Conjecture Given a prime number p, if you fill the pxp cells of a square array with the natural numbers 1 to p^2 from left to right and from top to bottom, you can always find a set of p primes such that no two of them share row & column. Example: Solution for p=181 Sum of primes= 2964961 p*(p^2+1)/2= 2964961 Prime Row Column 181 1 181 349 2 168 397 3 35 647 4 104 811 5 87 1069 6 164 1193 7 107 1433 8 166 1567 9 119 1787 10 158 1867 11 57 2089 12 98 2339 13 167 2437 14 84 2609 15 75 2843 16 128 3001 17 105 3217 18 140 3407 19 149 3557 20 118 3701 21 81 3911 22 110 4099 23 117 4297 24 134 4481 25 137 4673 26 148 4799 27 93 4973 28 86 5179 29 111 5323 30 74 5591 31 161 5737 32 126 5939 33 147 6143 34 170 6217 35 63 6353 36 18 6659 37 143 6857 38 160 7013 39 135 7121 40 62 7247 41 7 7573 42 152 7673 43 71 7933 44 150 8011 45 47 8291 46 146 8363 47 37 8623 48 116 8783 49 95 9043 50 174 9227 51 177 9311 52 80 9497 53 85 9631 54 38 9949 55 175 10091 56 136 10177 57 41 10369 58 52 10627 59 129 10771 60 92 10861 61 1 11131 62 90 11287 63 65 11579 64 176 11597 65 13 11887 66 122 12097 67 151 12281 68 154 12487 69 179 12517 70 28 12689 71 19 12917 72 66 13163 73 131 13313 74 100 13417 75 23 13681 76 106 13921 77 165 13963 78 26 14143 79 25 14407 80 108 14549 81 69 14737 82 76 14887 83 45 15077 84 54 15361 85 157 15497 86 112 15643 87 77 15787 88 40 16069 89 141 16141 90 32 16363 91 73 16649 92 178 16703 93 51 16903 94 70 17167 95 153 17209 96 14 17477 97 101 17659 98 102 17909 99 171 18013 100 94 18223 101 123 18353 102 72 18583 103 121 18773 104 130 18839 105 15 19051 106 46 19207 107 21 19423 108 56 19597 109 49 19843 110 114 19963 111 53 20113 112 22 20431 113 159 20477 114 24 20773 115 139 20947 116 132 21001 117 5 21193 118 16 21419 119 61 21599 120 60 21893 121 173 22073 122 172 22111 123 29 22273 124 10 22483 125 39 22721 126 96 22861 127 55 23131 128 144 23251 129 83 23357 130 8 23627 131 97 23789 132 78 24061 133 169 24229 134 156 24281 135 27 24439 136 4 24659 137 43 24917 138 120 25057 139 79 25321 140 162 25349 141 9 25609 142 88 25733 143 31 25919 144 36 26227 145 163 26387 146 142 26437 147 11 26641 148 34 26891 149 103 27017 150 48 27283 151 133 27361 152 30 27529 153 17 27751 154 58 27941 155 67 28097 156 42 28349 157 113 28597 158 180 28723 159 125 28843 160 64 29059 161 99 29191 162 50 29411 163 89 29641 164 138 29717 165 33 29867 166 2 30137 167 91 30271 168 44 30553 169 145 30713 170 124 30829 171 59 30971 172 20 31247 173 115 31319 174 6 31649 175 155 31687 176 12 31859 177 3 32119 178 82 32327 179 109 32467 180 68 32707 181 127