Weierstrass σ -function associated to the lattice L . Moreover,the classical n -division polynomial ψ n for the elliptic curve E evaluated at point P = ( x, y ) is defined using the initial values ψ ( P ) = 0 ,ψ ( P ) = 1 ,ψ ( P ) = 2 y + a x + a , ψ ( P ) = 3 x + b x + 3 b x + 3 b x + b , ψ ( P ) = ψ ( P )(2 x + b x + 5 b x + 10 b x + 10 b x +( b b − b b ) x + ( b b − b )),where the point P correspond to z ∈ C /L and b i are the usual quantities [29,Chapter III.1], and by the formulas ψ n +1 ( P ) = ψ n +2 ( P ) ψ n ( P ) − ψ n − ( P ) ψ n +1 ( P ) , for n ≥ ψ n ( P ) ψ ( P ) = ψ n − ( P ) ψ n ( P ) ψ n +2 ( P ) − ψ n − ( P ) ψ n ( P ) ψ n +1 ( P ) , for n ≥ P = ( x, y ) be a point of E ( K ) (with char ( K ) = 2), and n ≥
1. Thecoordinates of the point [ n ] P can be expressed in terms of the point P , that is, forsome γ ∈ K (1.2) [ n ] P = (cid:18) φ n ( P ) ψ n ( P ) , ω n ( P ) ψ n ( P ) (cid:19) = (cid:18) γ G n ( P ) F n ( P ) , γ H n ( P ) F n ( P ) (cid:19) where φ n , ψ n , ω n ∈ K [ x , y ], gcd( φ n , ψ n ) = 1, and(1.3) F n ( P ) = γ − n ψ n ( P ) , G n ( P ) = γ − n φ n ( P ) , H n ( P ) = γ − n ω n ( P )are suitably normalized division polynomials of E . Note that F ( P ) = 0 and F ( P ) = 1. Furthermore the polynomials φ n ( P ) and ω n ( P ) are given by the recur-sion formulas φ ( P ) = 1, φ ( P ) = x ,(1.4) ω ( P ) = 1, ω ( P ) = y ,and φ n ( P ) = xψ n ( P ) − ψ n +1 ( P ) ψ n − ( P ),(1.5) ω n ( P ) = ( ψ n − ( P ) ψ n +2 ( P ) − ψ n − ( P ) ψ n +1 ( P ) − ψ ( P ) ψ n ( P )( a φ n ( P ) + a ψ n ( P )))(2 ψ ( P )) − .for all n ≥
2. The normalized division polynomials G n ( P ) and H n ( P ) hold thefollowing relations for some γ ∈ K ∗ G ( P ) = 1, G ( P ) = γ − x ,(1.6) H ( P ) = 1, H ( P ) = γ − y ,(1.7) EQUENCES ASSOCIATED TO ELLIPTIC CURVES 3 and G n ( P ) = xγ − F n ( P ) − F n +1 ( P ) F n − ( P ) , (1.8) H n ( P ) = ( F n − ( P ) F n +2 ( P ) − F n − ( P ) F n +1 ( P ) (1.9) − γ − F ( P ) F n ( P )( a G n ( P ) + γ − a F n ( P )))(2 F ( P )) − for all n ≥ F n ( P )) n ≥ of the division polynomials of an elliptic curve E at a point P is purely periodic modulo prime powers. Cheon and Hahn [5] estimate valu-ations of division polynomials F n ( P ). Complete formulas for explicit valuationsof division polynomials at primes of good or bad reduction are given in [33]. Sil-verman [31] used sophisticated methods to study the arithmetic properties of thesequence ( F n ( P )) n ≥ . Silverman [31] also studied p -adic properties of the sequence( F n ( P )) n ≥ , and proved the existence and algebraicity of the p -adic limit of certainsubsequences of the sequence ( F n ( P )) n ≥ . More precisely, Silverman proved if theelliptic curve E has good reduction, then there is a power q = p e such that forevery m ≥
1, the limitlim i →∞ F mq i ( P ) converges in Z p and is algebraic over Q .The sequences ( G n ( P )) n ≥ and ( H n ( P )) n ≥ that are generated by the numera-tors of the x - and y -coordinates of the multiples of a point P on an elliptic curve E defined over a field K are also interesting and have properties similar to thesequence ( F n ( P )) n ≥ . In [12], the author and Bizim study periodicity propertiesand p -adic properties of the sequences ( G n ( P )) n ≥ and ( H n ( P )) n ≥ . The authorsshow that the sequences ( G n ( P )) n ≥ and ( H n ( P )) n ≥ are periodic when K is afinite field. Moreover, we prove that certain subsequences of these sequences areconverge in Z p and the limits are algebraic over Q .In this paper we continue to study the properties of the sequences ( G n ( P )) n ≥ and ( H n ( P )) n ≥ of values of the suitably normalized division polynomials of E evaluated at a point P ∈ E ( K ). Let L be a lattice in C , and let E be an ellipticcurve defined over C given by equation E : y = x − g ( L ) x − g ( L ) . Ward [36, equations 13.6, 13.7], proved that the modular invariants g ( L ) and g ( L ) associated to the lattice L and the Weierstrass values ℘ ( z, L ) and ℘ ′ ( z, L )associated to the point z on the elliptic curve C /L are rational functions of F , F ,and F , with F F = 0, see also [32, Appendix]. Our first main theorem shows that g ( L ), g ( L ), ℘ ( z, L ) and ℘ ′ ( z, L ) are all defined in the same field as the terms of thesequences ( G n ( P )) n ≥ and ( H n ( P )) n ≥ similar to that of the sequence ( F n ( P )) n ≥ .The proof of the theorem uses properties of elliptic functions. Theorem 1.1.
