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Solving a System of Linear Diophantine Equations with Lower and Upper Bounds on the Variables

Karen Aardal, Cor A. J. Hurkens, Arjen K. Lenstra
2000 Mathematics of Operations Research  
Ø Ö Ð ÒÙÑ Ö× ½ Ð Ö Ø ÖÑ Ò ÖÓÑ ½ Ð Ý Ø Ö ÙÖÖ Ò £ ½ ½ £ ½ ½ £ ¾ Ð ´ µ Ì £ ´ £ µ Ì £ ½ Ð Ä Ø ÒÓØ Ø Ù Ð Ò Ð Ò Ø Ò Á Ê Ò º Ä Ò×ØÖ ¸Ä Ò×ØÖ ¸ Ò ÄÓÚ ×Þ´½ ¾µ Ù× Ø ÓÐÐÓÛ Ò ¬Ò Ø ÓÒ Ó Ö Ù × × ¬Ò Ø ÓÒ ¾ × × ½ ¾ Ð ×  ...  ÒÓØ¸Û Û ÐÐ × ÓÛ Ò Ë Ø ÓÒ ÓÛ Û Ò ÒØ Ö Ð Ò Ö ÓÑ Ò Ø ÓÒ× Ó Ø Ú ØÓÖ× × Ø × Ý Ò Ü ¼ ØÓ Ü Ò ÓÖ Ö ØÓ Ó Ø Ò Ú ØÓÖ Ø Ø × Ø ×¬ × Ø ÓÙÒ × × Û ÐÐ ×Ù Ú ØÓÖ Ü ×Ø׺ Ä ÑÑ ¾´Ä Ò×ØÖ ¸Ä Ò×ØÖ ¸ Ò ÄÓÚ ×Þ¸½ ¾µº Ä Ø £ Ê Ò Ð  ...  Ñ × Ö Ú Ò Ò × ÓÒ × ÓÒ ½ ÅÀÞ ËÙÒ ÍÐØÖ ¹½º ÓÖ Ø Äȹ × Ö Ò ¹ Ò ¹ ÓÙÒ Û Ù× ÈÄ Ú Ö× ÓÒ º¼º ¸ Ò Ò ÓÙÖ Ð ÓÖ Ø Ñ Û Ù× Ä Á ¸ Ð Ö ÖÝ ÓÖ ÓÑÔÙØ Ø ÓÒ Ð ÒÙÑ ÖØ ÓÖݸ ÓÖ ÓÑÔÙØ Ò Ø Ö Ù × ×º  ... 
doi:10.1287/moor.25.3.427.12219 fatcat:5khn6lap4vhj7bagjy7b3eppru

Book Review: The algorithmic resolution of Diophantine equations

Maurice Mignotte
1999 Bulletin of the American Mathematical Society  
gives an upper bound on the number of solutions.  ...  The section on p-adic numbers contains Strassmann's theorem and examples where it enables us to solve a diophantine equation completely, and also where it does not lead to a complete conclusion, but just  ... 
doi:10.1090/s0273-0979-99-00861-7 fatcat:vplg7fz2svgkdkvbs57kffbjma

Page 2324 of Mathematical Reviews Vol. , Issue 99d [page]

1991 Mathematical Reviews  
Applying lower bounds for such linear forms leads to a large upper bound on the size of solutions of this inequality.  ...  Suppose that a =3 (mod 8), b is even and 4 does not divide b, (2) =—1, (4) =1 and a > Ab, where a y 7 -1/2 ia i{ exp (cars) a i} In the paper under review the author, using a lower bound for linear forms  ... 

Linear forms in logarithms and exponential Diophantine equations

Rob Tijdeman
2020 Hardy-Ramanujan Journal  
Firstly the importance of Alan Baker's work on linear forms in logarithms for the development of the theory of exponential Diophantine equations.  ...  Secondly how this theory is the culmination of a series of greater and smaller discoveries.  ...  I thank Lajos Hajdu, Tarlok Shorey and Cam Stewart for their suggestions to improve the paper.  ... 
doi:10.46298/hrj.2020.6458 fatcat:53okvgk6zfafzodweqqnv5oisu

Page 3849 of Mathematical Reviews Vol. , Issue 2002F [page]

2002 Mathematical Reviews  
Perhaps more intriguing is that this result along with a lower bound for y,(K) due to the reviewer can yield interesting lower bounds for y,,(Q).  ...  Here “better” means smaller than the original Minkowski upper bound.  ... 


