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19Вестник Сыктывкарского университета.
С ер Л. Вып. 13.2011
УДК 511.92
ON THE DIOPHANTINE EQUATION x2 - dy2 =
A. Grytczuk
In this Note we remark that there is some duality connected with the problem of solvability of the Diophantine equation (*) x2 dy2 = zn.
Namely, we prove that the equation (*) has no solution in positive integers x,y for every pime z = q* generated by an arithmetic progression and for every odd positive integer n if d is squarefree positive integer such that p \ d, where p is an odd prime. Keywords: solvability of the Diophantine equation.
1. Introduction. In 1770 Euler obtained integral solutions of the Diophantine equation
(1) ax2 — dy2 = z3.
Denoting by A, D the square roots of a and d ,respectively and assuming that
(2) Ax + Dy — (Au + Dvf
and replacing D by — D for the like equation we obtain the following formulas for the integer solutions of the equation (1):
(3) x — и (au2 + 3dv2), у — v (3au2 + dv2), z — au2 — dv2.
Euler remarked also that this method is fals to give integer solution with у — 1, when a — 2 and d — 5.Indeed, in this case the equation (1) reduces to the form:
(4) 2x2-5 = Z3,
but the formulas (3) we can't obtained the solution x — 4, z — 3 of the equation (4).
In 1769 Lagrange extended Euler's method by the following way; let the equation
© Grytczuk A., 2011.
(5) e-dr? = {ii + Drj){(i-Dri)
for d — D2 has the property that its product by u2 — dv2 is equal to x2 — dy2, where
(6) x + Dy = (f + Drj) (u + Dv),
whence
(7) x — dr/v, y = + r]u.
Putting ^ — u^r] — v and concluding that x2 — dy2 — z2 holds if x — u2 dv2, y — 2uv, z — u2 — dv2 then the factors in the second member of (6) are equal.
Next, we observe that these values of x and y are news values of £ and
m
(8) f = u2 + dv2, r] = 2uv, ^ + Dr)= (u + Dv)2 ,
and consequently we obtain that the Diophantine equation (1) has the solutions given by the formulas (3) for a = 1.
A repetition of this process leads to certain integer solutions of the Diophantine equation:
(*) x2 - dy2 = zn,
but this method rarely gives all integer solutions of (*) (Cf.[3]).Some further investigations concerning solvability of the Diophantine equation (*) are given by Ward [4], Czech [1] and Czech and Wieczorkiewicz [2].
In this paper we note that there is some duality connected with the problem of solvability of the Diophantine equation (*).
Namely,we prove, in contrast to the fact that the equation (*) has infinitely many solutions in positive integers x, y, z\ in general, that for some fixed squarefree positive integer d and prime p such that p \ d
there are infinitely many primes g*such that for every z = q* and every odd natural number n > 1, the Diophantine equation (*) has no solutions in integers x, y.The following theorem is true:
Theorem. Let p be an odd prime such that p | d , where d is a squarefree positive integer. Then for every prime q* = z from the arithmetic progression of the form; Spm+pjo+r, with pjo+r = 5 ( mod 8)
where = — 1 and every odd positive integer n, the Diophantine
equation (*) has no solutions in integers x, y.
2. Proof of the Theorem
Let p | d , where p is an odd prime and let r be quadratic non-residues
On the Diophantine equation x2 — dy2 = zn 39
for p, so = —1. it is easy to see that the numbers of the form: pj + r give distinct residues mod 8. Hence, for some j = jo, we have
(2.1) pj0 + r = 5 ( mod 8).
Now, we can consider the positive integers am of the following form:
(2.2) am= p (8m + j0) + r = 8pm + pj0 + r.
We oserve that the greatest common divisor of the numbers 8p and pjo + r is equal to one, so (8p.pjo + r) = 1.
Indeed, suppose that (8p,pjo + r) = k > l.Then there is a prime q such that q | k. Hence, from the property of the greatest common divisor and divisibility relation ,we get
(2.3) q\8p, q\ pj0 + r.
From (2.3) we obtain that q = p and q \ r, so p \ r,so is impossible, because (^j = —1.
Since (8p,pjo + r) = 1, then by Dirichlet theorem on arithmetic progressions it follows that the arithmetic progression given by (2.2) contains infinitely many primes.
Let for some positive integer m = mo the number anio generated by arithmetic progresson (2.2) is a prime number, so amo = g*.Then by (2.1) and (2.2) it follows that
(2.4) q* = 5 ( mod 8).
By the assumption of the Theorem and well-known properties of Legendre's symbol it follows that
(2.5) (f) =
Suppose that the Diophantine equation (*) has a solution in integers x, y and z = q* for some odd positive integer n.Hence,we have
(2.6) rr2 - dy2 = {q*)n ,
where p \ d for some odd prime p.
From (2.6) we obtain that
(2.7) x2 = (q*)n ( mod d) .
Since p | d then by (2.7) it follows that (q*)n is a quadratic residues mod p,so we have
(2-8) (0£)=+l.
/ 8pm+pjo+r \ _ (r j _
V P J ~ \PJ ~
From (2.5) and the assumption that n = 2k + 1 and well-known properties of the Legendre symbol ,we obtain
<2-s> = (f)" = (f)" (i) = (+»(-Ц —
We see that the equality (2.9) contrary to the equality (2.8) and the proof of the Theorem is complete.■
From the Theorem immediately follows of the following Corollary:
Corollary. There are infinitely many primes g* = 5 ( mod 8) such that each of them can't be representable by the quadratic form x2 — dy2 with some squarefree positive integer d.
References
1. J.Czech, On the equation x2 - Dy2 = zk with D = 2, 3, 5, 7,11,13// Fund. Approx. Comment. Math. 16(1988j, 77-79.
2. J.Czech and J.Wieczorkiewicz, An application of matrices to parametrization of the equation 3x2 — 2y2 — zh// Discuss. Math. 8(1986), 45-52.
3. L.E.Dickson, Introduction to the Theory of Numbers. Dower Publ. Inc. New York, 1957.
4. M.Ward, The Diophantine equation x2 — dy2 = zM// Trans. Amer. Math. Soc. 38 (1935), 447-457.
Faculty of Mathematics, Computer Science and Econometrics University of Zielona
Góra, ul.Prof.Szafrana 4a,65-516 Zielona Góra, Poland and
Western Higher School of Marketing and Internationale Finances pl.Slowianski 9,
65-069 Zielona Góra, Poland
e-mail: A.Grytczuk@wmie.uz.zgora.pl or
e-mail: algrytczuk@onet.pl Поступила 07.02.2011
