No full-text available
To read the full-text of this research,
you can request a copy directly from the authors.
Which Integers are Representable as the Product of the Sum of Three Integers with the Sum of their Reciprocals?
Abstract
For example, the integer 564 is so representable by the three integers 122 44200 50100028778116351171 17995213613513491867, -3460 69586 84255 04865 64589 22621 88752 08971 30654 24460 and 74807 1910153025 27837945836017146464 94820 59055 28060.
... In [2], Bremner, Guy and Nowakowski investigated Melvyn Knight's problem which asks for integers n representable as n = (x + y + z)( ...
... with integers x, y, z. They also briefly discussed the representation (2) in the set of positive integers. Integer solutions to equation (2) depend on the rank of the elliptic curve defined by (2). ...
... They also briefly discussed the representation (2) in the set of positive integers. Integer solutions to equation (2) depend on the rank of the elliptic curve defined by (2). When asking for positive integer solutions to (2), the situation becomes more subtle. ...
Melvyn Knight's problem asks for positive integers n that can be represented as n = (x + y + z)(1 x + 1 y + 1 z) with integers x, y, z. In this paper, we investigate integers n that can be represented as n = x + y + z a 2 b 2 c 2 (a 2 x + b 2 y + c 2 z) (1) with integers x, y, z, a, b, c. For integers n, a, b, c satisfying 4|n or 8|n − 5, a + b + c = −1, and abc is a square number, we show that the representation (1) is essentially unique if na 2 b 2 c 2 = (|a| + |b| + |c|) 2 and is impossible if na 2 b 2 c 2 = (|a| + |b| + |c|) 2 .
... According to Bremner, Guy, and Nowakowski [1], Melvyn Knight asked what integers can be represented in the form ...
... where , , are integers. In the same paper [1], the authors made an extension study of (1.1) in integers when is in the range | | ≤ 1000. Integer solutions are found except for 99 values of . ...
... Integer solutions are found except for 99 values of . The question becomes more interesting if we ask for positive integer solutions, which was also briefly discussed in [1,Section 2]. In this paper, we will prove the following theorem: This theorem gives the first parametric family when (1.1) does not have positive integer solutions. ...
... Theorem 5 gives a negative answer to a conjecture by Bremner, Guy, Nowakowski [4]. ...
... Different homogeneous forms in three variables have been studied: (x+y+z)( 1 x + 1 y + 1 z ) 2 by Bremner, Guy, Nowakowski [4]; (x+y+z) 3 xyz , x y + y z + z x by Bremner, Guy [5]; (x+y+z) 3 xyz by Brueggemen [10]; x y+z + y z+x + z x+y by Bremner, Macleod [7]. Theorem 5 is the first example on the four variable case. ...
... Theorem 7. Consider the surface S : x 4 + 7y 4 = 14z 4 + 18w 4 . Then S is everywhere locally solvable, and S has no rational points except (0, 0, 0, 0). ...
Diophantine arithmetic is one of the oldest branches of mathematics, the search for integer or rational solutions of algebraic equations. Pythagorean triangles are an early instance. Diophantus of Alexandria wrote the first related treatise in the fourth century; it was an area extensively studied by the great mathematicians of the seventeenth century, including Euler and Fermat. The modern approach is to treat the equations as defining geometric objects, curves, surfaces, etc. The theory of ellip-tic curves (or curves of genus 1, which are much used in modern cryptography) was developed extensively in the twentieth century, and has had great application to Dio-phantine equations. This theory is used in application to the problems studied in this thesis. This thesis studies some curves of high genus, and possible solutions in both rationals and in algebraic number fields, generalizes some old results and gives answers to some open problems in the literature. The methods involve known techniques together with some ingenious tricks. For example, the equations y 2 = x 6 + k, k = −39, −47, the two previously unsolved cases for |k| < 50, are solved using algebraic number theory and the elliptic Chabauty method. The thesis also studies the genus three quartic curves F (x 2 , y 2 , z 2) = 0 where F is a homogeneous quadratic form, and extend old results of Cassels, and Bremner. It is a very delicate matter to find such curves that have no rational points, yet which do have points in odd-degree extension fields of the rationals. The principal results of the thesis are related to surfaces where the theory is much less well known. In particular, the thesis studies some specific families of surfaces, and give a negative answer to a question in the literature regarding representation of integers n in the form n = (x+y +z +w)(1/x+1/y +1/z +1/w). Further, an example, the first such known, of a quartic surface x 4 + 7y 4 = 14z 4 + 18w 4 is given with remarkable properties: it is everywhere locally solvable, yet has no non-zero rational point, despite having a point in (non-trivial) odd-degree extension fields i of the rationals. The ideas here involve manipulation of the Hilbert symbol, together with the theory of elliptic curves.
