Question

prove that x^4+y^4=2020 has no integer solution

Answer 1

The right side must be an integer, so it must be 2Z2 , and any prime p cannot divide both x20−y202 and x20+y202(otherwise we would have some prime p dividing both x0 and y0. Then we have
x20−y20=u2
, and
x20+y20=2v2
for some positive integers u,v.

Now we solve for the Pythagorian triple in the first equation.
x0=s2+t2y0=s2−t2u=2st
for some positive integers s,t.

Then x20+y20=2(s4+t4)=2v2. Hence we obtain s4+t4=v2. However, this cannot have positive integer solution.

HOPE IT HELPS ✌
HOPE IT IS RIGHT

Answer 2

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