Problems & Puzzles: Puzzles

Puzzle 106. Prime-magic squares added up  to 2000

John Everett sent to me (last July 24, 2000) his third letter crowded of interesting and beauty Prime-Magical Squares solutions to different situations. I have selected only one of them before the current year ends.

(A₁ + A₂ = 2000)_4x4

Naively I tried to obtain one corresponding example for (A₁ + A₂ = 2000)_3x3 but for my surprise - and as far as my method works - it doesn't exists!... Nevertheless 2000 is not inaccessible from the 3x3 Prime-Magical squares...

Questions:

Regarding the Prime Magic "equation" (A₁ + A₂ + ... + A_k = 2000)_3x3
1. Find one minimal-k solution
2. Find one maximal-k-solution
3. Maybe it's time enough to start producing one solution to (A₁ + A₂ + A₃ = 2001)_3x3 - if any exist...

Solution