Integer Points on y 2 = x 3 - 7x + 10

Bremner, Andrew; Tzanakis, Nicholas

doi:10.2307/2007708

Cited by 4 publications

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“…Let us start out from an elliptic curve E : Y 2 + a 1 XY + a 3 Y = X 3 + a 2 X 2 + a 4 The invariant di erential is ! = dx 2y + a 1 x + a 3 = dy 3x 2 + 2a 2 x + a 4 a 1 y :…”

Section: Height Estimates and The Proof Of (3)mentioning

confidence: 99%

“…(ii) N 0 is usually very large (see the examples), but we also have the estimate (4) at one of the places q ∈ S. Some observations on the computation of rational approximations of the elliptic logarithms with the necessary very high precision will be made in the next Section. The solutions of the system of inequalities (3) and 4are then found by the reduction method of de Weger [28]. Since this method is well-described in [24], we omit the details here.…”

Section: Computational Aspects Of the Theoremmentioning

confidence: 99%

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Computing all S-integral points on elliptic curves

Pethö

Zimmer

Gebel

et al. 1999

Math. Proc. Camb. Phil. Soc.

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Let the elliptic curve E be defined by the equationformula herewith a1, …, a6 ∈ ℤ. Define a finite set of places S = {q1, …, qs−1, qs = ∞} of ℚ and put Q = max {q1, …, qs−1}. Let E(ℚ) denote the set of (x, y) ∈ ℚ2 satisfying (1) and the infinite point [Oscr ].The multiplicative height of a rational point P = (x, y) ∈ E(ℚ) is defined as the following product over all places q of ℚ (including q = ∞):formula herewhere the [mid ]x[mid ]qs are the normalized multiplicative absolute values of ℚ corresponding to the places q.

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“…Let us start out from an elliptic curve E : Y 2 + a 1 XY + a 3 Y = X 3 + a 2 X 2 + a 4 The invariant di erential is ! = dx 2y + a 1 x + a 3 = dy 3x 2 + 2a 2 x + a 4 a 1 y :…”

Section: Height Estimates and The Proof Of (3)mentioning

confidence: 99%

Section: Computational Aspects Of the Theoremmentioning

confidence: 99%

Computing all S-integral points on elliptic curves

Pethö

Zimmer

Gebel

et al. 1999

Math. Proc. Camb. Phil. Soc.

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“…This in turn amounts to solving a system of four linear inequalities in four unknowns. See [3]. Using Berwick's method, we showed that el=r/+l and et___=r/+l 5+7r/+6r/z_4r/3 e 2 2r/+ 3 satisfy (46) and (46').…”

mentioning

confidence: 96%

“…These are the parameters of the Witt design and its residual. 3. Finding all solutions to a 2 k(4k z llk+ 8).…”

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confidence: 99%

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Quasi-Symmetric 3-Designs and Elliptic Curves

Calderbank¹,

Morton²

1990

SIAM J. Discrete Math.

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A quasi-symmetric t-design is a t-design with two block intersection sizes p and q (where p < q). Quasi-symmetric 3-designs are classified with p 1. The only nontrivial examples are the 4-(23, 7, Witt design, and its residual, a 3- (22,7,4) design. This proves a conjecture of Sane and Shrikhande. The method is to reduce the classification problem to that of finding all integer points on the elliptic curves y2 X lX + 32X and y2 x 4x + 4.1. Quasi-symmetric designs. A t-(v, k, )) design is a collection 23 of subsets of a v-set (called blocks), such that every block contains k points, and any set of points is in exactly ), blocks. In a t-design, let ); denote the number of blocks containing a given set of points, with 0 < =< t. The identities (1) )k ), i=0,1 restrict the possible parameter sets. A symmetric design is a 2-(v, k, ,) design, such that 0 o, hi k, and any two blocks meet in , points. A t-design with two block intersection sizes is said to be quasi-symmetric. We shall consider quasi-symmetric 3-(v, k, X) designs, with block intersection sizes p and q, where p < q. Calderbank [5] and Sane and Shrikhande [15] proved that the parameters of a quasi-symmetric 2-design 23 satisfy(2) pq(v-1)(v-2)-k(k-1)(v-2)(p+q-1)+k(k-1)2(k-2) >=0, and that equality holds if and only if 23 is a 3-design. Neumaier [13], [14, Prop. 12] had earlier derived an inequality equivalent to (2), but that is not as simple to state. Sane and Shrikhande [15, Conj. 4.6] have conjectured that there exist only twonontrivial, quasi-symmetric 3-designs with p (we consider the designs formed by (v 2)-subsets of a v-set to be trivial). The first example is the 4-(23, 7, Witt design, with q 3, and the second example is its residual, a 3-(22, 7, 4) design (also with q 3). Both these designs are unique. The purpose of this paper is to prove this conjecture.One interesting property of quasi-symmetric 2-designs is that the block graph is strongly regular (see Goethals and Seidel [10], or Cameron and van Lint [7]). The vertices of this graph are the blocks of design, and two blocks are joined if they meet in q points. Recall that a strongly regular graph is regular, and that the number of vertices joined to two given vertices zl and z2 (zl 4 z2), depends only on whether or not Zl and

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A remark on a theorem of W. E. H. Berwick

Tzanakis¹

1986

Math. Comp.

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Abstract. We indicate and fill a gap in a theorem of W. E. H. Berwick concerning the computation of the fundamental units in a semireal biquadratic field.One of the standard methods for finding a pair of fundamental units in algebraic number fields that have two fundamental units is that The purpose of the present note is to indicate and fill a gap in [1] that occurs in the case of a semireal biquadratic field K, i.e., in the case of a field K that is generated over Q by an element /-. \ va + b{m , a,b,m e Z, m > 0, {m í Q, a + b{m>0, a-b{m 1, |e'| < 1, |e"| < 1}is a discrete nonempty set. Let ex be the minimum element of E and i > 1 the fundamental unit of Q(vm ). Then, exe[ = ±i or ±1. (A) // e1ej = ±t, then £,, i is a pair of fundamental units in R. (B) // e,ei = +1, then e,, i or e,, -fi is a pair of fundamental units in R, according as Jl £ R or 41 s R.Remark. In [1] only maximal orders (i.e., the rings of integers) of the various fields are considered. However, it is straightforward to see that the arguments in [1] are also valid if the maximal orders are replaced by orders containing the integers of Q(\/m ) (a simple but important fact is that for such orders R of K we have R' = R, where/?' = {a': a

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Integer Points on y 2 = x 3 - 7x + 10

Cited by 4 publications

References 0 publications

Computing all S-integral points on elliptic curves

Computing all S-integral points on elliptic curves

Quasi-Symmetric 3-Designs and Elliptic Curves

A remark on a theorem of W. E. H. Berwick

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