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| ON THE DIOPHANTINE EQUATION X4-Dy2=1(II) |
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Citation： |
ko chao,sun chi.ON THE DIOPHANTINE EQUATION X4-Dy2=1(II)[J].Chinese Annals of Mathematics B,1980,1(1):83~89 |
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Net amount： 744 |
Authors： |
ko chao; sun chi |
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| Abstract： |
For the Diophantine equation
x^4 — Dy^2 = 1 (1)
where D>0 and is not a perfect square, we prove the following theorems in this paper.
Theorem 1. If D\[{\not \equiv }\]7 (mod 8)，D=p1p2...ps，s≥2，where pi（i = 1，…，s) are distincyt primes，p1≡1(mod 4) such that either 2p1=a^2+b^2，а≡\[ \pm \]3(mod 8)，b三\[ \pm \]3(mod 8) or there is a j(2≤j≤s), for which Legendre
symbal \[\left( {\frac{{{p_j}}}{{{p_1}}}} \right) = - 1\],and pi≡7(mod8) (i=2,..., s) or pi≡3(mod 8) (i=2,..., s), then (1) has no solutions in positive integer x,y.
Theorem 2. If D=p1...ps,s≥2, where pi（i = 1，…，s) are distinct primes, and pi≡3(mod 4)（i = 1，…，s), then (1) has no solutions in positive integer x, y.
Theorem 3. The equation (1) with D=2p1...ps has no solutions in positive
integer x, y, if
(1) p1≡(mod 4), pi≡7(mod 8) (i = 2, ???, s), snch that either 2p1 = a^2+b^2
a≡\[ \pm \]3(mod 8)，b≡\[ \pm \]3(mod 8)or there is a j (2≤j≤s)，for which \[\left( {\frac{{{p_j}}}{{{p_1}}}} \right) = - 1\];
or
(2) p1≡5(mod8),pi≡3(mod8) (i = 2,..., s)；
or
⑶p1≡5(mod8)，pi≡7(mod 8) (i=2，…，s).
Corollary of theorem 3. If D = 2pq, p≡5(mod 8), q≡3(mod 4), where p, q
are distinct primes, then (1) has no solutions in positive integer x, y.
Theorem 4. If D=2p1...ps, pi≡3(mod 4)(0 = 1,...,s), then (1) has no solutions In positive integer x, y. |
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