SOME NOTES ON A GENERALIZED VERSION OF PYTHAGOREAN TRIPLES
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Abstract
A Pythagorean triple is a set of three positive integers a, b and c that satisfy the Diophantine equation a^2+b^2=c^2. The triple is said to be primitive if gcd(a, b, c)=1 and each pair of integers and are relatively prime, otherwise known as non-primitive. In this paper, the generalized version of the formula that generates primitive and non-primitive Pythagorean triples that depends on two positive integers k and n, that is, P_T=(a(k, n), b(k, n), c(k, n)) were constructed. Further, we determined the values of k and n that generates primitive Pythagorean triples and give some important results.
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Section
Combinatorics and Computational Mathematics
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References
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Shannon, A. G. & Horadam, A. F. (1973). Generalized Fibonacci number triples. American Mathematical Monthly, 80(2), 187-190.
Shannon, A. G. & Horadam, A. F. (1994). Arrowhead curves in a tree of Pythagorean triples. International Journal of Mathematical Education in Science and Technology, 25(2), 255-261.
Niven, I. and Zuckerman, H. S. (1980). An Introduction to the theory of numbers. 4th edition. John Wiley and Sons, Inc. Canada.
Rosen, K. H. 1993. Elementary Number Theory and its Application. 3rd edition. Addison-Wesley Publishing Company, USA.
Dickson, L. E. (1971). History of the Theory of Numbers. Chelsea Pub. Co., New York.
Erickson, M. and Vazzana, A. (2008). Introduction to Number Theory. Taylor and Francis Group, New York, USA.
Casinillo. L. F. (2020). Some new notes on Mersenne primes and perfect numbers. Indonesian Journal of Mathematics Education, 3(1), 15-22.
Casinillo, L. F. (2020). Odd and even repetition sequences of independent domination number. Notes on Number Theory and Discrete Mathematics, 26(1), 8-20.