1998 International Journal of Foundations of Computer Science  
parametrized by n, compute a lower bound on the number of processors required by the schedule as a function of n.  ...  In our formulation, the number of tasks that are scheduled for execution during any fixed time step is the number of non-negative integer solutions d n to a set of parametric linear Diophantine equations  ...  A lower bound on the number of processors needed to satisfy a schedule for a particular time step can be formulated as the number of solutions to a linear Diophantine equation, subject to the linear inequalities  ... 
doi:10.1142/s0129054198000295 fatcat:ic2n4sbstjeb7aezglob5gastq

Page 2796 of Mathematical Reviews Vol. , Issue 2001D [page]

2001 Mathematical Reviews  
In our study we use the algorithm of K. Aardal, C. Hurkens and A. K. Lenstra [“Solv- ing a system of linear Diophantine equations with lower and upper bounds on the variables”, Math. Oper.  ...  This algorithm is not restricted to dealing with market split instances only but is a general method for solv- ing systems of linear Diophantine equations with bounds on the variables.  ... 

On aggregating two linear Diophantine equations

Nan Zhu, Kevin Broughan
1998 Discrete Applied Mathematics  
The problem of aggregating a general system of two linear Diophantine equations with integct coeffkients and non-negative integer variables, to form a single linear Diophantine equation with the same solution  ...  New procedures, which generalize and nnprovc upon some results in the literature, are given. Some or all of the Lariables may be given upper bounds.  ...  The authors would also like to thank the referees for their helpful comments and suggestions.  ... 
doi:10.1016/s0166-218x(97)00123-6 fatcat:ys3oj6hxhfdwvkzfsaae6rkkzi

Page 413 of Mathematical Reviews Vol. , Issue 86a [page]

1986 Mathematical Reviews  
We also should note its closeness to the upper bound obtained earlier.” Lisovik, L. P. (2-KIEV) 86a:90041 Solution of systems of partially Diophantine linear equations and inequalities.  ...  number of infeasibilities does not increase). (1.3) The outgoing variable leaves the basis at a feasible level (lower or upper bound).  ... 

Page 4148 of Mathematical Reviews Vol. , Issue 89G [page]

1989 Mathematical Reviews  
Relying on these results, various lin- ear and nonlinear cases of moment conditions are described and the corresponding upper bounds for stochastic linear programs are  ...  The global rank is defined as the rank for which at least one of the integer nonnegative solutions of the Diophantine equation corresponds to a feasible integer point of the problem.  ... 


Shinto K.G .
2013 International Journal of Research in Engineering and Technology  
Equation (2) can be written: *A c,xi+ CL *zt .-+ CY'*n =22 (6) Equation (6) is to be treated and solved as a linear diophantine equation.  ...  linear pro-gramming problems which is based on· the idea of solving as a linear diophantine equation a constraint parallel to the objective function hyperplane and then ascertaining if the original constraint  ... 
doi:10.15623/ijret.2013.0207012 fatcat:qfzo5hjonjgzdktcmsp3laigua

Fibonacci and Lucas Numbers of the Form 2a + 3b + 5c + 7d

Yunyun Qu, Jiwen Zeng, Yongfeng Cao
2018 Symmetry  
In this paper, we find all Fibonacci and Lucas numbers written in the form 2 a + 3 b + 5 c + 7 d , in non-negative integers a , b , c , d , with 0 ≤ max { a , b , c } ≤ d .  ...  Acknowledgments: The authors would like to express their sincere gratitude to the referees for their valuable comments which have significantly improved the presentation of this paper.  ...  Conflicts of Interest: The authors declare no conflict of interest.  ... 
doi:10.3390/sym10100509 fatcat:be7kmtnjgja5xgh273bb7xlwwq

Page 4667 of Mathematical Reviews Vol. , Issue 2000g [page]

2000 Mathematical Reviews  
Solving a linear Diophantine equation with lower and upper bounds on the variables.  ...  Summary: “We develop an algorithm for solving a linear Dio- phantine equation with lower and upper bounds on the variables.  ... 


Pagdame Tiebekabe, Ismaïla Diouf
2021 JP Journal of Algebra Number Theory and Applications  
The tools used to solve our main result are properties of continued fractions, linear forms in logarithms, and a version of the Baker-Davenport reduction method in diophantine approximation.  ...  The goal of this paper is to find, in a simple and rigorous way, all powers of three as the difference of two Fibonacci numbers, that is, we study a diophantine equation is the Fibonacci sequence.  ...  But it also uses an upper bound on m n − which appears in the proof of Theorem 1 of [1] and rests on an application of lower bounds for linear forms in logarithms of algebraic numbers.  ... 
doi:10.17654/nt049020185 fatcat:2eeb4wqufbe5tfhjhjfz2edcne

Avoiding slack variables in the solving of linear diophantine equations and inequations

Farid Ajili, Evelyne Contejean
1997 Theoretical Computer Science  
(Clausen and Fortenbacher, 1989) for solving a single linear Diophantine equation.  ...  In this paper, we present an algorithm for solving &ire& linear Diophantine systems of both equations and inequations.  ...  Acknowledgements We thank the referees for carefully reading our paper.  ... 
doi:10.1016/s0304-3975(96)00195-8 fatcat:4jaelx75ozabpj24i5qwzohqem
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