mixedmath

A couple of years ago, I wrote a paper Congruent number triangles with the same hypotenuse with an appendix by Brendan Hassett. This paper details a computational experiment concerning triangles, related to the main term in my earlier paper A shifted sum for the congruent number problem from 2018.

For no deep reason, I hadn't given either of these papers a dedicated discussion page here. But to my surprise, I've received many emails about them asking some questions.

This post serves as the discussion page for both papers. Below, I briefly describe what these two papers prove and how they relate to each other.

If I get further questions or comments, I'll add them to this post.

A shifted sum for the congruent number problem

With my frequent collaborators Thomas Hulse, Chan Ieong Kuan, and Alexander Walker, I wrote a paper describing an explicit relationship between a certain shifted convolution sum and asymptotics for congruent numbers.^{1 1}Written in 2018, published in the Ramanujan Journal in 2020.

We call a number congruent if it appears as the area of a right triangle with rational sidelengths. For example, the triangle with sidelengths $(3, 4, 5)$ is a right triangle with area $6$, and thus $6$ is a congruent number. The decision problem for congruent numbers is famously hard.^{2 2}Here, we mean an algorithm that is guaranteed to terminate. Thus "try all triangles" isn't an acceptable algorithm.

Question

Given $q \in \mathbb{Q}$, does there exist an algorithm that determines if $q$ is congruent?

Scaling a rational right triangle with area $A$ by a rational $\lambda$ gives another rational right triangle with area $\lambda^2 A$. For example, we see that $3/2 = 6 \cdot 2^{-2}$ is congruent because we can scale the $(3, 4, 5)$ triangle by $2^{-1}$, giving $(3/2, 2, 5/2)$. Thus, after scaling, we lose no generality by only considering whether a given integer $t$ is congruent. (That is, nothing is lost in thinking only of integral areas instead of any potential rational area).

Further, scaling allows us to consider only squarefree integer areas $t$, as we can scale away the square factor. This leads to the following (equivalent) decision problem.

Equivalent Question

Given squarefree $t \in \mathbb{Z}$, does there exist an algorithm that determines if $t$ is congruent?

Thought of differently, the triangles $\lambda (3, 4, 5)$ all correspond to the "same" triangle and congruent number. Given any right triangle $(a, b, c)$ with rational sidelengths, there is a unique $\lambda$ such that $(\lambda a, \lambda b, \lambda c)$ are all integers and $\gcd(\lambda a, \lambda b, \lambda c) = 1$. We think of this right triangle as the primitive right triangle, and it is how we think about the whole class of triangle $(\lambda a, \lambda b, \lambda c)$. The area of a primitive right triangle will be a squarefree integer, and in fact these two descriptions ultimately offer two descriptions of the same class of fundamental elements.

There is a famous conjectural resolution due to Tunnell, which asserts that the answer is YES if we assume the Birch and Swinnerton-Dyer Conjecture.

Let $\tau(n) = 1$ if $n$ is a square and $0$ if $n$ is not a square. The primary theorem of our paper A shifted sum for the Congruent number problem is the following.

In particular, $C_t \neq 0$ if and only if $t$ is a squarefree congruent number. If we could bound $C_t$ from below when it was nonzero, and if we could bound the implicit constant in the $O_t(\cdot)$ error term, then we would solve the congruent number decision problem.

But we don't, and these are very hard. One can show that the set of hypotenuses of primitive triangles corresponding to a given congruent $t$ are logarithmically sparse (i.e. there are only powers of $\log(X)$ many up to size $X$). This means that the sum $C_t$ converges extremely quickly and is morally approximately equal to the first couple of summands. Thus approximating $C_t$ amounts to estimating the smallest hypotenuse of a potential congruent triangle for $t$.

There is a well-known correspondence between congruent number triangles $(a, b, c)$ with area $t$ and three term arithmetic progressions of rational squares with common difference $t$. Indeed, this paper was a warmup for our later paper on counting three term arithmetic progressions, which uses similar technology. There is also a well-known correspondence between three-term arithmetic progressions of rational squares with common difference $t$ and rational points on the elliptic curve $E_t$.

The whole game of this paper is tracking through these correspondences closely. In particular, the term \begin{equation*} \tau(m + n) \tau(m - n) \tau(m) \tau(tn) \end{equation*} detects three term arithmetic progressions of rational squares with common difference $t$.

The smallest hypotenuse $h_i$ or a primitive integer right triangle showing that the squarefree integer $t$ is congruent is then very closely related to upper bounds on the height of the first rational point on the elliptic curve $E_t$. This is hard to bound!

