This largest solution to this gorgeous equation is the first local extremum on a function related to the Fibonacci sequence:

$$x^2 \cdot \sin \left(\frac{2\pi}{x+1} \right) \cdot \left(3+2 \cos \left(\frac{2\pi}{x} \right) \right) = (x+1)^2 \cdot \sin \left(\frac{2\pi}{x} \right) \cdot \left(3+2 \cos \left(\frac{2\pi}{x+1} \right) \right)$$

This is as simplified as I could get it. The largest solution to this equation is around $x = 2.1392.$

It appears there is no closed-form solution for this; is there any way to prove if the solution is algebraic or transcendental?

P.S. Can anyone approximate this constant to more decimal places?

This largest solution to this gorgeous equation is the first local extremum on a function related to the Fibonacci sequence:

$$x^2 \cdot \sin \left(\frac{2\pi}{x+1} \right) \cdot \left(3+2 \cos \left(\frac{2\pi}{x} \right) \right) = (x+1)^2 \cdot \sin \left(\frac{2\pi}{x} \right) \cdot \left(3+2 \cos \left(\frac{2\pi}{x+1} \right) \right)$$

This is as simplified as I could get it. The largest solution to this equation is around $x = 2.1392.$

It appears there is no closed-form solution for this; is there any way to prove if the solution is algebraic or transcendental?

P.S. Can anyone approximate this constant to more decimal places? ANSWERED

This largest solution to this gorgeous equation is the first local extremum on a function related to the Fibonacci sequence:

$$x^2 \cdot \sin \left(\frac{2\pi}{x+1} \right) \cdot \left(3+2 \cos \left(\frac{2\pi}{x} \right) \right) = (x+1)^2 \cdot \sin \left(\frac{2\pi}{x} \right) \cdot \left(3+2 \cos \left(\frac{2\pi}{x+1} \right) \right)$$

This is as simplified as I could get it. The largest solution to this equation is around $x = 2.1392.$

It appears there is no closed-form solution for this; is there any way to prove if the solution is algebraic or transcendental?

This largest solution to this gorgeous equation is the first local extremum on a function related to the Fibonacci sequence:

$$x^2 \cdot \sin \left(\frac{2\pi}{x+1} \right) \cdot \left(3+2 \cos \left(\frac{2\pi}{x} \right) \right) = (x+1)^2 \cdot \sin \left(\frac{2\pi}{x} \right) \cdot \left(3+2 \cos \left(\frac{2\pi}{x+1} \right) \right)$$

This is as simplified as I could get it. The largest solution to this equation is around $x = 2.1392.$

It appears there is no closed-form solution for this; is there any way to prove if the solution is algebraic or transcendental?

P.S. Can anyone approximate this constant to more decimal places?