具有四项的指数丢番图方程(Ⅱ)

Exponential Diophantine Equations with Four Terms (II)

本文中，我们给出了丢番图方程的解ｘ，ｙ，ｚ，ｗ的上界，其中ｐ，ｑ是给定的互素的正整数，ａ，ｂ，ｃ，ｄ是给定的适合ａｂｅｄ≠０的整数，此外，我们将指出在具体情形下如何把上界降低到方程允许的实际的解．最后，我们将用这个方法来解方程１９．５ｘ·１７ｙ＝１２．５ｚ＋４１．１７ｗ＋１４， ５． ３ｘ· １３ｙ ＋ ２０＝ ７． ３ｚ ＋ １４． １３ｗ和 １３· ２ｘ＋ ５· ３ｙ＝ ２５． ２ｚ＋ １１． ３ｗ．

In this paper the upper bounds for the solutions x, y, z, w to the diophantine equations apxqy + bpz + cqw + d = 0 and apx + bqy + cpz + dqw = 0 are computed, where p, q are given to be fixed relatively prime positive integers and a, b, c, d integers with abed ≠ 0. Also, we show the upper bounds in a particular case can be reduced to allow the practical solution of the equation. Finally, we use the method to solve the equations 19× 5x17y = 12 × 5z +41 ×17w + 14, 5 × 3x13y + 20 = 7 ×3z + 14× 13w and 13 × 2x + 5× 3y = 25× 2z + 11× 3w.