Let L be a lattice in C , let E be an elliptic curve defined over C given by equation E : y = x − g ( L ) x − g ( L ) and let P ∈ E ( C ) . Let ( G n ( P )) n ≥ and ( H n ( P )) n ≥ be the sequences generatedby the numerators of the x - and y -coordinates of the multiples of P as in (1.2), BET¨UL GEZER respectively. Then the modular invariants g ( L ) and g ( L ) associated to the lattice L and the Weierstrass values ℘ ( z, L ) and ℘ ′ ( z, L ) associated to the point z on theelliptic curve C /L are in the field Q ( G , G , H , H ) . Section 2 provides background on elliptic divisibility sequences and elliptic curves.In Section 3, we give a representation of the sequences ( G n ( P )) n ≥ and ( H n ( P )) n ≥ by means of the elliptic functions and give the proof of Theorem 1.1. In Section 4and Appendix A, we consider the sequences ( G n ( P )) n ≥ and ( H n ( P )) n ≥ associ-ated to elliptic curves with a torsion point of order N . Ward [36, Theorem 23.1]studied the case N = 2 for elliptic divisibility sequences. It is a classical resultthat all elliptic curves with a torsion point of order N lie in a one parameter familywhere N ∈ { , ..., , } . In [9, Theorem 3.2], we use Tate normal form of anelliptic curve to give a complete description of elliptic divisibility sequences arisingfrom a point of order N . In Theorem 4.3, and Appendix A, we give a completedescription of sequences ( G n ( P )) n ≥ and ( H n ( P )) n ≥ arising from points of order N . We will also use Tate normal form of an elliptic curve to give the sequences( G n ( P )) n ≥ and ( H n ( P )) n ≥ arising from points of order N . As an application, inTheorem 5.1 and Appendix B, we determine square and cube terms in the sequences( G n ( P )) n ≥ and ( H n ( P )) n ≥ associated to a Tate normal form. Acknowledgements . This work was supported by the research fund of BursaUluda˘g University project no: KUAP(F)-2017/3.2.
Elliptic Divisibility Sequences An elliptic divisibility sequence (EDS) is a sequence ( h n ) n ≥ of integers satisfyinga recurrence relation of the form h m + n h m − n = h m +1 h m − h n − h n +1 h n − h m and the divisibility property h n | h m whenever n | m for all m ≥ n ≥
1. An elliptic divisibility sequence is called proper if h = 0, h = 1,and h h = 0. The discriminant of an EDS ( h n ) n ≥ is the quantity∆( h n ) = h h − h h + 3 h h − h h h + 3 h h + 16 h h + 8 h h h + h ,(this is the formula in [31] or [32], see also [36]). A proper EDS is called nonsingular if ∆( h n ) = 0. The arithmetic properties of EDSs were first studied by Morgan Wardin 1948 [36, 37]. For more details on EDSs, see also [6, 28, 35].Ward defined the division polynomials over the field C and using the complexanalytic theory of elliptic functions showed that nonsingular elliptic divisibilitysequences can be expressed in terms of elliptic functions. More precisely, Ward [36,Theorem 12.1] proved that if ( h n ) n ≥ is a nonsingular elliptic divisibility sequence,then there exist a lattice L ⊂ C and a complex number z ∈ C such that(2.1) h n = ψ n ( z, L ) = σ ( nz , L ) σ ( z , L ) n for all n ≥ ψ n ( z , L ) and σ ( z , L ) are the n -division polynomial and the Weierstrass σ -function associated to the lattice L , respectively. Further, Ward showed themodular invariants g ( L ) and g ( L ) associated to the lattice L and the Weierstrassvalues ℘ ( z ) and ℘ ′ ( z ) associated to the point z on the elliptic curve C /L can be EQUENCES ASSOCIATED TO ELLIPTIC CURVES 5 given by the terms h , h and h of the sequence ( h n ), see [36, equations 13.6,13.7, 13.5 and 13.1]. Silverman [31, Proposition 18] reformulated Ward’s result andshowed that if ( h n ) n ≥ is a nonsingular EDS associated to an elliptic curve E givenby a minimal Weierstrass equation over Q and a point P ∈ E ( Q ), then there is aconstant γ ∈ Q ∗ such that(2.2) h n = γ n − F n ( P ) for all n ≥ F n is the normalized n -division polynomial on E .3. The Representation of the Sequences ( G n ( P )) n ≥ and ( H n ( P )) n ≥ byElliptic Functions Let E be an elliptic curve defined over a field K with Weierstrass equation(3.1) E : y = x + ax + b .It is clear that the coefficients of the elliptic curve E can be defined in terms ofthe sequence ( F n ( P )) n ≥ of values of the division polynomials of E at a point P by using the relation (2.2) and Ward’s formulas for the modular invariants g ( L )and g ( L ) [36, equations 13.6, 13.7]; see also [32, Appendix]. In this section wegive a representation of the sequences ( G n ( P )) n ≥ and ( H n ( P )) n ≥ of values ofthe suitably normalized division polynomials of E evaluated at a point P ∈ E ( K )by means of the elliptic functions and prove the coefficients of the elliptic curve E can be defined in terms of these sequences. In this section we will also assume ψ ( P ) ψ ( P ) = 0 so that F ( P ) F ( P ) = 0.We first state some results from elliptic function theory that will be needed.Let L be a lattice in C . Recall from elliptic function theory that the Weierstrass ℘ -function associated to the lattice L and its derivative ℘ ′ satisfy(3.2) ℘ ′ ( z ) = 4 ℘ ( z ) − g ( L ) ℘ ( z ) − g ( L )where g ( L ) and g ( L ) are modular invariants associated to the lattice L . If wetake the derivative of the both sides of (3.2) we have the following relation(3.3) ℘ ′′ ( z ) = 6 ℘ ( z ) − g ( L ).Then we obtain(3.4) g ( L ) = 12 ℘ ( z ) − ℘ ′′ ( z ).by (3.3) and so(3.5) g ( L ) = 2 ℘ ( z )[ ℘ ′′ ( z ) − ℘ ( z ) ] − ℘ ′ ( z ) by (3.2) and (3.3). Moreover recall that(3.6) ψ ( z ) = − ℘ ′ ( z ),(3.7) ψ ( z ) = 3 ℘ ( z ) − g ( L ) ℘ ( z ) − g ( L ) ℘ ( z ) − g ( L ) ,and(3.8) ℘ (2 z ) − ℘ ( z ) = 14 (cid:18) ℘ ′′ ( z ) ℘ ′ ( z ) (cid:19) − ℘ ( z ),(3.9) ℘ (3 z ) − ℘ ( z ) = ℘ ′ ( z ) [ ℘ ′ ( z ) − ψ ( z ) ℘ ′′ ( z )] ψ ( z ) . BET¨UL GEZER
