Counterexamples to a theorem concerning solution of certain quadratic Diophantine equation.

[section]1. Introduction

In [2], R. A. Mollin considered the problem of giving necessary and sufficient conditions for the solvability of the Diophantine equation [x.sup.2] - [Dy.sup.2] = 4 and posed several theorems concerning the relations between fundamental solution of the Diophantine equation [x.sup.2] - [Dy.sup.2] = 4 with gcd(x, y) = 1 and central norm equals to 4 associated with a principal norm of 8, which is an analogue of the generalized Lagrange result.

The following theorem was given by R. A. Mollin in [2].

Theorem 1. If D = 4c, c is odd, l([square root of D]) = l is even with [Q.sub.l/2] = 4, and [Q.sub.j] = 8 for some j, then the following hold:

(1) c [equivalent to] 3, 7(mod 16) if and only if j is even.

(2) c [equivalent to] 11,15(mod 16) if and only if j is odd.

But the theorem is incorrect. In this paper we revise Theorem 1. Also, we describe an algorithm for calculate j with value of c and give some counterexamples to Theorem 1.

[section]2. Notation and preliminaries

We will be concerned with the simple continued fraction expansions of [square root of D] where D is an integer that is not perfect square. We denote this expansion

[square root of D] = <q0; [bar.[q.sub.1],[q.sub.2], ..., [q.sub.l-1], 2q0]>,

where l([square root of D]) = l is the period length, [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (the floor of [square root of D]) and [q.sub.1], [q.sub.2], ..., [q.sub.l-1] is a palindrome. The jth convergent of [square root of D] for j [greater than or equal to] 0 is given by,

[A.sub.j]/[B.sub.j] = <[q.sub.0]; [q.sub.1], [q.sub.2], ..., [q.sub.j],),

where

[A.sub.j] = [q.sub.j] [A.sub.j-1] + [A.sub.j-2], [B.sub.j] = [q.sub.j][B.sub.j-1] + [B.sub.j-2],

with [A.sub.-2] = 0, [A.sub.-1] = 1, [B.sub.-2] = 1, [B.sub.-1] = 0. The complete quotients are given by, ([P.sub.j] + [square root of D])/[Q.sub.j], where [P.sub.0] = 0,[Q.sub.0] = 0 and for j [greater than or equal to] 1,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

We will also need the following facts:

[A.sub.j][B.sub.j-i] - [A.sub.j-1] [B.sub.j] = [(-1).sup.j-1], [A.sup.2.sub.j-1] - [B.sup.2.sub.j-1]D = [(-1).sup.j] [Q.sub.j].

When l is even, [P.sub.l/2] = [P.sub.l/2+1] and [Q.sub.l/2]|2[P.sub.l/2], where [Q.sub.l/2]|2D and [Q.sub.l/2] is called the central norm. In general, the values [Q.sub.j] are called the principal norms, since they are the norms of principal reduced ideals in order Z[[square root of D]]. (Also, see [1] for a more advanced exposition)

We will be considering Diophantine equations [x.sup.2] - [Dy.sup.2] = 4. The fundamental solution of such an equation means the (unique) least positive integers (x,y) = ([x.sub.0],[y.sub.0]) satisfying it.

[section]3. Revision of Theorem 1

Firstly, we reformulate Theorem 1 as the following:

Theorem 2. If D = 4c, c is old, l([square root of D]) = t is even with [Q.sub.l/2] = 4 and [Q.sub.j] = 8 for some j, then the following hold:

(i) c [equivalent to] 7,15(mod 16) if and only if j is even.

(ii) c [equivalent to] 3, 11(mod 16) if and only if j is odd.

Proof. Theorem 1 is correct for c [equivalent to] 7(mod 16) and for c [equivalent to] 11(mod16) in part (1) and in part (2) of Theorem 1, respectively. Therefore, we only prove for c [equivalent to] 15(mod16) and c [equivalent to] 3(mod 16).

Let c [equivalent to] 15(mod 16). The solution [A.sup.2.sub.j-1] = [B.sup.2.sub.j-1]D = [(-1).sup.j]8 exists if and only if

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Since D = 4c and c is odd, there exists an integer k such that c = 16k + 15. If we calculate (4j(c - 1) + [c.sup.2] - 1)/8 with respec to the values of c except for j, then we have

(4j (c - 1) + [c.sup.2] - 1)/8 = j (8k + 7) + (32[k.sup.2] + 60k + 28).

Now we assume j is an odd integer. Then (4j(c - 1) + [c.sup.2] - 1)/8 is odd. Therefore, from equation (1), we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

This is a contradiction and so j is an even integer for c [equivalent to] 15(mod 16).

Conversely, suppose that j is an even integer. Then there exists an integer m such that j = 2m. From equation (1), we get for integers k

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and so (8m(c - 1) + [c.sup.2] - 1)|8 = 2k. Thus, we have 2[c.sup.2] - 2 [equivalent to] 0(mod16) and so c [equivalent to] 15(mod16).

Proof of (ii) is the analogue of (i).

Now we describe a procedure to calculate j with value of c.

1. P[0] = 0 : Q[0] = 1 : A[0] = 0 : A[1] = 1 : B[0] = 1 : B[1] = 0

2. For k = 0, ..., max d

3. c [left arrow] 16 * k + 15

4. D [left arrow] 4 * c

5. q[0] [left arrow] int(sqrt(D))

6. For n = 1, [left arrow], max d

7. P[n] [left arrow] q[n - 1] * Q[n - 1] - P[n - 1]

8. Q[n] [left arrow] (D - P[n] * P[n])/Q[n - 1]

9. q[n] [left arrow] int(P[n]+sqrt(float D)/Q[n]

10. For j = 1, ..., n

11. if q[n] equal 2*q[0] and q[j] equal q[j + n]

12. period [left arrow] period +1

13. End of For

14. For i = 1, ..., n

15. if (Q[??]n/2[??] equal 4) compute Q[i]

16. if (Q[i] equal 8) Display i

17. End of For

18. End of For

19. End of For

Using above algorithms, we seek some values of j on computer and get many counterexamples to Theorem 1.

Example 1. Let we take D = 204 = 4.51 where c = 51 [equivalent to] 3 + 316 [equivalent to] 3(mod 16). Then we have, l([square root of D]) = l = 8, [Q.sub.l/2] = 4 and [Q.sub.j] = 8. But, this case holds only for odd numbers j = 1 and j = 7.

Example 2. Now let we take D = 508 = 4127 where c = 127 [equivalent to] 15 + 716 [equivalent to] 15(mod 16). Then we have, l([square root of D]) = l = 32, [Q.sub.l/2] = 4 and [Q.sub.j] = 8. But, this case holds only for even numbers j = 6 and j = 26.

References

[1] R. A. Mollin, Quadratics, CRS Press Series on Discrete Mathematics and its Applications, CRS Press, Boca Raton, 1996.

[2] R. A. Mollin, Generalized Lagrange criteria for certain quadratic Diophantine equations, New York J. Mat., 11(2005), 539-545.

Ozen Ozer ([dagger]), Fitnat Karaali Telci ([double dagger]) and Aydin Carus #

([dagger])([double dagger]) Department of Mathematics, Faculty of Science and Arts, Trakya University, Edirne, 22030, Turkey

# Computer Engineering Department, Trakya University, Edirne, 22030, Turkey

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