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Number Theory Level 3

How many sets of solutions do the following equation have？

x 3 + y 4 + z 5 = w 7 x^{3}+y^{4}+z^{5}=w^{7}

note:x,y,z,t are all positive integer

6 1 13 ∞

Transformation Tang by transformation tang , 17, China

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1 solution

2 12 A + 2 12 A + 2 12 A + 1 = 2 12 A + 2 2^{12A}+2^{12A}+2^{12A+1}=2^{12A+2}

Thus,If 3|12A,4|12A,5|12A+1,7|12A+2, then the equation has a solution.

For example ,A=35k+22，k is an positive integer

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