Furthermore, one can derive a formula for ℘ ( nz ) in terms of ℘ ( z ), ψ n ( z ) and ψ n ± ( z ), more explicitly the following relation holds(3.10) ℘ ( nz ) = ℘ ( z ) − ψ n +1 ( z ) ψ n − ( z ) ψ n ( z ) for all n ≥
2. Thus substituting n = 2 and n = 3 into (3.10) we have(3.11) ℘ (2 z ) − ℘ ( z ) = − ψ ( z ) ψ ( z ) since ψ ( z ) = 1, and(3.12) ℘ (3 z ) − ℘ ( z ) = − ψ ( z ) ψ ( z ) ψ ( z ) .Now by (3.9), (3.6) and (3.12) we have − ψ ( z ) ψ ( z ) ψ ( z ) = ψ ( z ) [ ψ ( z ) − ψ ( z ) ℘ ′′ ( z )] ψ ( z ) and so(3.13) ℘ ′′ ( z ) = ψ ( z ) + ψ ( z ) ψ ( z ) ψ ( z ) .Let E be an elliptic curve over C . Then the points ( ℘ ( z ) , ℘ ′ ( z )) lie on the ellipticcurve(3.14) y = 4 x − g ( L ) x − g ( L )by (3.2). Now let ( F n ( P )) n ≥ , ( G n ( P )) n ≥ and ( H n ( P )) n ≥ be the sequences ofvalues of the normalized division polynomials of E at a point P . Then by secondpart of (1.6) we have(3.15) ℘ ( z ) = γ G ( P ).On the other hand by (1.8),(3.16) G ( P ) = xγ − F ( P ) − F ( P )since F ( P ) = 1. By first part of (1.3), and (3.11) we obtain(3.17) ℘ (2 z ) = ℘ ( z ) F ( P ) − γ F ( P ) F ( P ) .Hence by second part of (1.6), (3.15) and (3.16) we have ℘ (2 z ) = γ G ( P ) F ( P ) .Thus one can easily derive inductively that(3.18) ℘ ( nz ) = γ G n ( P ) F n ( P ) for all n ≥ G n and H n for G n ( P ) and H n ( P ), respectively, unless otherwisespecified. EQUENCES ASSOCIATED TO ELLIPTIC CURVES 7
Proof of Theorem 1.1.
By the first part of (1.3) we have(3.19) ψ ( z ) = γ F and(3.20) ψ ( z ) = γ F .Thus by (3.19) and (3.6) we obtain(3.21) ℘ ′ ( z ) = − γ F .On the other hand (3.18) implies that(3.22) ℘ (3 z ) = γ G /F .Now (3.13) and the first part of (1.3) imply that(3.23) ℘ ′′ ( z ) = γ ( F + F ) F F .On the other hand by (3.19) we have F = 2 γ − y since ψ = 2 y , for the elliptic curve E : y = x − g ( L ) x − g ( L ). Thus by thesecond part of (1.7) we derive that(3.24) F = 2 H .Now by putting n = 2 into (1.9) and then using (3.24) we obtain(3.25) F = 4 H H since F = 0 and F = 1. Thus(3.26) ℘ ′ ( z ) = − γ H ,by (3.21) and (3.24). Now by setting n = 2 in (1.8) and then using (3.21) and thesecond part of (1.6) we have(3.27) F = γ − ℘ ′ ( z ) G − G since F = 1. Thus(3.28) F = 4 G H − G by (3.26). Therefore by (3.23), (3.24), (3.25) and (3.28) we have(3.29) ℘ ′′ ( z ) = 2 γ (8 H + H )4 G H − G .On combining (3.4) with (3.15) and (3.29) we obtain the following formula for g ( L ),(3.30) g ( L ) = 4 γ (12 G H − G G − H − H )4 G H − G .Similarly combining (3.5) with (3.15), (3.26) and (3.29) we have(3.31) g ( L ) = 4 γ (4 G H + G H − G H + 2 G G + H G )4 G H − G .Now if E is an elliptic curve over Q given by a Weierstrass equation E : y = x + ax + b , BET¨UL GEZER then (3.14) imply that a = − g ( L )4 and b = − g ( L )4where g ( L ) and g ( L ) are the rational expressions in G , G , H and H byrelations (3.30) and (3.31) respectively. Finally rational expressions for ℘ ( z, L ) and ℘ ′ ( z, L ) are given by equations (3.15) and (3.26), which completes the proof of thetheorem. (cid:3) The Sequences ( G n ) n ≥ and ( H n ) n ≥ Associated to Tate NormalForms
The study of the group E ( Q ) has been playing important roles in number theory.The modern number theory originated in 1922 when L. J. Mordell proved that thegroup of rational points E ( Q ) is a finitely generated abelian group. This result wasgeneralized in 1928 to abelian varieties over number fields by A. Weil. Moreover,the characterization of torsion subgroups of E ( Q ) is always interesting. A uniformbound was studied for the order of the torsion subgroup E tors ( Q ) of E ( Q ) byShimura, Ogg, and others. The following result conjectured by Ogg, was proved byB. Mazur. Theorem 4.1 ([16]) . Let E be an elliptic curve defined over Q . Then the torsionsubgroup E tors ( Q ) is either isomorphic to Z /N Z for N = 1 , , ..., , or to Z / Z × Z / N Z for N = 1 , , , . Further, each of these groups does occur as an E tors ( Q ) . It is a classical result that all elliptic curves with a torsion point of order N liein a one parameter family where N ∈ { , ..., , } . The Tate normal form of anelliptic curve E with point P = (0 ,
0) is given by E N : y + (1 − c ) xy − by = x − bx where the point P has given order N .If an elliptic curve in normal form has a point of order N >
3, then admissiblechange of variables transforms the curve to the Tate normal form, in this case thepoint P = (0 ,
