Article regarding the Congruent Number Problem

Question

I am reading an article "Congruent Numbers and Elliptic curves" by Jasbir S.Cahal https://www.jstor.org/stable/27641916

I had difficulty understanding some details in this proof (theorem 3). (I added the related files as well).

1. In theorem 3:

• Why is it easy to see that $y>1$?

• What does he mean by the divisibility argument? And how to accomplish it?

2. In Corollary 1: What does he mean by "via addition of rational points on the elliptic curve defined by equation 4"?

Thanks a lot.

Answer 1

Why is it easy to see that $y>1$?

For $y=1$, one can see that $a\not\equiv 0\pmod 4$ since $a$ is a square-free integer.

In the following, note that $(\text{square})\equiv 0$ or $1\pmod 4$ which can be proven by noting that $(\text{even})^2\equiv 0\pmod 4$ and $(\text{odd})^2\equiv 1\pmod 4$.

• Suppose that $a\equiv 1\pmod 4$. Then, one has $z^2-x^2\equiv 1\pmod 4$. Suppose here that $x^2\equiv 1\pmod 4$. Then, one gets $z^2\equiv 2\pmod 4$, but there is no such $z$. So, one obtains $x^2\equiv 0\pmod 4$ which implies $t^2\equiv 3\pmod 4$, but there is no such $t$.

• Suppose that $a\equiv 2\pmod 4$. Then, one has $z^2-x^2\equiv 2\pmod 4$. If $z^2\equiv x^2\pmod 4$, then $z^2-x^2\equiv 0\pmod 4$. If $z^2\equiv 0\pmod 4$ and $x^2\equiv 1\pmod 4$, then $z^2-x^2\equiv 3\pmod 4$. If $z^2\equiv 1\pmod 4$ and $x^2\equiv 0\pmod 4$, then $z^2-x^2\equiv 1\pmod 4$. Therefore, there are no $z,x$ such that $z^2-x^2\equiv 2\pmod 4$.

• Suppose that $a\equiv 3\pmod 4$. Then, one has $z^2-x^2\equiv 3\pmod 4$. Suppose here that $x^2\equiv 0\pmod 4$. Then, one gets $z^2\equiv 3\pmod 4$, but there is no such $z$. So, one obtains $x^2\equiv 1\pmod 4$ which implies $t^2\equiv 2\pmod 4$, but there is no such $t$.

Therefore, one can say that $y\not=1$.

What does he mean by the divisibility argument? And how to accomplish it?

If I'm not mistaken, I think that it is false that if $u^2=s(s+at^2)(s-at^2)$ with $\gcd(s,t)=\gcd(u,t)=1$, then $s,s+at^2$, and $s-at^2$ are mutually coprime. Take $a=6,s=12,t=1$ and $u=36$.

In Q2, I think you're right, but then why they wrote each of them as a square?

It is true that if $s,s+at^2$ and $s-at^2$ are mutually coprime, then it follows from $u^2=s(s+at^2)(s-at^2)$ that each of $s,s+at^2$ and $s-at^2$ is a perfect square. However, my counterexample shows that it is not always true that $s,s+at^2$, and $s-at^2$ are mutually coprime. So, if I'm not mistaken, I think that the claim that each of $s,s+at^2$ and $s-at^2$ is a perfect square is not true.

What does he mean by "via addition of rational points on the elliptic curve defined by equation 4"?

I think that you can find explanations about the addition on the elliptic curve on page 311 ~ 313.

I ask about the concept of what how they are trying to obtain rational right traingle from a give one.

I think that you can find explanations about how to obtain rational right triangle from a given one in the proof of Theorem 1.