I am not genius in mathematics but i am capable of providing alternative solution for some problems in math using simple computer algorithm. Not to offend you, but the days of Fermat and Euler had past already, we are now in the age of Bill Gates and late Steve Jobs. :)

for a in range(-100,100):
for b in range(-100,100):
for c in range(-100,100):
if a*((b-c)**4)+b*((c-a)**4)+c*((a-b)**4)==836:
print a, b, c

yielded the results [(-2 ,1, 3),(-2, 3 ,1),(1 ,-2 ,3),(1 ,3, -2),(3, -2 ,1),(3, 1,-2)]
and the sum of all possible values of c is 4.
(Note: I ran the code in python 2.7 interpreter)

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I am not genius in mathematics but i am capable of providing alternative solution for some problems in math using simple computer algorithm. Not to offend you, but the days of Fermat and Euler had past already, we are now in the age of Bill Gates and late Steve Jobs. :)

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