2023 Number Challenge: Solve a Diophantine equation with 2023

Check out other 2023 Number Challenges from this list.
Share with your friends!!

sqrt(x) + sqrt(y) = sqrt(2023)

Here, we note that 2023 = 7 x 17 x 17.
By some simple analysis, the solutions are

x = 7m^2,
y = 7(17-m)^2
for m = 0, 1, 2, ..., 17

[angelmendez-rivera]

sqrt(x) + sqrt(y) = sqrt(2023) = sqrt(7•17^2) = 17•sqrt(7). Let x = 7•m^2 and y = 7•n^2, hence sqrt(x) + sqrt(y) = (m + n)•sqrt(7). Therefore, m + n = 17. There are no other possibilities, because square roots of different primes numbers are algebraically independent.