Prime Magic Squares

The number associated with my diurnal age today, \( \textbf{28209} \), is a member of OEIS A269324:

A269324: magic sums of 3 x 3 magic squares composed of primes in an arithmetic progression.

I was interested in finding out what these primes were and how they could be arranged so I sought help from Gemini. Here was what it came up with:

To construct a 3×3 magic square of primes in an arithmetic progression for the magic sum 28209 (a term from OEIS A269324), we need to determine the central prime, find the common difference of the progression, and then map those primes to the cells of the grid. Here is the complete step-by-step breakdown of the mathematics and the final arrangement.

1. Finding the Central Prime

In any 3×3 magic square, the central cell is always the average of the magic sum. Because the magic sum (\( S \)) is the total of any given row, column, or diagonal, the center prime is exactly \( S / 3 \).

$$ \text{Center} = \frac{28209}{3} = 9403 $$

Because the 9 primes must form an arithmetic progression, the central cell of the magic square must also be the median (the 5th term) of the sorted sequence. Therefore, our arithmetic progression of primes is centered at 9403.

2. Determining the Common Difference

Let \( d \) be the common difference. Our 9 primes will take the form of \( 9403 + kd \), where \( k \) ranges from \( -4 \) to \( 4 \).

According to prime arithmetic progression properties, an arithmetic progression of \( k \) primes must have a common difference divisible by every prime \( q \le k \) (unless the progression actually begins with \( q \)). Since our sequence contains 9 primes and does not start with 2, 3, 5, or 7, the common difference \( d \) must be a multiple of their primorial:

$$ d = 2 \times 3 \times 5 \times 7 = 210 $$

By testing the multiples of 210, we can find the exact difference where all 9 resulting terms are prime:

• \( d = 210 \) fails because the term \( 9403 - 210 = 9193 \), which is \( 29 \times 317 \).
• \( d = 420 \) fails because \( 9403 + 420 = 9823 \), which is \( 11 \times 19 \times 47 \).
• \( d = 630 \) fails because \( 9403 - 3(630) = 7513 \), which is \( 11 \times 683 \).
• \( d = 840 \) succeeds.

3. The Nine Primes

Using \( d = 840 \) and our center prime of 9403, we can generate the 9 primes for the magic square by stepping out by 840 in both directions:

• \( 9403 - 4(840) = \) 6043
• \( 9403 - 3(840) = \) 6883
• \( 9403 - 2(840) = \) 7723
• \( 9403 - 1(840) = \) 8563
• 9403 (Center)
• \( 9403 + 1(840) = \) 10243
• \( 9403 + 2(840) = \) 11083
• \( 9403 + 3(840) = \) 11923
• \( 9403 + 4(840) = \) 12763

4. Arranging the Magic Square

To arrange these into a valid 3×3 magic square, we map our ordered sequence to the classic Lo Shu magic square structure. In the traditional 1–9 Lo Shu square, the numbers are placed so that opposites balance out across the center (5). Substituting our 1st through 9th primes directly into the equivalent positions of the Lo Shu configuration yields the following layout:

11923	6043	10243
7723	9403	11083
8563	12763	6883

Every row, column, and main diagonal in this grid perfectly sums to 28209. Keep in mind that any of the 8 standard symmetries (rotations or reflections) of this exact grid will also produce a valid magic square for this progression.