0) is a torsion point of maximal order. Kubert [14] gives a list ofparameterizable torsion structures, which includes one parameter family of ellipticcurves E defined over Q with a torsion point of order N where N = 4 , ...,
10, 12.Some algorithms are given by using the existence of such a family, see [7] for moredetails. In order to describe when an elliptic curve defined over Q has a point ofgiven order N , we need the following result on parametrization of torsion structures.Most cases of the following parameterizations are proved by Husem¨oller [13]. Theorem 4.2 ([7]) . Every elliptic curve with point P = (0 , of order N = 4 , ... , , can be written in the following Tate normal form E N : y + (1 − c ) xy − by = x − bx , with the following relations:1. If N = 4 , then b = α and c = 0 , α = 0 .2. If N = 5 , then b = α and c = α , α = 0 .3. If N = 6 , then b = α + α and c = α , α = − , .4. If N = 7 , then b = α − α and c = α − α , α = 0 , .5. If N = 8 , then b = (2 α − α − and c = b/α , α = 0 , , .6. If N = 9 , then c = α ( α − and b = c ( α ( α −
1) + 1) , α = 0 , . EQUENCES ASSOCIATED TO ELLIPTIC CURVES 9
7. If N = 10 , then c = (2 α − α + α ) / ( α − ( α − ) and b = cα / ( α − ( α − ) , α = 0 , , .8. If N = 12 , then c = (3 α − α + 1)( α − α ) / ( α − and b = c ( − α + 2 α − / ( α − , α = 0 , , . Theorem 4.2 says that every elliptic curve with a point of order N is birationallyequivalent to one of the Tate normal forms given above. We will assume that theparameter α ∈ Z and the coefficients of E N are chosen to lie in Z . Hence for N = 8,10, 12, we transform E N into a birationally equivalent curve E ′ N having an equationwith integral coefficients. The equations of the birationally equivalent curves for N = 8, 10, 12 are given, respectively, as follows: E ′ : y +( α − β ) xy − α βy = x − α βx , E ′ : y +( ζ − αβζ ) xy − α βζ y = x − α βζ x , E ′ : y +( α − α − − λ ) xy − ( α − λθy = x − ( α − λθx ,where α = 0, 1,(4.1) β = (2 α − α − ζ = − α + 3 α − λ = (3 α − α + 1)( α − α ), θ = 2 α − α − E , E , E for E ′ , E ′ , E ′ ,respectively.In [9, Theorem 3.2], we give the general terms of the elliptic divisibility sequences( h n ) n ≥ associated to a Tate normal form E N of an elliptic curve for some integerparameter α . For example, the general term of ( h n ) n ≥ for N = 8 is(4.3) h n = εα { (15 n − p ) / } ( α − { (7 n − q ) / } (2 α − { (3 n − r ) / } where ε = (cid:26) +1, if n ≡ , , , , , ,
14 (16) −
1, if n ≡ , , , , , ,
15 (16),and p =
15, if n ≡ , n ≡ , n ≡ , n ≡ , q =
7, if n ≡ , n ≡ , n ≡ , n ≡ , r =
3, if n ≡ , , , , if n ≡ , n ≡ . In Section 2, we recall that if ( h n ) n ≥ is a nonsingular EDS associated to anelliptic curve E given by a minimal Weierstrass equation over Q and a point P ∈ E ( Q ), then there is a constant γ ∈ Q ∗ such that h n = γ n − F n ( P ) for all n ≥ F n ) n ≥ associatedto a Tate normal form E N , by using the relation above.In this section we consider ( G n ) n ≥ and ( H n ) n ≥ sequences associated to a Tatenormal form E N with torsion point P = (0 ,
0) and give the general terms of thesesequences. We take γ = 1 in (1.3) so that G n = φ n , H n = ω n , and(4.4) F n = h n for all n ≥ In the following theorem we determine general terms of the sequences ( G n ) n ≥ and ( H n ) n ≥ associated to an elliptic curve in Tate normal form with a torsionpoint P = (0 ,
0) of order 8. For the convenience of the reader, we have given theother cases in Appendix A. The proof uses the general terms of sequences in [9,Theorem 3.2].
Theorem 4.3.
Let E be a Tate normal form of an elliptic curve with a torsionpoint P = (0 , of order . Let ( G n ) n ≥ and ( H n ) n ≥ be the sequences generatedby the numerators of the x - and y -coordinates of the multiples of P as in (1.2).Then the general terms of the sequences ( G n ) n ≥ and ( H n ) n ≥ can be given by thefollowing formulas: (4.5) G n = ( , if n ≡ , α { (15 n + a ) / } ( α − { (7 n − b ) / } (2 α − { (3 n + c ) / } , otherwise,and (4.6) H n = ( , if n ≡ , εα { (45 n + a ) / } ( α − { (21 n − b ) / } (2 α − { (9 n − c ) / } , otherwise,where α = 0 , , a = , if n ≡ , if n ≡ , , if n ≡ , , if n ≡ , b = , if n ≡ , if n ≡ , , if n ≡ , , if n ≡ , c = (cid:26) , if n ≡ , , , , if n ≡ , .and ε = (cid:26) +1 , if n ≡ , , , ,
13 (16) − , if n ≡ , , , , , ,
15 (16) , a = , if n ≡ − , if n ≡ , if n ≡ , if n ≡ − , if n ≡ , if n ≡ , b = , if n ≡ , if n ≡ , if n ≡ , , if n ≡ , if n ≡ , c = , if n ≡ , , if n ≡ , if n ≡ , − , if n ≡ .Proof. We give the proof only for the sequence ( G n ) n ≥ as the proof for ( H n ) n ≥ is similar.Let n ≡ G k +1 = − F k +2 F k for all k ≥ x = 0. We note that F n = 0 if and only if the order N of the point P divides n . It follows that F k = 0 for all k ≥
0, hence G k +1 = 0.Now let n ≡ G k +2 = − F k +3 F k +1 for all k ≥ F k +1 = α k +15 k ( α − k +7 k (2 α − k +6 k for all k ≥ EQUENCES ASSOCIATED TO ELLIPTIC CURVES 11 and F k +3 = − α k +45 k +8 ( α − k +21 k +3 (2 α − k +18 k +3 for all k ≥ G k +2 = α k +60 k +8 ( α − k +28 k +3 (2 α − k +24 k +3 .for all k ≥
0. On the other hand by the general term formula in (4.5) we have G k +2 = α k +60 k +8 ( α − k +28 k +3 (2 α − k +24 k +3 ,which completes the proof for n ≡ (cid:3) Remark 4.1.
