On The Sum of Three Square Formula
DOI:
https://doi.org/10.51867/scimundi.3.1.11Keywords:
Diophantine Equation, Sums of Three SquaresAbstract
Let n,x,y,z be any given integers. The study of n for which n = x2 +y2 + z2 is a very long-standing problem. Recent survey of sizeable literature shows that many researchers have made some progress to come up with algorithms of decomposing integers into sums of three squares. On the other hand, available results on integer representation as sums of three square is still very minimal. If a,b,c,d,k,m,n,u,v and w are any non-negative integers, this study determines the sum of three-square formula of the form abcd+ka2+ma+n = u2 +v2 +w2 and establishes its applications to various cases.
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References
Arenas, A., and Bayer, P. (1987). Some Arithmetic Behaviour of the Sums of Three Squares. Journal of Number Theory, 27(3), 273-284.
https://doi.org/10.1016/0022-314X(87)90067-9 DOI: https://doi.org/10.1016/0022-314X(87)90067-9
Deshouillers, J. M., and Luca, F. (2010). How often is n! a sum of three squares?. The legacy of Alladi Ramakrishnan in the mathematical sciences, 243-251.
https://doi.org/10.1007/978-1-4419-6263-8_14 DOI: https://doi.org/10.1007/978-1-4419-6263-8_14
Gauss, C. (1965). Disquisitiones Arithmeticae, Yale University Press, ISBN 0-300-09473-6, 1(293), 342.
Hirschhorn, M. D., and Sellers, J. A. (1999). On representations of a number as a sum of three squares. Discrete Mathematics, 199(1-3), 85-101.
https://doi.org/10.1016/S0012-365X(98)00288-X DOI: https://doi.org/10.1016/S0012-365X(98)00288-X
Lao, H., (2022). On The Diophantine Equation ab(cd + 1) + L = u 2 + v 2 , Asian Research Journal of Mathematics , Article no.ARJOM.88102,ISSN: 2456-477X, 18(9): 8-13.
Lao, H. M., Zachary, K. K., and Kinyanjui, J. N. U. (2023). Some Generalized Formula For Sums of Cube. Journal of Advances in Mathematics and Computer Science, Article no.JAMCS.101314,ISSN: 2456-9968, 37(8): 47-52.
https://doi.org/10.9734/jamcs/2023/v38i81789 DOI: https://doi.org/10.9734/jamcs/2023/v38i81789
Legendre. M., (1785).Hist. et Mem. Acad. Roy. Sci. Paris, ' , 514-515.
Mude, L. H. (2022). Some formulae for integer sums of two squares. Journal of Advances in Mathematics and Computer Science, 37(4), 53-57.
https://doi.org/10.9734/jamcs/2022/v37i430448 DOI: https://doi.org/10.9734/jamcs/2022/v37i430448
Rabin, M. O., and Shallit, J. O. (1986). Randomized algorithms in number theory. Communications on Pure and Applied Mathematics, 39(S1), S239-S256.
https://doi.org/10.1002/cpa.3160390713 DOI: https://doi.org/10.1002/cpa.3160390713
Rob B. (2021). Factorials and Legendre's three-square theorem ,arXiv, 1(2).
Yingchun, C. (2012). Gauss Three Square Theorem Almost Involving Primes , Rocky Mountain Journal of Mathematics, 42(4).
https://doi.org/10.1216/RMJ-2012-42-4-1115 DOI: https://doi.org/10.1216/RMJ-2012-42-4-1115
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Copyright (c) 2023 Lao Hussein Mude, Maurice Owino Oduor, Michael Onyango Ojiema

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