Prime Magic Square/Examples/Order 12/Smallest with Consecutive Primes from 3/Mistake
Source Work
1986: David Wells: Curious and Interesting Numbers:
- The Dictionary
- $144$
1997: David Wells: Curious and Interesting Numbers (2nd ed.):
- The Dictionary
- $144$
Mistake
- The smallest magic square composed of consecutive primes comprises the $144$ odd primes from $3$ upwards. The magic constant is $4515$.
The prime magic square being referred to is actually composed of the $143$ odd primes from $3$ upwards, along with $1$.
- $\begin{array}{|c|c|c|c|c|c|c|c|c|c|c|c|}
\hline 1 & 823 & 821 & 809 & 811 & 797 & 19 & 29 & 313 & 31 & 23 & 37 \\ \hline 89 & 83 & 211 & 79 & 641 & 631 & 619 & 709 & 617 & 53 & 43 & 739 \\ \hline 97 & 227 & 103 & 107 & 193 & 557 & 719 & 727 & 607 & 139 & 757 & 281 \\ \hline 223 & 653 & 499 & 197 & 109 & 113 & 563 & 479 & 173 & 761 & 587 & 157 \\ \hline 367 & 379 & 521 & 383 & 241 & 467 & 257 & 263 & 269 & 167 & 601 & 599 \\ \hline 349 & 359 & 353 & 647 & 389 & 331 & 317 & 311 & 409 & 307 & 293 & 449 \\ \hline 503 & 523 & 233 & 337 & 547 & 397 & 421 & 17 & 401 & 271 & 431 & 433 \\ \hline 229 & 491 & 373 & 487 & 461 & 251 & 443 & 463 & 137 & 439 & 457 & 283 \\ \hline 509 & 199 & 73 & 541 & 347 & 191 & 181 & 569 & 577 & 571 & 163 & 593 \\ \hline 661 & 101 & 643 & 239 & 691 & 701 & 127 & 131 & 179 & 613 & 277 & 151 \\ \hline 659 & 673 & 677 & 683 & 71 & 67 & 61 & 47 & 59 & 743 & 733 & 41 \\ \hline 827 & 3 & 7 & 5 & 13 & 11 & 787 & 769 & 773 & 419 & 149 & 751 \\ \hline \end{array}$
The magic constant of this square is in fact $4514$, not $4515$.
Elementary arithmetic tells us that the magic constant, being the sum of $12$ (an even number of) odd primes, will in fact be even.
Hence it cannot be $4515$.
It also needs to be pointed out that the smallest magic square composed of consecutive primes is actually of order $3$:
- $\begin{array}{|c|c|c|}
\hline 1 \, 480 \, 028 \, 159 & 1 \, 480 \, 028 \, 153 & 1 \, 480 \, 028 \, 201 \\ \hline 1 \, 480 \, 028 \, 213 & 1 \, 480 \, 028 \, 171 & 1 \, 480 \, 028 \, 129 \\ \hline 1 \, 480 \, 028 \, 141 & 1 \, 480 \, 028 \, 189 & 1 \, 480 \, 028 \, 183 \\ \hline \end{array}$
$\blacksquare$
Sources
- 1986: David Wells: Curious and Interesting Numbers ... (previous) ... (next): $144$
- 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $144$