There is no Tate normal form of an elliptic curve with the torsionpoint of order two or three, but in [14] , Kubert gives a list of elliptic curves withtorsion point of order two or three are (4.8) E : y = x + a x + a x , a = 0 ,and (4.9) E : y + a xy + a y = x , a = 0 ,respectively. The following theorem gives the general term of the sequence ( G n ) n ≥ associatedto E and E , respectively and the general term of the sequence ( H n ) n ≥ associatedto elliptic curve E . We note the sequence ( H n ) n ≥ associated to elliptic curve E is not defined since F = 0; see relation (1.9). The proof of the theorem is similarto the proof of Theorem 4.3. Theorem 4.4.
Let E N be an elliptic curve with the torsion point P = (0 , of order N as in (4.8) and (4.9). Let ( G n ) n ≥ and ( H n ) n ≥ be the sequences generated by thenumerators of the x - and y -coordinates of the multiples of P as in (1.2). Then thegeneral term of the sequences ( G n ) n ≥ and ( H n ) n ≥ can be given by the followingformulas:1. If N = 2 , then G n = ( , if n is odd a { n / } , if n is even.2. If N = 3 , then G n = ( , if n ≡ , a { n / } , if n ≡ .and H n = (cid:26) , if n ≡ εa n , if n ≡ , where ε = (cid:26) +1 , if n ≡ , − , if n ≡ , . Squares and Cubes in ( G n ) n ≥ and ( H n ) n ≥ Sequences
The problem of determining square and cube terms in linear sequences has beenconsidered by various authors, see [21], [24], [25], [26], and see also, [3], [4]. Similarproblem has also been considered for non-linear sequences, see [10], [11], [9], seealso [23], [17]. In this section we determine square and cube terms in the sequences( G n ) n ≥ and ( H n ) n ≥ associated to a Tate normal form E N of an elliptic curve witha torsion point P = (0 ,
0) of order N . Throughout this paper the symbol (cid:3) meansa square of a non-zero integer, i.e., (cid:3) = ± β where β is a non-zero integer, and C means a cube of a non-zero integer. Determining square and cube terms in thesesequences leads to some equations and these equations are similar to equations in[9, Table 3]. Therefore we use similar techniques in [9] for determining square andcube terms in these sequences. We observe that the irreducible factors appearing inthe left hand side (if they are at least two) of these equations are pairwise relativelyprime (for example, one can easily show that the irreducible factors in the α ( α − α − α − α + 1)(3 α − α + 1) = (cid:3) , are pairwise relatively prime, see [9], p. 498). It follows that, if the right hand sideof the equation is (cid:3) (or C ), then every irreducible factor is (cid:3) (or C ). It turns outthat it is not necessary to consider all irreducible factors in the left hand side. Forexample, the equation α (2 α − α − α + 1) = C implies that all three α , 2 α −
1, and 2 α − α + 1 are C , we only use the fact thatthe third one is C .We use the tables in [22] when our equations turned into Mordell’s equation. Insome cases we will apply Elliptic Logarithm Method to find all integral solutions ofour equations (this method has been developed in [34] and, independently, in [8]and now is implemented in MAGMA [15]; see also [2]) . Theorem 5.1 answers the following three questions:(1) Which terms of the sequence ( G n ) n ≥ (or ( H n ) n ≥ ) can be (cid:3) (or C ) inde-pendent of α ?(2) Which terms of the sequence ( G n ) n ≥ (or ( H n ) n ≥ ) can be (cid:3) (or C ) withadmissible choice of α ?(3) Which terms of the sequence ( G n ) n ≥ (or ( H n ) n ≥ ) can not be (cid:3) (or C )independent of α ?Here again we only consider the case N = 8, for the convenience of the reader,we have given the other cases in Appendix B. Theorem 5.1.
