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

2000
*
Mathematics of Operations Research
*

Ø Ö Ð ÒÙÑ Ö× ½ Ð Ö Ø

doi:10.1287/moor.25.3.427.12219
fatcat:5khn6lap4vhj7bagjy7b3eppru
*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×º*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 ÄÈ¹ × 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 ÓÖÝ¸ ÓÖ ÓÑÔÙØ Ò Ø Ö Ù × ×º ...##
###
Book Review: The algorithmic resolution of Diophantine equations

1999
*
Bulletin of the American Mathematical Society
*

gives an

doi:10.1090/s0273-0979-99-00861-7
fatcat:vplg7fz2svgkdkvbs57kffbjma
*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 ...##
###
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

2020
*
Hardy-Ramanujan Journal
*

Firstly

doi:10.46298/hrj.2020.6458
fatcat:53okvgk6zfafzodweqqnv5oisu
*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. ...##
###
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*. ...##
###
PROCESSOR LOWER BOUND FORMULAS FOR ARRAY COMPUTATIONS AND PARAMETRIC DIOPHANTINE SYSTEMS

1998
*
International Journal of Foundations of Computer Science
*

parametrized by n, compute

doi:10.1142/s0129054198000295
fatcat:ic2n4sbstjeb7aezglob5gastq
*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 ...##
###
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

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. ...

##
###
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. ...##
###
AN ALGORITHM FOR SOLVING INTEGER LINEAR PROGRAMMING PROBLEMS

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 ...

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

2018
*
Symmetry
*

In this paper, we find all Fibonacci

doi:10.3390/sym10100509
fatcat:be7kmtnjgja5xgh273bb7xlwwq
*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. ...##
###
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*. ...

##
###
POWERS OF THREE AS DIFFERENCE OF TWO FIBONACCI NUMBERS

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. ...

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

1997
*
Theoretical Computer Science
*

(Clausen

doi:10.1016/s0304-3975(96)00195-8
fatcat:4jaelx75ozabpj24i5qwzohqem
*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. ...
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