Magic Constant of Smallest Prime Magic Square with Consecutive Primes from 3

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Theorem

The magic constant of the smallest prime magic square whose elements are consecutive odd primes from $3$ upwards is $4514$.


Proof

The smallest prime magic square whose elements are the first consecutive odd primes is:

$\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 fact that this is the smallest is proven here.


The sum of the first $144$ prime numbers can either be calculated or looked up: it is $54 \, 169$.

We see that $2$ is not included in this prime magic square, but instead $1$ is used.

So the total of all the elements of this prime magic square is $54 \, 169 - 2 + 1 = 54 \, 168$.

There are $12$ rows, and $12$ columns, each with the same magic constant.

This magic constant must therefore be $\dfrac {54 \, 168} {12} = 4514$.


As can be seen by inspection, the sums of the elements in the rows, columns and diagonals is $4514$.

$\blacksquare$


Sources

but beware: he makes a mistake and reports it as $4515$.