Let E be a Tate normal form of an elliptic curve with a torsionpoint P = (0 , of order . Let ( G n ) n ≥ and ( H n ) n ≥ be the sequences generatedby the numerators of the x - and y -coordinates of the multiples of P as in (1.2). Let G n and H n = 0 .1. ( i ) • If n ≡ , then G n = (cid:3) for all α = 0 , , • if n ≡ , , then G n = (cid:3) iff ( α − α −
1) = (cid:3) , • otherwise G n = (cid:3) for all α = 0 , . ( ii ) • If n ≡ , then G n = C for all α = 0 , , • if n ≡ , , ,
22 (24) , then G n = C iff α = C , • if n ≡ ,
16 (24) , then G n = C iff α − C , EQUENCES ASSOCIATED TO ELLIPTIC CURVES 13 • otherwise G n = C for all α = 0 , .2. ( i ) • If n ≡ , , ,
12 (16) , then H n = (cid:3) for all α = 0 , , • if n ≡ , then H n = (cid:3) iff α − (cid:3) , • if n ≡ , , then H n = (cid:3) iff α − (cid:3) , • if n ≡
11 (16) , then H n = (cid:3) iff α = (cid:3) , • otherwise H n = (cid:3) for all α = 0 , . ( ii ) • If n ≡ , then H n = C for all α = 0 , , • otherwise H n = C for all α = 0 , .Proof. We give the proof only for the sequence ( G n ) n ≥ as the proof for ( H n ) n ≥ is similar.1. i . We note that G n = 0 for n ≡
1, 7 (mod 8), by (4.5). It can easily be seenthat G n = (cid:3) for every α = 0, 1 for n ≡ n ≡
2, 6 (8), then G n = (cid:3) iff( α − α −
1) = (cid:3) by (4.5). This equation leads to(5.1) (4 α − − β = 1or(5.2) (4 α − + 8 β = 1,where β is a non zero-integer. The last equation is trivial equation and the solutionsof this equation do not provide any acceptable α . The first equation leads to Pellequation τ − β = 1where τ = 4 α −
3. The solutions of this equation are (3 , , ... . Note thatonly the solutions of the form τ + 3 ≡ α , and their numberis infinite.If n ≡
3, 5 (8), then G n = (cid:3) iff α ( α − α −
1) = (cid:3) ,and if n ≡ G n = (cid:3) iff α ( α −
1) = (cid:3) by (4.5). These last two equations lead to trivial equations(5.3) (2 α − ± β = 1where β is a non-zero integer. The solutions of these trivial equations do not provideany acceptable α , which completes the proof of ( i ). ii . If 1, 7, 9, 15, 17, 23 (24), then G n = 0, if n ≡ G n = C for every α = 0, 1, if n ≡
2, 10, 14, 22 (24), then G n = C iff α = C , and if n ≡
8, 16 (24),then G n = C iff α − C , by (4.5).If n ≡
3, 21 (24), then G n = C iff α ( α − α −
1) = C ,and if n ≡
4, 20 (24), then G n = C iff α ( α −
1) = C , if n ≡
6, 18 (24), then G n = C iff α ( α −
1) = C ,and if n ≡
12 (24), then G n = C iff α ( α − = C by (4.5). These equations lead to α ( α −
1) = C .This equation leads to trivial equation(5.4) β − β = 1,where α = β , α − β , and β , β are non-zero integers. The solutions of thisequation do not provide any acceptable α .If n ≡
5, 11, 19, 13 (24), then G n = C iff α (2 α −
1) = C by (4.5), or equivalently α (2 α −
1) = C .The last equation leads to classical equation (5.5) 2 β + ( − β ) = 1,where α = β , 2 α − β , and β , β are non-zero integers. The solution of thisequation does not provide any acceptable α , which completes the proof of ( ii ). (cid:3) In the following theorem we determine square and cube terms in the sequence( G n ) n ≥ associated to elliptic curves E and E , respectively, and square and cubeterms in the sequence ( H n ) n ≥ associated to elliptic curve E . The proof is similarto the proof of Theorem 5.1. Theorem 5.2. E N be an elliptic curve with the torsion point P = (0 , of order N as in (4.8) and (4.9). Let ( G n ) n ≥ and ( H n ) n ≥ be the sequences generated bythe numerators of the x - and y -coordinates of the multiples of P as in (1.2), andlet G n = 0 .1. Let N = 2 . ( i ) • G n = (cid:3) for every non-zero a . ( ii ) • If n ≡ , then G n = C for every non-zero a , • otherwise G n = C iff a = C .2. Let N = 3 . ( i ) • G n = (cid:3) for every non-zero a . ( ii ) • G n = C for every non-zero a . ( iii ) • If n ≡ , , then H n = (cid:3) for every non-zero a , • otherwise H n = (cid:3) iff a = (cid:3) , ( iv ) • If n ≡ , then H n = C for every non-zero a , • otherwise H n = C iff a = C . The equation x + 2 y = 1 has the integer solution ( x, y ) = ( − , , hence, by Theorem , Chapter 24 of [19] can not have further solutions with xy = 0 . EQUENCES ASSOCIATED TO ELLIPTIC CURVES 15
Appendix
A.In the following theorems we determine general terms of the sequences ( G n ) n ≥ and ( H n ) n ≥ associated to an elliptic curve in Tate normal form with a torsionpoint P = (0 ,
0) of order N . The proofs are similar to the proof of Theorem 4.3. Theorem A.1.
Let E N be a Tate normal form of an elliptic curve with a torsionpoint P = (0 , of order N . Let ( G n ) n ≥ be the sequence generated by the nu-merators of the x -coordinates of the multiples of P as in (1.2). Let ζ, λ, θ be as in(4.1) and (4.2). Then the general term of the sequence ( G n ) n ≥ can be given bythe following formulas:1. If N = 4 , then (A.1) G n = (cid:26) , if n is odd α { n / } , if n is even,where α = 0 .2. If N = 5 , then (A.2) G n = (cid:26) , if n ≡ , α { (4 n − a ) / } , otherwise,where α = 0 , and a = (cid:26) , if n ≡ , if n ≡ , .3. If N = 6 , then (A.3) G n = ( , if n ≡ , α { (5 n − a ) / } ( α + 1) { (2 n + b ) / } , otherwise,where α = − , , and a = , if n ≡ , if n ≡ , , if n ≡ , b = (cid:26) , if n ≡ , , if n ≡ , .4. If N = 7 , then (A.4) G n = (cid:26) , if n ≡ , α { (10 n + a ) / } ( α − { (6 n − b ) / } , otherwise,where α = 0 , , and a = , if n ≡ , if n ≡ , , if n ≡ , , b = , if n ≡ , if n ≡ , , if n ≡ , .5. If N = 9 , then (A.5) G n = ( , if n ≡ , α { (14 n − a ) / } ( α − { (8 n − b ) / } η { (2 n + c ) / } , otherwise where α = 0 , , η = α − α + 1 , and a = , if n ≡ , , , if n ≡ , − , if n ≡ , , b = , if n ≡ , if n ≡ , , if n ≡ , , if n ≡ , , c = (cid:26) , if n ≡ , , , if n ≡ , , , .6. If N = 10 , then (A.6) G n = , if n ≡ , α { (21 n + a ) / } ( α − { (9 n − b ) / } × (2 α − { (4 n − c ) / } ζ { (5 n + d ) / } , otherwise,where α = 0 , , a = , if n ≡ , if n ≡ , , if n ≡ , , if n ≡ , , if n ≡ , b = , if n ≡ , , if n ≡ , , if n ≡ , , if n ≡ , , if n ≡ ,and c = , if n ≡ , , if n ≡ , , , − , if n ≡ , , d = (cid:26) , if n ≡ , , , , , if n ≡ , , .7. If N = 12 , then (A.7) G n = , if n ≡ ,
11 (12) εα { ( n − a ) / } ( α − { (59 n + b ) / } × (2 α − { ( n − c ) / } λ { (3 n + d ) / } θ { (2 n + e ) / } , otherwise,where α = 0 , , ε = (cid:26) +1 , if n ≡ , , , ,
10 (12) − , if n ≡ , , , , , a = , if n ≡ , if n ≡ ,
10 (12)9 , if n ≡ , , if n ≡ , , if n ≡ , , if n ≡ , b = , if n ≡ , if n ≡ ,
10 (12)9 , if n ≡ , , if n ≡ , , if n ≡ , , if n ≡ , c = , if n ≡ , , , if n ≡ , , ,
10 (12)9 , if n ≡ , , if n ≡ , , d = (cid:26) , if n ≡ , , , , ,
10 (12)1 , if n ≡ , , , ,and e = (cid:26) , if n ≡ , , , , if n ≡ , , , , ,
10 (12) . EQUENCES ASSOCIATED TO ELLIPTIC CURVES 17
Theorem A.2.
