complicated Diophantine equations. I will demonstrate one approach to solve such equations under natural conditions coming from

the combinatorial nature of the problem, in particular, solving $ - 16384 k^{12} v + 65536 k^{12} + 98304 k^{11} v^2 - 393216 k^{11} v - 253952 k^{10} v^3$ $+ 786432 k^{10} v^2 + 1744896 k^{10} v - 3309568 k^{10} + 368640 k^9 v^4 - 327680 k^9 v^3 - 8724480 k^9 v^2 + 16547840 k^9 v -328320 k^8 v^5$ $- 1102464 k^8 v^4 + 17194752 k^8 v^3 - 21567744 k^8 v^2 - 49810560 k^8 v + 62323584 k^8 + 182784 k^7 v^6 + 2050560 k^7 v^5 -16432128 k^7 v^4$ $- 13016064 k^7 v^3 + 199242240 k^7 v^2 - 249294336 k^7 v - 61184 k^6 v^7 - 1642240 k^6 v^6 + 6536960 k^6 v^5 + 58253568 k^6 v^4$ $- 293538048 k^6 v^3 + 209662720 k^6 v^2 + 511604992 k^6 v - 488998144 k^6 + 10752 k^5 v^8 + 698880 k^5 v^7 + 1258752 k^5 v^6 - 59703552 k^5 v^5$ $+ 183266304 k^5 v^4 + 243542016 k^5 v^3 - 1534814976 k^5 v^2 + 1466994432 k^5 v - 640 k^4 v^9 - 143664 k^4 v^8 - 2296192 k^4 v^7$ $+27050224 k^4 v^6 - 7038496 k^4 v^5 - 582955856 k^4 v^4 + 1856597696 k^4 v^3 - 1428764528 k^4 v^2 - 1015706784 k^4 v + 974873344 k^4$ $+ 7520 k^3 v^9 + 772608 k^3 v^8 - 2875616 k^3 v^7 - 58917568 k^3 v^6 + 469164960 k^3 v^5 - 1155170432 k^3 v^4 + 412538336 k^3 v^3$ $+ 2031413568 k^3 v^2 - 1949746688 k^3 v + 336 k^2 v^{10} - 52816 k^2 v^9 - 1582560 k^2 v^8 + 27560816 k^2 v^7 - 127930016 k^2 v^6 + 28759472 k^2 v^5$ $+ 1497511456 k^2 v^4 - 4944873072 k^2 v^3 + 6922441360 k^2 v^2 - 4733985888 k^2 v + 1506333312 k^2 - 2352 k v^{10} + 203472 k v^9$ $- 764688 k v^8 - 24513072k v^7 + 293023248 k v^6 - 1459281552 k v^5 + 3929166288 k v^4 - 5947568016 k v^3 + 4733985888 k v^2$ $- 1506333312 k v + 45 v^{11} + 972 v^{10} - 191952 v^9 + 2961396 v^8 - 14780538 v^7 - 18769932 v^6 + 544096980 v^5 - 2755473732 v^4$ $+ 7281931941 v^3 - 11097146016 v^2 + 9310949028 v - 3408102864 = 0$.