As x and y are positive integers, we are looking for factor pairs of 2019 that could be the values of x + y -1 and x - y + 3.

2023 = 1 x 2023 or 7 x 289 or 17 x 119 are the only possibilities.

x + y - 1 > x - y + 3 for any y >2, so let's check what happens if y = 1 or y = 2

y = 1 would make (x + 2)(x + 2) = 2023, which is not possible as 2023 is not a square number.

y = 2 would make (x + 1)(x + 1) = 2023, which is not possible as 2023 is not a square number.

Therefore we must now consider the following simultaneous equations:

(i) x + y - 1 = 2023 and x - y + 3 = 1

(ii) x + y - 1 = 289 and x - y + 3 = 7

(iii) x + y - 1 = 119 and x - y + 3 = 17

These are solved by adding the equations to find x and subtracting them to find y.

For (i) x = 1011 and y = 1013

For (ii) x = 147 and y = 143

For (iii) x = 67 and y = 53

We are told that neither x nor y is prime, so must discard (i) (as 1013 is prime) and (iii) (as both 67 and 53 are prime).

In (ii) both 147 and 143 are composite (147 = 3 x 49 and 143 = 11 x 13), so the required value is xy = 147 x 143 = 21, 021

Challenge 349: Integer Inquiry