... We refer to [1,4,2,3,5]. For m ≥ 2, it is interesting to find nonzero integers x 1 , · · ·, x m such that (1.1) n = (x 1 + · · · + x m ) 1 x 1 + · · · + 1 x m . ...
... which is known as Melvyn Knight's problem: which integers n can be represented as the product of the sum of three integers with the sum of their reciprocals. In 1993, A. Bremner, R.K. Guy and R.J. Nowakowski [1] studied this problem and gave the necessary and sufficient conditions for the integers with a positive representation. In 2010, R. Kozuma [9] got two sufficient conditions for an integer n which cannot be expressed in the form of Eq. (1.2) by considering the elliptic curve E n : y 2 + (n − 3)xy + (n − 1)y = x 3 over Q. ...
... When m = 4, the authors of [1] gave two remarks about Eq. (1.1): (1) "If ...
By the theory of elliptic curves, we study the integers representable as the product of the sum of four integers with the sum of their reciprocals and give a sufficient condition for the integers with a positive representation.
... In [2] we discussed Melvyn Knight's problem of finding those integers representable in the form n = (x + y + z)(-+ -+and exhibited triples of integers (x, y, z) for all but 99 of the possible n in the range -1000 < n < 1000. Analogous problems have arisen elsewhere, and in [8] it was asked to find all integers representable by (x + y + zf xyz but it turns out that only a small finite number of positive n are so representable, while * Supported by Nat. ...
... From Konhauser, Velleman and Wagon [11], one learns that Underwood Dudley had given the solutions (9,162,4), (72,162,4) and (350,196,5), each for n = 41. Peter Montgomery observes that if x, y, z have no factor in common, then (2) implies that there exist integers a, b, c such that In the form (3), the representation problem at (2) has been studied by Dofs [5,6], who gives some parametric solutions and a table of numerical solutions for some n in the range -81 < n < 80. See also Craig [5], Mohanty [12], and Thomas and Vasquez [15]. Notice that (a 3 , b 3 , c\ n 3 ) at (3) is now a solution to the first representation problem (1). ...
... From Konhauser, Velleman and Wagon [11], one learns that Underwood Dudley had given the solutions (9,162,4), (72,162,4) and (350,196,5), each for n = 41. Peter Montgomery observes that if x, y, z have no factor in common, then (2) implies that there exist integers a, b, c such that In the form (3), the representation problem at (2) has been studied by Dofs [5,6], who gives some parametric solutions and a table of numerical solutions for some n in the range -81 < n < 80. See also Craig [5], Mohanty [12], and Thomas and Vasquez [15]. Notice that (a 3 , b 3 , c\ n 3 ) at (3) is now a solution to the first representation problem (1). ...
We discuss the problem of finding those integers which may be represented by (x + y + z)3/xyz, and also those which may be represented by x/y + y/z + z/x, where x, y, z are integers. For example,
satisfy (x + y + z)3/(xyz) = -47, and
satisfy x/y + y/z + z/x = -86.
