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

 67 241 577 571 1933 1759 1423 1429 2000 2000 2000 2000 547 769 127 13 1453 1231 1873 1987 2000 2000 2000 2000 223 139 421 673 + 1777 1861 1579 1327 = 2000 2000 2000 2000 619 307 331 199 1381 1693 1669 1801 2000 2000 2000 2000

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 + ... + Ak = 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

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