# Bounds on largest possible square in sum of two squares

Suppose we are given integers $$k,c$$ such that $$k=1+c^2$$.

Let $$n$$ be an odd integer and suppose that $$k^n=a_i^2+b_i^2$$ for distinct positive integers $$a_i and $$i\le d$$. That is, there are $$d$$ different ways to express $$k^n$$ as a sum of two squares.

For instance, $$(a_1,b_1)=(k^{(n-1)/2},ck^{(n-1)/2})$$ is a valid pair.

What can be said about $$\max b_i$$? Are there good bounds (as a function of $$c,n$$) on the magnitude of largest possible square when writing an integer as a sum of two squares?

Addendum: As mentioned below, we can rephrase this problem in terms of the irrationality measure of $$\arctan(1/c)/\pi$$. I'm having a lot of trouble finding results on irrationality measures of values of inverse trigonometric functions in general, but I could be missing something.

• if $\theta=\arctan 1/c$, we need an odd multiple of $\theta$ which is close to 0 modulo $2\pi$, this may be tricky May 17, 2022 at 15:13

Rather than discuss $$\max b_{i}$$, I'll discuss the equivalent question of bounding $$\min a_{i}$$. The ABC conjecture implies that for all $$\epsilon > 0$$, $$\min a_{i} \gg (c^{2}+1)^{n/2 - 1 - \epsilon}$$. This is because if we have a solution to $$a^{2} + b^{2} = (c^{2}+1)^{n}$$ with $$a \ll (c^{2} + 1)^{n/2 - 1 - \epsilon}$$, set $$A = a^{2}$$, $$B = b^{2}$$ and $$C = (c^{2}+1)^{n}$$. Assume for simplicity that $$\gcd(a,b) = 1$$. (It doesn't change much if $$\gcd(a,b) > 1$$.) Then $$C \ll {\rm rad}(ABC)^{1+\delta}$$ for all $$\delta > 0$$. This gives $$(c^{2}+1)^{n} < (ab(c^{2}+1))^{1+\delta} \ll ((c^{2}+1)^{n/2 - 1 - \epsilon} (c^{2}+1)^{n/2} (c^{2}+1))^{1+\delta} = ((c^{2}+1)^{n-\epsilon})^{1+\delta}$$ which is a contradiction if $$\delta < \frac{\epsilon}{n-\epsilon}$$.
For $$n = 3$$, it is possible to construct a sequence of integers $$c$$ which getse close to this bound. In particular, let $$c_{k} = \frac{(2 + \sqrt{3})^{k} - (2 - \sqrt{3})^{k}}{\sqrt{3}} \quad a_{k} = \frac{(2 + \sqrt{3})^{k} + (2 - \sqrt{3})^{k}}{2}.$$ It is easy to see that $$a_{k}, c_{k} \in \mathbb{Z}$$, $$c_{k}$$ is even, and a somewhat tedious calculation shows that $$(c_{k}^{2} + 1)^{3} = a_{k}^{2} + \left(c_{k}^{3} + \frac{3}{2} c_{k}\right)^{2}.$$ In particular $$\min a_{i} \leq a_{k} \approx \frac{\sqrt{3}}{2} c_{k}$$.
One could hope to generalize this construction by finding $$c_{k}^{2} + 1 = d_{k}^{2} + e_{k}^{2}$$, and $$\frac{e_{k}}{c_{k}} \approx \sin\left(\frac{\pi}{2n}\right)$$. Letting $$\frac{e_{k}}{\sqrt{d_{k}^{2} + e_{k}^{2}}} = \sin(\theta_{k})$$, this makes $$(c_{k} + i)^{n} = (d_{k} + ie_{k})^{n} \approx (c_{k}^{2} + 1)^{n/2} \left(\cos(\theta_{k}) + i \sin(\theta_{k})\right)^{n} = (c^{2} + 1)^{n/2} \left(\cos(n \theta_{k}) + i \sin(n \theta_{k}\right))$$ and $$\sin(n \theta_{k}) \approx \sin(\pi/2) = 1$$. This boils down to finding points on the hyperboloid $$x^{2}+y^{2} = z^{2} + 1$$ that lie close to the line $$y = \sin\left(\frac{\pi}{2n}\right) z$$.