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Integral Solution of Linear Indeterminate Equations of n Variables:Generalized Matrix Kuttaka Method


Affiliations
1 Department of Mathematics, Cochin University of Science and Technology Cochin 682022, India
2 International School of Photonics,Cochin University of Science and Technology, Cochin 682022, India
     

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Problems of indeterminate equations first appeared in Baudhayana Sulbasutra (800-500 B.C.). But a general method of integral solutions of linear indeterminate equations is not described in it, except some geometrical solutions. Aryabhata I (476 A.D.) first gave a general method (kuttaka) of integral solution of linear indeterminate equations of two variables. The kuttaka method was subsequently discussed with modifications by several ancient and medieval Indian mathematicians. However, a general method of solving indeterminate equations of n variables is not available in kuttaka method. The present paper reviews the method used by earlier writers, and describes kuṭṭaka in terms of matrices and determinants. Generalizing this matrix kuttaka method, we present a general method of integral solutions of linear indeterminate equations of n variables. Using this method, we can evaluate all positive integral solutions of the indeterminate equations in Sulbasutras (800-200 B.C.).


Keywords

Linear Indeterminate Equations, Sulbasutras, Kuttaka, Karanapadhati, Aryabhaṭiya, Generalized Matrix Kuttaka.
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  • S.N.Sen and A.K.Bag, (ed) The Sulbasutras of Baudhayana, Apastamba,Katyayana and Manava, Indian National Science Academy, New Delhi

  • K.S.Shukla (ed), Aryabhaṭiya of Aryabhaṭa with the commentary of Bhaskara I and Somesvara, Indian National Science Academy,1976.

  • T.S. Kuppanna Sastri (ed), Mahabhaskariya of Bhaskaracarya, Government Oriental Manuscript Library, Madras,1957.

  • Henry Thomas Colebrooke (ed), Classics of Indian Mathematics: Algebra with Arithmetic and Mensuration, From the Sanskrit of Brahmagupta and Bhāskara, Sharada Publishing House, Delhi, 2005.

  • M.Rangacarya (ed), The Ganita-Sāra-Sangraha of Mahaviracarya, Cosmo Publications, New Delhi, 2009.

  • Sudhakara Dvivedi (ed), Mahāsiddhānta of Āryabhaṭa II , Benares Sanskrit Series, no.36, Benares 1910

  • Padmanabha Rao (ed.), Bhaskaracarya’s Lilavati (Part II), Chinmaya International Foundation Shoda Sansthan, Ernakulam, 2014.

  • K.V.Sarma, K.Ramasubramanian, M.S.Sriram, M.D.Srinivas (ed), Gaṇita – Yukti – Bhāṣā of Jyeṣṭhadeva Hindustan Book Agency, New Delhi, 2008.

  • Venketeswara Pai, Ramasubramanian, M.S.Sriram, M.D.Srinivas(ed.), Karaṇapaddhati of Puthumana Somayājῑ, Hindustan Book Agency, New Delhi,2017.

  • J.Needham , Science and Civilization in China , Cambridge University Press,1959

  • G.Sarton , Introduction to History of Science,1, “I-Hsing”, New York, Krieger,1975

  • L.E. Dickson, History of the theory of Numbers, Vol. I, Chelsea Publishing Company, New York,1934.

  • L.E. Dickson, History of the theory of Numbers, Vol. II, Chelsea Publishing Company, New York, 1952.

  • Bibhutibhusan Datta, The science of Śulba, Cosmo Publications, New Delhi,2009


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  • Integral Solution of Linear Indeterminate Equations of n Variables:Generalized Matrix Kuttaka Method

Abstract Views: 399  |  PDF Views: 3

Authors

P. J. Sindhurani
Department of Mathematics, Cochin University of Science and Technology Cochin 682022, India
V. P. N. Nampoori
International School of Photonics,Cochin University of Science and Technology, Cochin 682022, India

Abstract


Problems of indeterminate equations first appeared in Baudhayana Sulbasutra (800-500 B.C.). But a general method of integral solutions of linear indeterminate equations is not described in it, except some geometrical solutions. Aryabhata I (476 A.D.) first gave a general method (kuttaka) of integral solution of linear indeterminate equations of two variables. The kuttaka method was subsequently discussed with modifications by several ancient and medieval Indian mathematicians. However, a general method of solving indeterminate equations of n variables is not available in kuttaka method. The present paper reviews the method used by earlier writers, and describes kuṭṭaka in terms of matrices and determinants. Generalizing this matrix kuttaka method, we present a general method of integral solutions of linear indeterminate equations of n variables. Using this method, we can evaluate all positive integral solutions of the indeterminate equations in Sulbasutras (800-200 B.C.).


Keywords


Linear Indeterminate Equations, Sulbasutras, Kuttaka, Karanapadhati, Aryabhaṭiya, Generalized Matrix Kuttaka.

References





DOI: https://doi.org/10.24906/isc%2F2019%2Fv33%2Fi2%2F183892