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Integral solutions of quartic Diophantine equation 2[x.sup.2][z.sup.2] = [y.sup.2]([x.sup.2] + [z.sup.2]).

Abstract

We search for nonzero integral triples (x, y, z) satisfying the quartic Diophantine equation 2[x.sup.2][z.sup.2] = [y.sup.2]([x.sup.2] + [z.sup.2]). A few interesting relations among the solutions are also given.

Introduction

In [1, 2] Quadratic Diophantine equations with three unknowns have been considered for its parametric integral solutions. In [3, 4, 5, 6, 7, 8, 9, 10, 11] typical ternary quadratic Diophantine equations have been analysed for their integral solutions.

In this communication, the quartic Diophantine equation with three variables represented by 2[x.sup.2][z.sup.2] = [y.sup.2]([x.sup.2] + [z.sup.2]) is considered for its parametric integral solutions. A few interesting relations among the solutions are also given.

Method of Analysis

The equation under consideration to be solved is

2[x.sup.2][z.sup.2] = [y.sup.2]([x.sup.2] + [z.sup.2]) (1)

The substitution

x = u + v, y = u-v, u [not equal to] v (2)

in (1) reduces it to

[u.sup.4] - [u.sup.2](2[v.sup.2] + [y.sup.2]) + [v.sup.4] - [y.sup.2][v.sup.2] = 0(3)

Solving (2) as a quadratic in [u.sup.2], we get

[u.sup.2] = 1-2 [(2[v.sup.2] + [y.sup.2]) + y[square root of (8[v.sup.2] + [y.sup.2]] (4)

or

[u.sup.2] = 1/2 [(2[v.sup.2] + [y.sup.2]) - y[square root of (8[v.sup.2] + [y.sup.2])] (5)

Assume 8[v.sup.2] + [y.sup.2] = [[alpha].sup.2] (6)

This is satisfied by the following two choices of v, y and [alpha]:

v = 2 pq, y = 8[p.sup.2] - [q.sup.2], [alpha] = 8[p.sup.2] + [q.sup.2] (7)

v = 2 pq, y = [p.sup.2] - 8[q.sup.2], [alpha] = [p.sup.2] + 8[q.sup.2] (8)

Case (1)

The substitution of (7) in (4) gives

[u.sup.2] = [q.sup.2] [[q.sup.2] - [(2p).sup.2]]

whose solution is

p = rs, q = [r.sup.2] + [s.sup.2], u = [r.sup.4] - [s.sup.4]

and hence

v = 2rs([r.sup.2] + [s.sup.2])

Using the values of p, q, u and v in (2) and (7), we get

x = [r.sup.4] - [s.sup.4] + 2rs([r.sup.2] + [s.sup.2]) y = 6[r.sup.2][s.sup.2] - [r.sup.4] - [s.sup.4] z = [r.sup.4] - [s.sup.4] - 2rs([r.sup.2] + [s.sup.2])

Some relations among the solutions are:

1) The triple ([x.sup.2], [y.sup.2], [z.sup.2]) form an H.P.

2) x+z-2y can be written as the difference of two squares.

3) 2(x-z) represents the difference of two fourth powers.

A few numerical illustrations are given below:
x y z

35 7 -5
140 -28 20
221 119 -91
391 161 -119
560 112 -120
775 527 -425


4) When r = s, x [equivalent to] 0(mod y) and x [equivalent to] 0(mod z).

5) -(xy/z) is a perfect square.

6) yz [equivalent to] 0(mod x)

Case (2)

Using (7) in (5), we get

[u.sup.2] = 4[p.sup.2][16[q.sup.2] - [q.sup.2]

Proceeding as before, we get the solution of (1) as

x = 8[r.sup.4] - 8[s.sup.4] + 16rs([r.sup.2] + [s.sup.2]) y = 8[r.sup.4] + 8[s.sup.4] - 48[r.sup.2][s.sup.2] z = 8[r.sup.4] - 8[s.sup.4] - 16rs([r.sup.2] + [s.sup.2])

In a similar manner, employing (8) and following a procedure similar to the above we can obtain other patterns of solutions of (1).

References

[1] L.E. Dickson, 1952, History of Theory of Numbers, Vol. 2, Chelsea Publishing Company, New York.

[2] L.J. Mordell, 1969, Diophantine Equations, Academic Press, New York.

[3] M.A. Gopalan, S. Vidhyalakshmi and S. Devibala, 2006, Integral solutions of 49[X.sup.2] + 50[Y.sup.2], Acta Ciencia Indica, Vol. XXXII M, No:2, pp. 839.

[4] M.A. Gopalan, S. Vidhyalakshmi and A. Krishnamoorthy, 2005, Integral solutions of Ternary Quadratic a[X.sup.2] + b[Y.sup.2] = c(a + b)[Z.sup.2], Bulletin of Pure and Applied Sciences, Vol. 24E (No: 2), pp. 443-446,.

[5] M.A. Gopalan, S. Vidhyalakshmi and S. Devibala, 2006, On the Ternary Quadratic Equation [X.sup.2] + B[Y.sup.2] = [B.sup.2][Z.sup.2], Antartica. J. Math., 3 (2), pp. 139-142,.

[6] M.A. Gopalan, 2000, Note on the Diophantine equation [X.sup.2] + aXY + B[Y.sup.2] = [Z.sup.2], Acta Ciencia Indica, Vol. XXVIM, No:2, 105.

[7] M.A. Gopalan, 2000, Note on the Diophantine equation [X.sup.2] + XY + [Y.sup.2] = 3[Z.sup.2], Acta Ciencia Indica, Vol. XXVIM, No:3, pp. 265.

[8] M.A. Gopalan, 2000, A Remark of the equation [D.sup.[alpha]] [X.sup.2] + [Y.sup.2] = [Z.sup.2], Acta Ciencia Indica, Vol. XXVIM, No:3, pp. 277.

[9] M.A. Gopalan, A. Krishnamoorthy and S. Vidhyalakshmi, 2006, On the Integral solutions of [X.sup.2] + pXY + [Y.sup.2] = [Z.sup.2], Acta Ciencia Indica, Vol. XXXII M, No:1, pp. 209.

[10] M.A. Gopalan and R. Anbuselvi, 2005, On Ternary Quadratic Homogeneous Diophantine Equation [X.sup.2] + pXY + [Y.sup.2] = [Z.sup.2], Bulletin of Pure and Applied Sciences, Vol.24E (No:2), pp. 405-408.

[11] M.A. Gopalan, Manju Somanath and N. Vanitha, 2006, On Ternary Cubic Diophantine Equation [x.sup.2] + [y.sup.2] = 2[z.sup.3], Advances in Theoretical and Applied Mathematics (ATAM) Vol. 1(3), pp. 227-231.

Manju Somanath and N. Vanitha

Department of Mathematics,

Cauvery College for Women, Trichy, India

manjuajil@yahoo.com

vanittha_1978@yahoo.co.in

M.A. Gopalan

Department of Mathematics,

National College, Trichy, India

E-mail: gopalanma@yahoo.com
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Author:Somanath, Manju; Vanitha, N.; Gopalan, M.A.
Publication:Advances in Theoretical and Applied Mathematics
Article Type:Report
Geographic Code:9INDI
Date:May 1, 2007
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