OnSpecialPairsofPythagoreanTriangles

Transcription

1 Global Journal of Science Frontier Research: F Mathematics and Decision Sciences Volume 5 Issue 3 Version.0 Year 05 Type : Double Blind Peer Reviewed International Research Journal Publisher: Global Journals Inc. (USA Online ISSN: & Print ISSN: On Special Pairs of Pythagorean Triangles By M. A. Gopalan, S. Vidhyalakshmi, N. Thiruniraiselvi & R. Presenna Shrimati Indira Gandhi College, India Abstract- We search for pairs of Pythagorean triangles such that, in each pair, twice the difference between their perimeters is expressed in terms of special polygonal numbers. Keywords: pair of pythagorean triangles, special polygonal numbers. GJSFR-F Classification : FOR Code : MSC 00: D09, Y50. OnSpecialPairsofPythagoreanTriangles Strictly as per the compliance and regulations of : 05. M. A. Gopalan, S. Vidhyalakshmi, N. Thiruniraiselvi & R. Presenna. This is a research/review paper, distributed under the terms of the Creative Commons Attribution-Noncommercial 3.0 Unported License permitting all non commercial use, distribution, and reproduction in any medium, provided the original work is properly cited.

2 Ref 8. A. A. K. Majumdar, On some pseudo smarandache function related triangles, pythagorean triangles hold a special fascination, scientia majna, 4(3, 95-05, 008 On Special Pairs of Pythagorean Triangles M. A. Gopalan α, S. Vidhyalakshmi σ, N. Thiruniraiselvi ρ & R. Presenna Ѡ Abstract- We search for pairs of Pythagorean triangles such that, in each pair, twice the difference between their perimeters is expressed in terms of special polygonal numbers. Keywords: pair of pythagorean triangles, special polygonal numbers. Notations used: t m, n (n (m = n + s n = 6n(n + csn = n + (n I. Introduction Number is the essence of mathematical calculation. Varieties of numbers have variety of range and richness, many of which can be explained very easily but extremely difficult to prove. One of the varieties of numbers that have fascinated mathematicians and the lovers of maths is the pythagorean number as they provide limitless supply of exciting and interesting properties. In other word, the method of obtaining three nonzero integers x,y and z under certain relations satisfying the equation x + y = z has been a matter of interest to various mathematicians. For an elaborate review of various properties are may refer [-7]. In [8], the author proves existence of an infinite family of pairs of dissimilar Pythagorean triangles that are pseudo smarandache related. For problems on pairs of Pythagorean triangles one may refer [,5]. In this communication concerns with the problem of determining pairs of Pythagorean triangles wherein each pair -times the difference between the perimeters is expressed interms of special polygonal numbers. II. Method of Analysis Let T ( α, β, γ and T ( α, β, γ be two distinct Pythagorean triangles where mq, β = m q γ = m + q and pq, β = p q γ = p + q, m>q>0;p>q>0,, Global Journal of Science Frontier Research F Volume XV Issue III V ersion I Year Author α σ: Professor, Department of Mathematics, SIGC, Trichy-6000, Tamilnadu,India. s: mayilgopalan@gmail.com, vidhyasigc@gmail.com Author ρ: Research Scholar, Department of Mathematics, SIGC, Trichy-6000, Tamilnadu, India. nts.maths.ig@gmail.com Author Ѡ: M.Phil student, Department of Mathematics, SIGC, Trichy-6000, Tamilnadu, India. presennateddy9@gmail.com 05 Global Journals Inc. (US

3 Let P = m(m + q,p = p(p + q be their perimeters respectively. We illustrated below the process of obtaining pair of Pythagorean triangles such that times the difference between their perimeters is expressed in terms of a special polygonal number. Illustration (P P = (3 ( m q = (p + q + (3 The assumption α ( gives + α ( which is in the form of well know Pythagorean equation and it is satisfied by Global Journal of Science Frontier Research F Volume XV Issue III V ersion I Year α = RS,p + q = R S,m + q = R + S,R > S > 0 (3 Solving the above system of equations (3, we have RS +,m = p + S,q = R S p 3 As our main thrust is on integers, note that α is an integer when and thus m = p + (3k q = 9k ( k + k (6k + 8k k R = 3(k + k,s = 3k 4 4k k p The conditions m>q>0, p>q>0 lead to of + 3 (4 p < 9k + 8kk k < 3p (5 Choose k,k, p such that (5 is satisfied. Substituting the corresponding values,k, p in (4 and in the value of m,q, α are obtained. Thus, it is observed that k A few examples are given below in Table - 3t8, α + (P P = sα + 3t4, α k k p m q α ( P P 3 t 8, α ( ( ( (045 + s α + 3t 4, α ( ( ( ( ( ( ( ( ( ( ( (47 05 Global Journals Inc. (US

