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NOTES ON MELVYN KNIGHT'S PROBLEM
Abstract
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 .
... This idea has been applied to different problems. For the problem on the presentation of positive integers n in the form n = (x + y + z)(a 2 /x + b 2 /y + c 2 /z), n = (x + y + z) 3 /(xyz), or n = (x + y + z + w)(1/x + 1/y + 1/z + 1/w), where x, y, z, w ∈ Z + see [4,[16][17][18]. For the problem on the presentation of positive integers n in the form n = x/y +y/z +dz/x, n = x/y +y/z +z/w+w/x, or n = x/y +py/z +z/w+pw/x, where x, y, z, w, d ∈ Z + and p is a prime, see [6,14,15]. ...
In this paper, we study the problem on the sum of four rational numbers with a fixed product. We will show that when d in an odd positive integer, the equation xy+dyz+zw+dwx=4dndoes not have solutions in positive integers if 2|n or dn2=1+22mq, where m,q∈Z+, m≥2, and 8|q+1. This extends the recent results of the author and Dofs and Tho (Int J Number Theory, 18 (1): 75–87, 2022 and Tho (Vietnam J Math 50 (1):183–194, 2022).
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.
This paper proves that for all positive integers n, the equation
where p = 1 or p is a prime congruent to 1 (mod8), does not have solutions in positive integers.
In the 1993 Western Number Theory Conference, Richard Guy proposed Problem 93:31, which asks for integers n representable by , where are integers, preferably with positive integer solutions. We show that the representation is impossible in positive integers if , where are such that and .
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 the rst of two papers on Magma, a new system for computational algebra, we present the Magma language, outline the design principles and theoretical background, and indicate its scope and use. Particular attention is given to the constructors for structures, maps, and sets. c 1997 Academic Press Limited Magma is a new software system for computational algebra, the design of which is based on the twin concepts of algebraic structure and morphism. The design is intended to provide a mathematically rigorous environment for computing with algebraic struc- tures (groups, rings, elds, modules and algebras), geometric structures (varieties, special curves) and combinatorial structures (graphs, designs and codes). The philosophy underlying the design of Magma is based on concepts from Universal Algebra and Category Theory. Key ideas from these two areas provide the basis for a gen- eral scheme for the specication and representation of mathematical structures. The user language includes three important groups of constructors that realize the philosophy in syntactic terms: structure constructors, map constructors and set constructors. The util- ity of Magma as a mathematical tool derives from the combination of its language with an extensive kernel of highly ecient C implementations of the fundamental algorithms for most branches of computational algebra. In this paper we outline the philosophy of the Magma design and show how it may be used to develop an algebraic programming paradigm for language design. In a second paper we will show how our design philoso- phy allows us to realize natural computational \environments" for dierent branches of algebra. An early discussion of the design of Magma may be found in Butler and Cannon (1989, 1990). A terse overview of the language together with a discussion of some of the implementation issues may be found in Bosma et al. (1994).
The Diophantine equation
- E Dofs
- N X Tho
E. DOFS and N. X. THO, The Diophantine equation, Int. J. Number Theory, 18 (1), pp. 75-87,
2021. https://doi.org/10.1142/S1793042122500075