... In Bremner, Guy and Nowakowski [1], the authors investigate the Diophantine problem of representing integers n in the form (x+y+z)(1/x+1/y+1/z) for rationals x, y, z, which is equivalent to studying rational points on a parametrized family of elliptic curves. Solutions for x, y, z depend upon the rational rank of the curve being positive. ...
... The question arises as to whether there are always positive solutions to (1.1) for w, x, y, z when n ≥ 16. Quoting the previous authors: ". . . it seems likely that for n ≥ 16 there is always such a representation, e.g., 16(1, 1, 1, 1); 17 (2,3,3,4); 18(1, 1, 2, 2); 19 (8,9,18,21); 20 (1,3,3,3)." For a fixed rational ratio y/z, Eq. (1.1) represents the equation of an affine cubic curve in w, x. For given n, it seems that an appropriate specialization of y/z to positive rational values may give rise to an elliptic curve of positive rank containing points with x, w > 0. Thus, the remark of Bremner, Guy and Nowakowski, seems plausible. ...
Bremner, Guy and Nowakowski [Which integers are representable as the product of the sum of three integers with the sum of their reciprocals? Math. Compos. 61(203) (1993) 117–130] investigated the Diophantine problem of representing integers n in the form (x + y + z)(1/x + 1/y + 1/z) for rationals x,y,z. For fixed n, the equation represents an elliptic curve, and the existence of solutions depends upon the rank of the curve being positive. They observed that the corresponding equation in four variables, the title equation here (representing a surface), has infinitely many solutions for each n, and remarked that it seemed plausible that there were always solutions with positive w,x,y,z when n ≥ 16. This is false, and the situation is quite subtle. We show that there cannot exist such positive solutions when n is of the form 4m2, 4m2 + 4, where m≢2(mod 4). Computations within our range seem to indicate that solutions exist for all other values of n.
... In Bremner, Guy and Nowakowski [1], the authors investigate the Diophantine problem of representing integers n in the form (x+y+z)(1/x+1/y+1/z) for rationals x, y, z, which is equivalent to studying rational points on a parametrized family of elliptic curves. Solutions for x, y, z depend upon the rational rank of the curve being positive. ...
... The question arises as to whether there are always positive solutions to (1.1) for w, x, y, z when n ≥ 16. Quoting the previous authors: ". . . it seems likely that for n ≥ 16 there is always such a representation, e.g., 16(1, 1, 1, 1); 17(2, 3, 3, 4); 18(1, 1, 2, 2); 19 (8,9,18,21); 20 (1,3,3,3)." For a fixed rational ratio y/z, Eq. (1.1) represents the equation of an affine cubic curve in w, x. For given n, it seems that an appropriate specialization of y/z to positive rational values may give rise to an elliptic curve of positive rank containing points with x, w > 0. Thus, the remark of Bremner, Guy and Nowakowski, seems plausible. ...
There seem few examples in the literature of quartic surfaces defined over ℚ that are everywhere locally solvable, yet which have no global point. It is a delicate question as to whether such surfaces can possess points defined over an odd-degree number field, and to our knowledge no previous example is known. We give here an example of such a diagonal quartic surface which contains a point defined over a cubic extension field (and it follows that there exist number fields of every odd degree greater than 1 in which the surface has points). This surface is one member of a more general family of surfaces, each of which is also everywhere locally solvable but with no rational point.
... In Bremner, Guy and Nowakowski [1], the authors investigate the Diophantine problem of representing integers n in the form (x+y+z)(1/x+1/y+1/z) for rationals x, y, z, which is equivalent to studying rational points on a parametrized family of elliptic curves. Solutions for x, y, z depend upon the rational rank of the curve being positive. ...
