1979 USAMO Problems/Problem 1
Determine all non-negative integral solutions if any, apart from permutations, of the Diophantine Equation .
Recall that for all integers . Thus the sum we have is anything from 0 to 14 modulo 16. But , and thus there are no integral solutions to the given Diophantine equation.
In base , this equation would look like:
We notice that the unit digits of the LHS of this equation should equal to . In base , the only unit digits of fourth powers are and . Thus, the maximum of these terms is 14 or . Since is less than , there are no integral solutions for this equation.
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