A peculiar Diophantine equation

Question

Solve the following equation in the variables $m\in\{1,2,\dots\}, n\in \{2,3,\dots \}$ and $k$ $\in\{1,\dots \lceil{ \frac{n}{2} } \rceil \}$

$$(m+\sqrt{m^2+4})^2(1-\cos\frac{2\pi}{n}) = 4(1-\cos\frac{2\pi k}{n}).$$

There are solutions $(m, n, k) = (1, 5, 2)$ and $(2, 8, 3)$ but are there any others?

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Disclaimer: I got this equation from the still(?) open question posed on this video.

Answer 1

I would imagine your list is all. I did the easy search I mentioned; if any others in the following list are actual solutions, then $m = \lfloor x \rfloor \; .$ I would guess all the others are false, just lucky approximations. In the first three lines, we see the evident golden ratio and $1 + \sqrt 2.$ If there were other genuine $n,$ I really would expect it to factor as a possible power of two times distinct Fermat primes. That does not seem to work out, I do not see $17$

Fri Jan 26 14:46:14 PST 2018
  n : 5   k :  2  x :  1.618033988749895  y :  1
  n : 5   k :  3  x :  1.618033988749895  y :  1
  n : 8   k :  3  x :  2.414213562373095  y :  1.000000000000001
  n : 558   k :  83  x :  80.01248621068363  y :  0.9990527601478923
  n : 677   k :  104  x :  100.009997880325  y :  0.9998879901098201
  n : 825   k :  181  x :  167.0059888610697  y :  1.000175665093749
  n : 899   k :  440  x :  286.003493087486  y :  0.9990352226702742
  n : 1016   k :  362  x :  291.00343498031  y :  0.9995910693092293
  n : 1192   k :  95  x :  94.01063249462001  y :  0.9995675442229549
  n : 1368   k :  253  x :  239.0041879453035  y :  1.000936466415433
  n : 1371   k :  304  x :  280.00356898271  y :  0.9993278964276524
  n : 1422   k :  135  x :  133.0075143732494  y :  0.9994681079707861
  n : 1525   k :  112  x :  111.0090028305573  y :  0.9993952428222878
  n : 1773   k :  259  x :  250.0039990719964  y :  0.9997839916742329
  n : 1776   k :  124  x :  123.0081300506843  y :  1.000062331894867
  n : 1782   k :  278  x :  267.0037488734972  y :  1.000963277793927
  n : 1891   k :  601  x :  506.0019763456762  y :  1.000034818105543
  n : 1959   k :  659  x :  543.0018409490152  y :  0.9996387043566668
  n : 2080   k :  389  x :  367.0027257683346  y :  1.000364408605766
  n : 2091   k :  327  x :  314.0031857528351  y :  1.000336539231729
  n : 2322   k :  846  x :  673.0014848226627  y :  0.9992878567266738
  n : 2328   k :  1158  x :  741.0013494149186  y :  0.9999182756402187
  n : 2358   k :  558  x :  508.0019685863887  y :  1.000045760805963
  n : 2443   k :  590  x :  535.0018674153106  y :  0.9990706784028501
  n : 3464   k :  650  x :  613.0016316564793  y :  1.000208084099048
  n : 3944   k :  336  x :  332.0030095815791  y :  0.9991901418354138
Fri Jan 26 14:46:15 PST 2018