Let E N be a Tate normal form of an elliptic curve with a torsionpoint P = (0 , of order N . Let ( H n ) n ≥ be the sequence generated by the nu-merators of the y -coordinates of the multiples of P as in (1.2). Let ζ, λ, θ be as in(4.1) and (4.2). Then the general term of the sequence ( H n ) n ≥ can be given bythe following formulas:1. If N = 4 , then (A.8) H n = (cid:26) , if n ≡ , εα { (9 n − a ) / } , otherwise,where α = 0 , and ε = (cid:26) +1 , if n ≡ − , if n ≡ , , , a = (cid:26) , if n ≡ , if n ≡ .2. If N = 5 , then (A.9) H n = (cid:26) , if n ≡ , , εα { (6 n − a ) / } , otherwise,where α = 0 , and ε = (cid:26) +1 , if n ≡ , , − , if n ≡ , , , a = , if n ≡ − , if n ≡ , if n ≡ .3. If N = 6 , then (A.10) H n = (cid:26) , if n ≡ , , εα { (5 n − a ) / } ( α + 1) n , otherwise,where α = − , , and ε = (cid:26) +1 , if n ≡ , , , − , if n ≡ , , ,
11 (12) , a = (cid:26) , if n ≡ , , if n ≡ , .4. If N = 7 , then (A.11) H n = ( , if n ≡ , εα { (15 n − a ) / } ( α − { (9 n − b ) / } , otherwise,where α = 0 , , ε = (cid:26) +1 , if n ≡ , , ,
11 (14) − , if n ≡ , , , , ,
13 (14) ,and a = , if n ≡ − , if n ≡ , if n ≡ − , if n ≡ , if n ≡ , b = , if n ≡ , if n ≡ , if n ≡ , , if n ≡ .5. If N = 9 , then (A.12) H n = ( , if n ≡ , εα { (7 n + a ) / } ( α − { (4 n − b ) / } η ( n + c ) , otherwise where α = 0 , , η = α − α + 1 , ε = (cid:26) +1 , if n ≡ , , , , , ,
15 (18) − , if n ≡ , , , , , ,
17 (18) ,and a = , if n ≡ , , if n ≡ , − , if n ≡ , , if n ≡ , b = , if n ≡ , if n ≡ , , if n ≡ , , if n ≡ , , c = (cid:26) , if n ≡ , otherwise.
6. If N = 10 , then (A.13) H n = , if n ≡ , εα { (63 n + a ) / } ( α − { (27 n − b ) / } × (2 α − { (6 n + c ) / } ζ { (15 n + d ) / } , otherwise,where α = 0 , , ε = (cid:26) +1 , if n ≡ , , , , , , ,
17 (20) − , if n ≡ , , , , , , ,
19 (20) ,and a = , if n ≡ , if n ≡ − , if n ≡ , if n ≡ , if n ≡ − , if n ≡ , if n ≡ − , if n ≡ , b = , if n ≡ , , if n ≡ , if n ≡ , , if n ≡ , , if n ≡ , if n ≡ , c = , if n ≡ , , if n ≡ , , − , if n ≡ , , if n ≡ , d = (cid:26) , if n ≡ , , , , otherwise.7. If N = 12 , then (A.14) H n = , if n ≡ ,
10 (12) εα { ( n − a ) / } ( α − { (59 n + b ) / } × (2 α − { ( n − c ) / } λ { (9 n − d ) / } θ ( n + e ) , otherwise,where α = 0 , , ε = (cid:26) +1 , if n ≡ , , , , , , , , ,
21 (24) − , if n ≡ , , , , , , , , ,
23 (24) , EQUENCES ASSOCIATED TO ELLIPTIC CURVES 19 and a = , if n ≡ , if n ≡ , if n ≡ , if n ≡ , , if n ≡ , if n ≡ , if n ≡ , , if n ≡
11 (12) , b = , if n ≡ , − , if n ≡ , if n ≡ , if n ≡ ,
11 (12)12 , if n ≡ − , if n ≡ , , if n ≡ , c = , if n ≡ , , , if n ≡ , , if n ≡ ,
11 (12)9 , if n ≡ , − , if n ≡ , d = , if n ≡ , , − , if n ≡ , , if n ≡ , ,
11 (12) − , if n ≡ , ,and e = (cid:26) , if n ≡ , , otherwise. Appendix
B.In the following theorems we determine square and cube terms in the sequences( G n ) n ≥ and ( H n ) n ≥ associated to an elliptic curve in Tate normal form with atorsion point P = (0 ,
0) of order N . The proofs are similar to the proof of Theorem5.1. Theorem B.1.