... The question arises as to whether there are always positive solutions to (1.1) for w, x, y, z when n ≥ 16. Quoting the previous authors: ". . . it seems likely that for n ≥ 16 there is always such a representation, e.g., 16(1, 1, 1, 1); 17(2, 3, 3, 4); 18(1, 1, 2, 2); 19 (8,9,18,21); 20 (1,3,3,3)." For a fixed rational ratio y/z, Eq. (1.1) represents the equation of an affine cubic curve in w, x. For given n, it seems that an appropriate specialization of y/z to positive rational values may give rise to an elliptic curve of positive rank containing points with x, w > 0. Thus, the remark of Bremner, Guy and Nowakowski, seems plausible. ...
... which bears a very strong resemblance to the integer representation problems in [1] and [2]. In all cases we look to express N as a ratio of two homogeneous cubics in 3 variables. ...
... We have shown that integer solutions to equation (4) are related to rational points on the curves E N defined in equation (9). The problem is that equation (4) can be satisfied by integers which could be negative as in the representation problems of [1] and [2]. ...
We consider the problem of finding integer-sided triangles with R/r an integer, where R and r are the radii of the circumcircle and incircle respectively. We show that such triangles are relatively rare.
... In [9], Andrew Bremner, Richard Guy and Richard Nowakowski discussed the problem of (if possible) finding integers X, Y, Z such that ...
Several problems which could be thought of as belonging to recreational mathematics are described. They are all such that solutions to the problem depend on finding rational points on elliptic curves. Many of the problems considered lead to the search for points of very large height on the curves, which (as yet) have not been found.
... Unfortunately, the formula used involves an uncomputable term for the size of the Tate–Safarevic group, so there is no guarantee that the height value is the height of the generator of the infinite order points, though it usually is correct. For heights less than 5, a simple search is effective, while for heights up to about 18, the well-known method of 4-descent can be used, as described in [1]. Beyond this value the quartic searches needed start to take a significantly long time. ...
We consider the problem of finding two integer right-angled triangles, having a common base, where the respective heights are in the integer ratio n:1. By considering an equivalent elliptic curve problem, we find parametric solutions for certain values of n. Extensive computational resources are then employed to find those integers which do appear with 2⩽n⩽999.
... Many Diophantine problems can be reduced to determining points on elliptic curves. A fascinating example is from Bremner et al [3], where finding possible representations of n as ...
We consider the practical computation of rational points on y^2=x(x^2+ax+b). The algebra necessary for a 4-descent procedure is described. A simple further descent is then described which only uses integer arithmetic. Numerous examples are given to illustrate the effectiveness of this extra descent. Currently, the largest point found has height 51.15 or 102.3, depending on which height normalisation you use.
For a non-zero integer N, we consider the problem of finding 3 integers (a, b, c) such that (formula presented). We show that the existence of solutions is related to points of infinite order on a family of elliptic curves. We discuss strictly positive solutions and prove the surprising fact that such solutions do not exist for N odd, even though there may exist solutions with one of a, b, c negative. We also show that, where a strictly positive solution does exist, it can be of enormous size (trillions of digits, even in the range we consider).
We determine reduction types and images of local connecting homomorphisms of elliptic curves with a rational 3-torsion point over a number field in which 3 is unramified. These results are shown to be useful for the explicit calculation of Selmer groups.
We discuss the problem of nding integer-sided triangles with the ratio base/altitude or altitude/base an integer. This problem is mentioned in Richard Guy's book "Unsolved Problems in Number Theory". The problem is shown to be equivalent to nding rational points on a family of elliptic curves. Various computa- tional resources are used to nd those integers in (1; 99) which do appear, and also nd the sides of example triangles. A.M.S. (MOS) Subject Classication Codes. 11D25 , 11Y50
It has long been known that every positive semidefinite function of R(x, y) is the sum of four squares. This paper gives the first example of such a function which is not expressible as the sum of three squares. The proof depends on the determination of the points on a certain elliptic curve defined over C(x). The 2-component of the Tate-Šafarevič group of this curve is nontrivial and infinitely divisible.