Note that there are two unknowns in the equation. With no constraint being imposed there are many solution. If (x,y) is a pair of integer the trivial solutions: (x^y)-y=77 =78¹-1 --> (x,y) --> (78,1) =1^(-76)-(-76) --> (1,-76) Other solution: (x^y)-y=81-4 (x^y)-y=3⁴-4 --> (x,y)=(3,4) by comparing the structure. The general approach is [x^(½y)]²-(y^½)²=77 [x^(½y)+(y^½)][x^(½y)-(y^½)]=7×11 x^(½y)+y^½=11 x^(½y)-y^½=7 Half of the sum is x^(½y)=9 Half of the difference is y^½=2 --> y=4. Hence x=3 --> (x,y)=(3,4 ) Thus (x,y)={(-1,76),(3,4),(78,1)}
There are infinite solutions. (x, y) = (78, 1), (√79, 2), (80^(1/3), 3), (3, 4), (82^(1/5), 5), (83^(1/6), 6), ・・・ [ (-3, 4) is also one of the solutions. ]
