Another Look At Elliptic Curves

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It was on the 24th February 2021 that I made a post title Elliptic Curves and I've not made mention of them since despite their importance in high level mathematics. However, I was reminded of them again by a recent article in Quanta Magazine titled Big Advance on Simple-Sounding Math Problem Was a Century in the Making on the 14th October 2024. The article begins:

One morning last November, the mathematician Hector Pasten(opens a new tab) finally solved the problem that had been dogging him for more than a decade by using a time-tested productivity hack: procrastination.

He was supposed to be writing a final exam for his number theory class at the Pontifical Catholic University of Chile in Santiago. To avoid the task, he started pondering, for the umpteenth time, one of his favorite sequences: 2, 5, 10, 17, 26 and so on, the list of all numbers of the form \(n^2 + 1\) (where \(n\) is a whole number).

Mathematicians have used this sequence for over a century to probe the fraught relationship between addition and multiplication, a tension that lies at the heart of number theory. Fundamental problems about multiplication — about, say, how numbers factor into primes — suddenly become much deeper and more challenging as soon as addition enters the picture. One of math’s biggest open questions, for example, asks whether every even number larger than 2 is the sum of two primes; another asks whether there are infinitely many pairs of primes that differ by only 2, such as 11 and 13.

The \(n^2 + 1\) sequence offers a good starting point for investigating the relationship between addition and multiplication, because it combines one of the simplest types of multiplication (squaring a number) with one of the simplest types of addition (adding 1). That doesn’t mean the sequence itself is simple. Mathematicians still can’t answer elementary questions about it, such as whether it contains infinitely many primes. “It doesn’t take far to get to the boundary of our knowledge,” said Andrew Granville(opens a new tab) of the University of Montreal. When mathematicians do manage to shift this boundary even a little, the techniques they develop often illuminate much broader questions about addition and multiplication.

Pasten was trying to show that the numbers in the sequence must always have at least one prime factor that is fairly large. On the morning when he should have been writing his final exam, he finally succeeded, by figuring out how to embed information about the prime factors of \(n^2 + 1\) in the structure of an equation called an elliptic curve.

Over lunch that day, he described his proof to his wife, the mathematician Natalia Garcia-Fritz(opens a new tab). Given the surprising strength of his result, she “told me that I should probably check this many times,” Pasten said. “That afternoon I did so, and the theorems were still there.”

The elliptic curve that Pasten was focused on is \(y^2 = x^3 + 3x + 2n \) with the variable \(n\) being any integer. Curves of the form:$$ y^2=x^3+Ax+B$$have a discriminant given by:$$ \Delta=-16(4A^3+27B^2)$$Thus when \(A=3\) and \(B=2n\) we have:$$ \Delta=16 \times 108 (1+n^2)$$It was the \(n^2+1\) factor in the discriminant that caught Pasten's attention. The level of mathematics that he then applied to make his discovery is beyond my comprehension unless I take the Krell Brain Boost but here is an excerpt from the article that explains what he found:

Applying his theory of Shimura curves to this particular elliptic curve, he could show that the product of the exponents of \(n^2 + 1\) must be fairly small. This didn’t necessarily mean that all the exponents must be small, but it gave him enough control over them to be able to bring in Stewart and Yu’s older method from transcendence theory. By using the two techniques together, he was able to prove that the largest prime factor of \(n^2 + 1\) must be at least about \( (\log(\log n))^2\) — the square of the estimate Chowla and Mahler discovered in the 1930s. Pasten’s new growth rate is much higher than the previous record, though mathematicians suspect the true growth rate is higher still.

Figure 1 shows what the curve looks like when \(n=7\):

Figure 1

In my earlier post on Elliptic Curves, I mentioned that SageMath provides a way of finding the integer values of \(x\) and \(y\) that satisfy the equation of elliptic curves of the form \(y^2=x^3+ax+b\), namely:

E=EllipticCurve([a,b])

E.integral_points()

In the case of \(n=7\) there is only the point where the curve \(y^2 = x^3 + 3x + 14 \) crosses the \(x\)-axis and thus \(x=-2\) and \(y=0\). However, other values of \(n\) yield more points. For example, if \(n=11\) we have the curve \(y^2 = x^3 + 3x + 22 \) and the points:$$(2 : \pm \, 6) \text{ and } (6 : \pm \, 16)$$Figure 2 shows the graph of this curve:

Figure 2

So that's just a brief refresher on elliptic curves and a reminder of how important they are in higher mathematics.