4 ( ( ( ( ( ( ( ( ( ( ( (47 It is worth to note that there are only finitely many pairs of Pythagorean triangles such that in each pair, twice the difference between their perimeters is expressed in terms of a special polygonal number. Illustration- Assume (P P = (5α Proceeding as in illustration, it is seen that there are 3 values for α satisfying (RS + the equation and thus, there are 3 sets of triples ( m,q, α which are 5 represented as follows The condition to be satisfied by k, k and p is p < (0k + 5k 4(5k < 3p From each of the above 3 sets it is observed that (P P = 5t, α + 4 Illustration 3 SET : SET : The condition to be satisfied by k, k Assume (P P = ( α For this choice, the values for m,q, α satisfying the equation ( m q = (p + q + ( (k + k m = p + (5k q = (0k + α are given by and p is (k + k m = p + (5k (5k q = (5k + 0k 4 4 k k 0(5k p The condition to be satisfied by k, k and p is p < (0k + 5k 0(5k < 3p SET 3: (k + k m = p + (5k q = (5k + 0k (5k 3 3 6k 6(5k p (5k 4k 4(5k p + 4 p < (0k + 5k 6(5k < 3p + Global Journal of Science Frontier Research F Volume XV Issue III V ersion I Year Global Journals Inc. (US

5 The conditions m>q>0,p>q>0 lead to Choose r,s,p satisfying (6, it is seen that rs +,m = s + p,q = r s p p < r s < 3p (6 (P P = (csα t4, α Global Journal of Science Frontier Research F Volume XV Issue III V ersion I Year III. Conclusion In this paper, we have obtained finitely many pairs of Pythagorean triangles where each pair connects -times the difference between the perimeters with special polygonal numbers. To conclude, one may search for the connections between special numbers and the other characterizations of Pythagorean triangle. IV. Acknowledgement The financial support from the UGC, New Delhi (F-MRP-5/4(SERO/UGC dated march 04 for a part of this work is gratefully acknowledged. References Références Referencias. L. E. Dickson, History of Theory of numbers, vol., Diophantine analysis, New York, Dover, L. J. Mordell, Diophantine Equations, Academic press, New York, 969. B. M. Stewart, Theory of Numbers, nd edit, Mac Millan Company, New York, Carl B. Boyer and Utah C.Merzlach, A History of Mathematics, John wiley and sons, Waclaw sierpiriski, Pythagorean triangles, Dover publication, New York, M. A. Gopalan and A. Gnanam, pairs of pythagorean triangles with equal perimeters, impact J.sci.Tech, (,67-70, M. A. Gopalan and G. Janaki, Pythagorean triangle with area/perimeter as a special polygonal number, Bulletin of Pure and Applied Science, Vol.7E (No., , M. A. Gopalan and S. Leelavathi, Pythagorean triangle with area/perimeter as a square integer, International Journal of Mathemtics, Computer sciences and Information Technology, Vol., No., 99-04, M. A. Gopalan and S. Leelavathi, Pythagorean triangle with area/perimeter as a cubic integer, Bulletin of Pure and Applied Science, Vol.6E(No., 97-00, M. A. Gopalan and B. Sivakami, Special Pythagorean triangles generated through the integral solutions of the equation yy = (kk + kkxx +, Diophantus J.Math., Vol (, 5-30, 03.. M. A. Gopalan and A. Gnanam, Pythagorean triangles and Polygonal numbers, International Journal of Mathematical Sciences, Vol 9, No. -,-5,00.. K. Meena, S. Vidhyalakshmi, B. Geetha, A. Vijayasankar and M. A. Gopalan, Relations between special polygonal numbers generated through the solutions of Pythagorean equation, IJISM, Vol 5(, 5-8, M. A. Gopalan and G. Janaki, Pythagorean triangle with perimeter as Pentagonal number, AntarticaJ.Math., Vol 5(, 5-8, M. A. Gopalan, Manjusomanath and K. Geetha, Pythagorean triangle with area/perimeter as a special polygonal number,iosr-jm, Vol.7(3,5-6, Global Journals Inc. (US

6 5. M. A. Gopalan and V. Geetha, Pythagorean triangle with area/perimeter as a special polygonal number,irjes,vol.(7,8-34, M. A. Gopalan and B. Sivakami, Pythagorean triangle with hypotenuse minus (area/perimeter as a square integer,archimedesj.math., Vol (,53-66,0. 7. M. A. Gopalan and S. Devibala, Pythagorean triangle with triangular number as a leg, ImpactJ.Sci.Tech., Vol (4, 95-99, A. A. K. Majumdar, On some pseudo smarandache function related triangles, pythagorean triangles hold a special fascination, scientia majna, 4(3, 95-05, 008 Global Journal of Science Frontier Research F Volume XV Issue III V ersion I Year Global Journals Inc. (US

7 Global Journal of Science Frontier Research F Volume XV Issue III V ersion I Year 05 6 This page is intentionally left blank 05 Global Journals Inc. (US