This paper introduces Gravitational Search Algorithm (GSA) to find numerical solutions of Diophantine equations, for which there exists no general method of finding solutions. This algorithm finds upon introducing randomization concept along with the two of the four primary parameters `velocity' and `gravity' in physics. The performance of this algorithm has been evaluated on a set of random values. Computed result shows that the gravitational search algorithm - based heuristic is capable of producing high quality solutions, can offer many solutions of such equations.
AbrahamS. and SanglikarM., A diophantine equation solver-a genetic algorithm application, Mathematical Colloquium Journal15(3) (2001), 16-20.
2.
AbrahamS. and SanglikarM., Finding solution to a hard problem: An evolutionary and co-evolutionary approach, Proceedings of the International Conference on Soft Computing and Intelligent Systems, Jabalpur, India, (Dec 27-29 2007).
3.
AbrahamS. and SanglikarM., Finding numerical solution to a diophantine equation: Simulated annealing as a viable search strategy, Proceedings of the International Conference on Mathematical Sciences, organized by United Arab Emirates University, Al Ain, UAE, (March 3-6 2008).
4.
BahrololoumA., Nezamabadi-PourH., BahrololoumH. and SaeedM., A prototype classifier based on gravitational search algorithm, Applied Soft Computing12(2) (2012), 819-825.
5.
BarzegarB., RahmaniA.M. and KzF., Gravitational emulation local search algorithm for advanced reservation and scheduling in grid computing systems, Fourth International Conference on Computer Sciences and Convergence Information Technology (2009), 1240-1245.
6.
BehrangM.A., AssarehE., GhalambazM., AssariM.R. and NoghrehabadiA.R., Forecasting future oil demand in Iran using GSA (Gravitational Search Algorithm), Energy36(9) (2011), 5649-5654.
7.
BurtonD.M., Elementary Number Theory Second Edition, Universal Book Stall, W.L. Brown Publishers, 1995.
8.
GopalanM.A., SrikanthR. and SankaranarayananM.G., The diophantine equation, Applied Science Periodical2(2) (1999), 72-73.
9.
GopalanM.A., GanapathyR. and SrikanthR., On the diophantine equation, Pure & Applied Mathematika Sciences2(1-2) (2000), 15-17.
10.
GopalanM.A. and SrikanthR., On a binary quadratic diophantine equation, Bulletin of Pure & Applied Sciences20E (2001), 7-11.
11.
GopalanM.A. and SrikanthR., On finding integral solution of CZ^2 = AX^2 + BY^2, Pure & Applied Mathematika Sciences4(1-2) (2002), 23-25.
12.
GopalanM.A. and SrikanthR., A remark on the square root of a complex binomial quadratic surd, Bulletin of Pure & Applied Sciences22(1) (2003), 165-168.
13.
KumanduriR. and ChristinaR., Number Theory with Computer Applications, Prentice Hall upper saddle river, New Jersey, First edition, 1997.
14.
LiC. and ZhouJ., Parameters identification of hydraulic turbine governing system using improved gravitational search algorithm, Energy Conversion and Management52(1) (2011), 374-381.
15.
MollinR.A., All solutions of the diophantine equation, Far East J Math Sci, special volume, Part III, (1998), 257-293.
16.
BalachandarS.R. and KannanK., Randomized gravitational emulation search algorithm for symmetric traveling salesman problem, Applied Mathematics and Computation192 (2007), 413-421.
17.
BalachandarS.R. and KannanK., A meta-heuristic algorithm for vertex covering problem based on gravity, International Journal of Computational and Mathematical Sciences3(7) (2009), 332-336.
18.
BalachandarS.R. and KannanK., A meta-heuristic algorithm for set covering problem based on gravity, International Journal of Computational and Mathematical Sciences4(5) (2010), 223-228.
19.
BalachandarS.R. and KannanK., Newton's law of gravity-based search algorithms, Indian Journal of Science and Technology6(2) (2013), 4141-4150.
20.
RamaswamyA.M.S., Some identities for Pell's equation, J Ramanujan Mathematics Society9(2) (1994), 103-108.
21.
RashediE., Gravitational Search Algorithm, M.c. Thesis, Shahid Bahonar University of Kerman, Kerman, Iran, 2007.
22.
RashediE., Nezamabadi-PourH. and SaryazdiS., GSA: A gravitational search algorithm, Information Sciences179 (2009), 2232-2248.
RashediE., Nezamabadi-PourH. and SaryazdiS., Filter modeling using gravitational search algorithm, Engineering Applications of Artificial Intelligence24 (2011), 117-122.
25.
SarafraziS., Nezamabadi-PourH. and SaryazdiS., Disruption: A new operator in gravitational search algorithm, Scientia Iranica, Transactions D: Computer Science & Engineering and Electrical Engineering18 (2011), 539-548.
26.
SarafraziS. and Nezamabadi-PourH., Facing the classification of binary problems with a GSA-SVM hybrid system, Mathematical and Computer Modeling57(1-2) (2013), 270-278.
27.
SchutzB., Gravity from the Ground Up, Cambridge University Press, 2003.
28.
ShawB., MukherjeeV. and GhoshalS.P., A novel opposition-based gravitational search algorithm for combined economic and emission dispatch problems of power systems, Electrical Power and Energy Systems35(1) (2012), 21-33.
29.
SmartN.P., The Algorithmic Resolution of Diophantine Equations, Cambridge University Press, 1999.
30.
TelangS.G., Number Theory, Tata McGraw-Hill Publishing Company Ltd., New Delhi, 1996.
31.
VenkateshT., On integer solutions of EMBED Equation, Indian J Pure Appl math26(9) (1995), 837-839.
32.
WebsterB.L., Solving Combinatorial Optimization Problems using a New Algorithm Based on Gravitational Attraction, Ph.D. thesis, Florida Institute of Technology, Melbourne, 2004.
33.
WiegandP., An Analysis of Co-operative Co-evolutionary Algorithms, PhD dissertation, George Mason University, 2003.
34.
YinM., HuY., YangF., LiX. and GuW., A novel hybrid K-harmonic means and gravitational search algorithm approach for clustering, Expert Systems with Applications38 (2011), 9319-9324.
