Magic Constant of Smallest Prime Magic Square with Consecutive Primes

Theorem

The magic constant of the smallest prime magic square whose elements are consecutive odd primes is $4 \, 440 \, 084 \, 513$.

Proof

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

$\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}$

As can be seen by inspection, the sums of the elements in the rows, columns and Diagonal of Array is $4 \, 440 \, 084 \, 513$:

\(\ds 1 \, 480 \, 028 \, 159 + 1 \, 480 \, 028 \, 153 + 1 \, 480 \, 028 \, 201\)

\(=\)

\(\ds 4 \, 440 \, 084 \, 513\)

\(\ds 1 \, 480 \, 028 \, 213 + 1 \, 480 \, 028 \, 171 + 1 \, 480 \, 028 \, 129\)

\(=\)

\(\ds 4 \, 440 \, 084 \, 513\)

\(\ds 1 \, 480 \, 028 \, 141 + 1 \, 480 \, 028 \, 189 + 1 \, 480 \, 028 \, 183\)

\(=\)

\(\ds 4 \, 440 \, 084 \, 513\)

\(\ds 1 \, 480 \, 028 \, 159 + 1 \, 480 \, 028 \, 213 + 1 \, 480 \, 028 \, 141\)

\(=\)

\(\ds 4 \, 440 \, 084 \, 513\)

\(\ds 1 \, 480 \, 028 \, 153 + 1 \, 480 \, 028 \, 171 + 1 \, 480 \, 028 \, 189\)

\(=\)

\(\ds 4 \, 440 \, 084 \, 513\)

\(\ds 1 \, 480 \, 028 \, 201 + 1 \, 480 \, 028 \, 129 + 1 \, 480 \, 028 \, 183\)

\(=\)

\(\ds 4 \, 440 \, 084 \, 513\)

\(\ds 1 \, 480 \, 028 \, 159 + 1 \, 480 \, 028 \, 171 + 1 \, 480 \, 028 \, 183\)

\(=\)

\(\ds 4 \, 440 \, 084 \, 513\)

\(\ds 1 \, 480 \, 028 \, 141 + 1 \, 480 \, 028 \, 171 + 1 \, 480 \, 028 \, 201\)

\(=\)

\(\ds 4 \, 440 \, 084 \, 513\)

This theorem requires a proof. In particular: It remains to be demonstrated that there are no prime magic squares that are any smaller than this. You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by crafting such a proof. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{ProofWanted}} from the code. If you would welcome a second opinion as to whether your work is correct, add a call to {{Proofread}} the page.