First, a bit of a digression regarding the writing of this series and this blog. In fact, the content of this blog represents the research results of the author around this period. That is to say, the fact that I am writing this article now indicates that I am currently studying this problem. As for whether the subsequent research will yield results, or what kind of results they might be, it is completely unknown. Therefore, when I write this article, I am not certain what the next article will be about. I will group similar topics into the same series. Regardless, these articles may not follow conventional pedagogical or learning logic, and some content may differ significantly from mainstream thinking and methods. I ask for the readers’ understanding and hope for your continued support!
In the previous article, we discussed the tangent method for finding rational points on quadratic and cubic curves. The tangent method is very successful in finding rational points on curves of degree no higher than three, but is there a similar method for higher-degree curves? In other words, is there a possibility for generalization? Let us review the reason why the tangent method works from a purely algebraic perspective. The tangent method (and more generally, the secant method) works primarily because if a cubic equation with rational coefficients has two rational roots, then the third root must also be rational. If there is only one known rational root, one can make two roots coincide at that known root, thereby turning the secant line into a tangent line.
Next, we consider quartic curves. It should be noted that for a quartic equation with rational coefficients, we must have three rational roots prepared in advance to ensure that the fourth root is rational. Three rational roots correspond to three rational points. We know that two points determine a line; therefore, three points generally determine a parabola (or other types of curves, provided such a curve has three free parameters to "interpolate" these three points. The fitting curve must also satisfy the condition that substituting a rational independent variable must yield a rational dependent variable. For example, in y=x^2, substituting a rational x yields a rational y; but for y^2=x^2+1, substituting a rational x does not necessarily yield a rational y. This latter type of curve is unsuitable). What if only one rational point is known? That is equivalent to making all three points coincide! In this case, the parabolic secant becomes a parabolic tangent, which is equivalent to the second-order Taylor expansion of the original curve at that rational point! That is to say, we still consider tangents, but not just "tangent lines"—we also consider more precise "tangent curves."
Here, we take the problem of finding rational points for the Diophantine equation 2x^4=y^2+1 as an example to demonstrate the above technique. It is known that there is an obvious rational point (1,1). Performing a Taylor expansion at this point, we calculate: \left\{\begin{aligned} &8x^3=2yy',\quad y'=\frac{4x^3}{y}=4\\ &12x^2=\left(y'\right)^2+y y'',\quad y''=\frac{12x^2-(y')^2}{y}=-4 \end{aligned}\right. Thus, the second-order Taylor expansion is: y=\frac{1}{2}(-4)(x-1)^2+4(x-1)+1=-2x^2+8x-5 The intersection points of this curve with the original curve are found by: 2x^4=(-2x^2+8x-5)^2+1 Which simplifies to x^4-16x^3+42x^2-40x+13=0. By our construction, we know this quartic equation has a triple root at x=1. Therefore, the final root must be x=13, from which we can calculate y=-239. This means (13, -239) is another integer solution. (Coincidentally, this Diophantine equation has only these two positive integer solutions.)
This example might make our generalization look successful, but it is perhaps better described as an illustration of why the tangent method fails for higher-degree curves. First, we know the original curve is of degree four, and we must use at least a parabola (i.e., a curve of the type y=ax^2+bx+c) to complete the "interpolation." After substituting this into the original curve, to ensure the resulting equation remains a quartic equation, the degree of y cannot exceed 2! This means that if we use the parabolic secant or parabolic tangent method, it is only applicable to specific quartic Diophantine equations such as 2x^4=y^2+1 or 3x^4=y^2+xy+1. This is very restrictive. If we consider quintic equations, this technique no longer works because we would need four known rational points, and fitting four rational points requires a cubic curve. Even when squared, a cubic curve becomes degree six, which exceeds the original degree of the equation. It seems that the parabolic secant and parabolic tangent methods reach their limit with quartic Diophantine equations.
Of course, we look forward to the emergence of more techniques for finding rational points with clear geometric significance.
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