Annals of Pure and Applied Mathematics Vol. 18, No. 2, 2018, 185-188 ISSN: 2279-087X (P), 2279-0888(online) Published on 24 November 2018 www.researchmathsci.org DOI: http://dx.doi.org/10.22457/apam.v18n2a7 185   On the Non-Linear Diophantine Equations 31 x + 41 y = z 2 and 61 x + 71 y = z 2 Satish Kumar 1 , Deepak Gupta 2 and Hari Kishan 3 1 Department of Mathematics, D.N.College Meerut, U.P., India email: 1 skg22967@gmail.com ; 3 harikishan10@rediffmail.com 1 Corresponding author. Email: deepakgupta1763@gmail.com Received 10 October 2018; accepted 22 November 2018 Abstract. In this paper, we discussed all the solutions of non-linear Diophantine equations 31 x + 41 y = z 2 and 61 x + 71 y = z 2 , where x, y and z are non-negative integers and proved that these non-linear Diophantine equations have no non-negative integer solution. Keywords: Diophantine Equations, Exponential Equations, Catalan’s Conjecture AMS Mathematics Subject Classification (2010): 11D61 1. Introduction The Diophantine equation plays a major role in number theory. There is no general method to determine that a given Diophantine equation has how many solutions. In [5], Catalan conjectured that the Diophantine equation a x - b y = 1, where a, b, x and y are non- negative integers under condition min {a, b, x, y}>1 has a unique solution (a, b, x, y) = (3, 2, 2, 3). In [10], Sroysang proved that the Diophantine equation 3 x + 5 y = z 2 where x, y and z are non-negative integers has a unique solution (x, y, z) = (1, 0, 2). In [1], Acu proved that the Diophantine equation 2 x + 5 y = z 2 , where x, y and z are non-negative integers has only two solutions (x, y, z) = (3, 0, 3) and (2, 1, 3). In [8], B. Sroysang proved that the Diophantine equation 8 x + 19 y = z 2 , where x, y and z are non-negative integers has a unique solution (x, y, z) = (1, 0, 3). In [11], Sroysang proved that the Diophantine equation 2 x + 3 y = z 2 where x, y and z are non-negative integers, has only three solutions (x, y, z) = (0, 1, 2), (3, 0, 3) and (4, 2, 5). In [9], Sroysang proved that the Diophantine equation 31 x + 32 y = z 2 has no non-negative integer solution. In [3], Burshtein proved that the Diophantine equation 2 a + 7 b = c 2 when a and b both are odd integers, has no solution. In [2], Burshtein discussed an open problem of Chotchaisthit, on the Diophantine equation 2 x + p y = z 2 , where p is particular prime and y = 1. In [4], Burshtein also discussed on the Diophantine equation 2 x + p y = z 2 for odd prime p and x, y and z are positive integers. In [6], S. Kumar et.al. proved that the Diophantine equation 61 x + 67 y = z 2 and 67 x + 73 y = z 2 have no non-negative integer solution. In most of these papers, the authors used theory of congruence and/or Catalan’s conjecture [7] to find or to show the non-existence of the solutions of the Diophantine equations of form p x + q y = z 2 . In this paper, we discussed about the solutions of non-linear Diophantine equations 31 x + 41 y = z 2 (1)