Let E N be a Tate normal form of an elliptic curve with a torsionpoint P = (0 , of order N . Let ( G n ) n ≥ be the sequence generated by the numer-ators of the x -coordinates of the multiples of P as in (1.2), and let G n = 0 .1. Let N = 4 . ( i ) • If n ≡ , then G n = (cid:3) for all non-zero α , • otherwise G n = (cid:3) iff α = (cid:3) . ( ii ) • G n = C for all non-zero α .2. Let N = 5 . ( i ) • If n ≡ , then G n = (cid:3) for all non-zero α , • otherwise G n = (cid:3) iff α = (cid:3) . ( ii ) • If n ≡ , , , ,
13 (15) , then G n = C for all non-zero α , • otherwise G n = C iff α = C .3. Let N = 6 . ( i ) • If n ≡ , then G n = (cid:3) for all α = − , , • if n ≡ , then G n = (cid:3) iff α = (cid:3) , • otherwise G n = (cid:3) for all α = − , . ( ii ) • If n ≡ , , , ,
16 (18) , then G n = C for all α = − , , • if n ≡ , ,
15 (18) , then G n = C iff α = C , • otherwise G n = C for all α = − , .4. Let N = 7 . ( i ) • If n ≡ , then G n = (cid:3) for all α = 0 , , • if n ≡ , , then G n = (cid:3) iff α − (cid:3) , • otherwise G n = (cid:3) for all α = 0 , . ( ii ) • If n ≡ , , , ,
19 (21) , then G n = C for all α = 0 , , • if n ≡ , , ,
14 (21) , then G n = C iff α = C , • otherwise G n = C for all α = 0 , .5. Let N = 9 . ( i ) • If n ≡ , then G n = (cid:3) for all α = 0 , , • if n ≡ , , then G n = (cid:3) iff α − (cid:3) , • otherwise G n = (cid:3) for all α = 0 , . ( ii ) • If n ≡ , , , ,
25 (27) , then G n = C for all α = 0 , , • if n ≡ ,
22 (27) , then G n = C iff α − α + 1 = C , • otherwise G n = C for all α = 0 , .6. Let N = 10 . ( i ) • If n ≡ , then G n = (cid:3) for all α = 0 , , • if n ≡ , , then G n = (cid:3) iff ( α − α −
1) = (cid:3) , • otherwise G n = (cid:3) for all α = 0 , . ( ii ) • If n ≡ , then G n = C for all α = 0 , , • if n ≡ , , ,
28 (30) , then G n = C iff α − α + 1 = C , • if n ≡ ,
18 (30) , then G n = C iff α − C , • otherwise G n = C for all α = 0 , .7. Let N = 12 . ( i ) • If n ≡ , then G n = (cid:3) for all α = 0 , , • otherwise G n = (cid:3) for all α = 0 , . ( ii ) • If n ≡ , ,
24 (36) , then G n = C for all α = 0 , , • if n ≡ ,
20 (36) , then G n = C iff α = C , • if n ≡ ,
34 (36) , then G n = C iff α − C , • otherwise G n = C for all α = 0 , . Theorem B.2.
Let E N be a Tate normal form of an elliptic curve with a torsionpoint P = (0 , of order N . Let ( H n ) n ≥ be the sequence generated by the numer-ators of the y -coordinates of the multiples of P as in (1.2), and let H n = 0 .1. Let N = 4 . ( i ) • If n ≡ , , , then H n = (cid:3) for all non-zero α , • otherwise H n = (cid:3) iff α = (cid:3) . ( ii ) • If n ≡ , then H n = C for all non-zero α , • otherwise H n = C iff α = C .2. Let N = 5 . ( i ) • If n ≡ , then H n = (cid:3) for all non-zero α , • otherwise H n = (cid:3) iff α = (cid:3) . ( ii ) • If n ≡ , then H n = C for all non-zero α , • otherwise H n = C iff α = C .3. Let N = 6 . ( i ) • If n ≡ , , then H n = (cid:3) for all α = − , , • if n ≡ , , then H n = (cid:3) iff α = (cid:3) , • otherwise H n = (cid:3) for all α = − , . ( ii ) • If n ≡ , then H n = C for all α = − , , • if n ≡ , then H n = C iff α = C , • otherwise H n = C for all α = − , .4. Let N = 7 . ( i ) • If n ≡ , ,
10 (14) , then H n = (cid:3) for all α = 0 , , • if n ≡ , , then H n = (cid:3) iff α = (cid:3) , EQUENCES ASSOCIATED TO ELLIPTIC CURVES 21 • if n ≡ ,
13 (14) , then H n = (cid:3) iff α − (cid:3) , • otherwise H n = (cid:3) for all α = 0 , . ( ii ) • If n ≡ , then H n = C for all α = 0 , , • if n ≡ , then H n = C iff α − C , • otherwise H n = C for all α = 0 , .5. Let N = 9 . ( i ) • If n ≡ , ,
14 (18) , then H n = (cid:3) for all α = 0 , , • if n ≡ ,
12 (18) , then H n = (cid:3) iff α − (cid:3) , • otherwise H n = (cid:3) for all α = 0 , . ( ii ) • If n ≡ , then H n = C for all α = 0 , , • if n ≡ , then H n = C iff α − C , • otherwise H n = C for all α = 0 , .6. Let N = 10 . ( i ) • If n ≡ , , ,
16 (20) , then H n = (cid:3) for all α = 0 , , • if n ≡ ,
12 (20) , then H n = (cid:3) iff α − (cid:3) , • if n ≡ ,
13 (20) , then H n = (cid:3) iff ( α − α −
1) = (cid:3) , • otherwise H n = (cid:3) for all α = 0 , . ( ii ) • If n ≡ , then H n = C for all α = 0 , , • otherwise H n = C for all α = 0 , .7. Let N = 12 . ( i ) • If n ≡ , , , ,
20 (24) , then H n = (cid:3) for all α = 0 , , • if n ≡ ,
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