I suspect that with a lot of work, it actually would be possible to handle the implicit constant (depending on $t$) in the error term $O_t(\cdot)$. But without being able to estimate the complexity of the smallest rational points on elliptic curves, this wouldn't suffice.

Aside

We had an idea for how to approach the congruent number problem through this perspective, studying the natural multiple Dirichlet series

\begin{equation*} D(s, w) = \sum_{m, n \geq 1} \frac{\tau(m + n) \tau(m - n) \tau(m) \tau(nt)}{m^s n^w}. \end{equation*}

Of note is that $D(s, s)$ would have a pole at $s = 1/2$ if and only if $t$ was congruent. So the decision problem would reduce to making sense of this multiple Dirichlet series. Presumably, having a pole would reduce precisely to behavior of a particular elliptic curve. In fact, Tunnell proved his conjectured algorithm for congruent number detection by constructing particular products of four theta functions and studying their behavior; the construction of $D(s, w)$ would also involve products of four theta functions. Perhaps they're deeply connected in ways I don't understand.

Actually making sense of this proved very hard. We don't know how to proceed and are not actively pursuing this. Our paper on arithmetic progressions can be thought of as a successful stepping stone in that direction though.

Congruent Number Triangles with the Same Hypotenuse

I wrote Congruent number triangles with the same hypotenuse later.^{3 3}Written in 2020. Published in 2021 in Simon's Symposia.

The main question had to do with a potential complication in applying the shifted-sum asymptotic above to actually determine if a number is congruent.

Heuristically, it seems that $C_t$ is very well-approximated by exactly the reciprocal of the smallest hypotenuse. (This refers to the previous paper, described above). All other hypotenuses of relevant primitive congruent triangles are much, much smaller.

It would be a bit complicated if there were several hypotenuses all approximately as small as the smallest. For deep reasons, there can't be more than a fixed small multiple of the rank of the corresponding elliptic curve $E_t$. In the worst case, there would be several different triangles that all had the exact same hypotenuse.

In this new paper, we set out to investigate if this worst case can ever happen.

Question: can two dissimilar primitive right integer triangles have the same hypotenuse and the same squarefree part of their area?

I conjecture that this cannot happen. In an appendix to our paper, Brendan Hassett explains geometrically why this is probably an intractable problem to resolve. This has to do with the variety defined by the equations \begin{align*} s^2 + t^2 &= S^2 + T^2 \\ u^2 st(s^2 - t^2) &= v^2 ST (S^2 - T^2), \end{align*} considered in $\mathbb{P}^3_{[s, t, S, T]} \times \mathbb{P}^1_{[u, v]}$. (The point is that this admits a resolution of singularities that is of general type and simply connected. It's too complicated!).

Observe that each adjective is essential. The triangles $(3, 4, 5)$ and $(4, 3, 5)$ obviously have the same hypotenuse and area, but are similar.

Primitivity also matters. Otherwise, one could choose any two dissimilar triangles with the same squarefree part of the area and scale them to produce a counterexample.^{4 4}And in fact if there is one triangle, there are actually infinitely many dissimilar triangles with the same squarefree part of the area. They are logarithmically dense. For example, the triangles $(20, 21, 29)$ and $(12, 35, 37)$ both have area $210$, but different hypotenuseses. However we can scale the first by $37$ and and second by $29$ to get the triangles $(740, 777, 1073)$ and $(348, 1015, 1073)$. These have the same hypotenuse. The first has area $210 \cdot 37^2$ and the second has area $210 \cdot 29^2$, and thus correspond to the same congruent number. But neither is primitive, and so this doesn't give a counterexample.

(This gives an infinite family of near-counterexamples in a sense, but none of them work).

The rest of the paper goes towards an efficient experimental investigation of this problem. I investigate two potential sources of problems: small congruent numbers (the law of small numbers suggests that distributions of small numbers should have more coincidental overlaps; this is even more true given that everything is only logarithmically dense here) and congruent number elliptic curves with high rank. Thus I investigated all congruent numbers up to $1000$ and every known congruent number elliptic curve found of rank $\geq 6$.^{5 5} I have separately and since done the same for all congruent numbers up to $2000$ and every congruent number elliptic curve of rank $4$ and $5$.

From a purely statistical standpoint, if the first several hypotenuses don't collide, then we shouldn't expect any to collide. The points are too sparse for random collisions. I limited my experiments by size: I considered all triangles whose side-lengths took less than one megabyte each to store (approximately). That is, I considered all triangles with sidelengths up to $2^{1000000}$, approximately.^{6 6}But to reiterate, the triangles themselves are logarithmically dense, i.e. they grow exponentially. This doesn't actually lead to that many triangles. It does lead to some very large triangles though.

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