Square of Primes
 

Construct a 5 x 5 square containing distinct primes
such that each row, column and diagonal sums to a distinct prime.

[QUOTE=davar55;132471]Construct a 5 x 5 square containing distinct primes
such that each row, column and diagonal sums to a distinct prime.[/QUOTE]

Is it a magic square where every sum is the same?

A magic square of primes (where every sum is the same) is
solved elsewhere (although it would be a perfectly good
puzzle to re-solve).

Here every sum is a different prime.

The original problem was perhaps too computationally simple
to be interesting. The following additional condition adds an
iota of complexity:

The 25 distinct primes in the square should be the first 25
odd primes {3,5,7,...,97,101}.

(I have a solution which wasn't hard to find by trial and error,
so there must be many solutions; but plain brute force on the 25!
such possible squares is obviously too computationally costly.)

Here's one:[CODE][SPOILER] 239

/
/
/

3, 5, 7, 11, 17 --- 43

13, 19, 29, 23, 43 --- 127

31, 67, 61, 47, 71 --- 277

53, 59, 41, 73, 37 --- 263

79, 83, 89, 97, 101 --- 443

| | | | | \
| | | | | \
| | | | | \

179 233 227 251 269 257[/SPOILER][/CODE]

There is an error in your calculations : the last row total is of by 6. But the right number is prime so the solution stands :-)

Jacob

Hm, I think I worked the total out right, but wrote it down wrong.

It wasn't just a fluke, honest! ;)

Does the sequence

a(n) = number of square matrices containing the first (2n+1)x(2n+1) odd primes, such that row, column and diagonal sums are distinct primes

exist on OEIS ?

It would appear that there are an awful lot of these out there, so perhaps the challange should be to find a square with the lowest standard deviation of column/row/diagonal totals.

I'll start the ball rolling with a slightly modified version of the last square I posted, with an s.d. of 84.51:[code][SPOILER] 239

/
/
/

3, 5, 7, 11, 17 --- 43

13, 19, 29, 23, 43 --- 127

31, 67, 61, 47, 71 --- 277

53, 59, 89, 73, 37 --- 311

79, 83, 41, 97, 101 --- 401

| | | | | \
| | | | | \
| | | | | \

179 233 227 251 269 257[/SPOILER][/code]

Here's another solution:

041 005 007 071 003
013 023 029 031 067
059 053 043 047 037
019 011 089 061 017
079 101 083 097 073

Rows: 127,163,239,197,433
Columns: 211,193,251,307,197
Diagonals: 241,167

(Standard deviation: 76.7)

An alternative measure is simply mini-max: minimize the largest sum.
By that measure, lavalamp's solution is a better one.