Problems & Puzzles: Puzzles

Puzzle 69.- Primeful Heterosquares

Let’s define a “Primeful heterosquare” as a matrix nxn, n=odd, such that:

a. The nxn inside numbers

b. The sum of every column, row & main diagonals, &

c. The sum of all the nxn inside numbers

are distinct prime numbers.

I have found (Set/98) what - I think - is the smallest 3x3 of this kind of squares, in two varieties:

a) Non consecutive primes inside

 19 5 41 13 59 17 3 47 67 7 83 11 101 23 29 127 71 227

b) Consecutive primes inside

 137 31 37 41 109 53 59 61 173 67 43 47 157 167 151 139 149 439

Questions:

1. Can you find the smallest 5x5 & 7x7 of both kind of heterosquares?

See other Unusual Magic Squares at always interesting Harvey Heinz pages

Solution

Well, a good surprise!... Jean-Charles Meyrignac found (24/12/2000) that my first claim was wrong. He found another and smaller 3x3 primeful-heterosquare, using non-consecutive primes:

 41 3 5 11 19 23 31 29 83 17 37 7 61 59 43 73 47 163

He has sent also his "best" solution for the 5x5 and the 7x7 cases:

 271 3 5 7 11 17 43 13 19 23 29 47 131 31 37 41 53 61 223 71 79 59 107 67 383 73 89 97 83 101 443 239 191 229 227 283 293 1223

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 757 3 5 7 11 13 17 23 79 19 29 31 37 41 43 71 271 47 53 59 61 67 73 83 443 89 97 101 103 107 109 113 719 127 131 137 139 149 151 157 991 163 167 173 179 199 181 227 1289 211 191 193 239 197 223 233 1487 751 659 673 701 769 773 797 907 5279

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Are these the least in its corresponding type?

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Jean-Charles added: "I verified if your consecutive prime solution for puzzle 69 was the best, and it is !"

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Five years later, Anurag found better results than JCM:

Least 3x3 non-Consecutive primes: sum = 139
17 5 7
3 23 11
41 19 13

Least 5x5 non-Consecutive :sum = 1171 [A.S. sent several; only one shown by me, CR.]
67 97 71 19 3
5 61 59 47 79
53 23 113 13 37
31 43 29 89 41
17 83 11 73 7

Least 5x5 Consecutive: sum= 1259 (5..103)

89 5 17 7 31
19 13 67 29 23
11 59 37 73 53
61 79 83 71 43
47 41 103 97 101

Least 7x7 non-Consecutive: sum= 5119

157 13 151 131 71 41 23
197 67 109 163 101 61 179
11 29 113 211 3 37 83
181 97 191 173 193 89 17
73 47 139 167 53 79 19
7 107 5 149 127 233 199
227 103 31 223 59 137 43

Least 7x7 Consecutive: sum= 5813
19 103 41 131 127 241 71
173 193 83 137 139 13 149
113 211 53 229 37 167 73
61 197 31 109 97 79 43
151 23 59 157 47 199 107
181 163 101 67 89 29 17
239 227 179 191 233 11 223

...

A. S. sent larger solutions both kind...

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