New Year Countdown - 10
Find the sum of all integral solutions of the equation 2 x 4 + 1 4 0 2 = y 4 .
The answer is 0.
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2 solutions
Did the same! By the way, there are no integral solutions too. So, @Aditya Tiwari, care to change the problem to "all positive integral solutions".
PROOF
Taking m o d 6 both sides,
( y 4 − 2 x 4 ) m o d 6 = 1 4 0 2 m o d 6
y 4 − 2 x 4 = 4 m o d 6
We know that ϕ ( 6 ) = 2 , and so a 4 = 1 o r 0 m o d 6
Hence, we arrive at a conclusion that there are no integral solutions because we will never be able to get 4 on the RHS(can easily be seen).
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well, since the question said all integral solutions i didn't bother solving for the integral solutions, but nice proof hahahaha
Yeah that's a good logical way to solve it.
Upvoted :)
The exponent of both x and y is 4, which is even so if (a,b) is an integral solution of 2x^4+1402=y^4, then (-a,-b) is also an integral solution. If the integral solutions are added, all positive integral solutions will be cancelled out by negative integral solutions. Therefore, the sum of all integral solutions of 2x^4+1402=y^4 is 0