Prime Number Pattern in Magic Sum of Pseudo / Semi Magic Squares of Squares

Given infinitely many primes $p\equiv1\pmod6$, each corresponds to at least one “3×3 pseudo-magic square of squares” (PMSS), and therefore there are infinitely many such PMSS in this family.

After working through the very lengthy equations, two key observations establish that there are infinitely many 3×3 pseudo‐magic squares of squares in this family (abbreviated as PMSS):

1. Infinitely many primes $p\equiv1\pmod6$

By Dirichlet’s theorem on arithmetic progressions, there are infinitely many primes of the form $6k+1$. Each such prime can potentially generate a PMSS using the construction below.

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2. Parametrization of 3×3 PMSS

Fix any prime $p \equiv 1 \pmod 6$. This construction relies on solving a specific quadratic in order to define two integers $m$ and $n$. These will later be used to generate the entries of the square. The roots of the quadratic govern the choice of entries such that the resulting square satisfies the pseudo-magic condition with a constant magic sum.

The quadratic is:

$$ x^2 - (p + v)x + v(2p + v) = 0 $$

This arises from the symmetric relationships we want among the square’s entries, and it has discriminant:

$$ \Delta = (p + v)^2 - 4v(2p + v) = p^2 - 6pv - 3v^2 $$

If this discriminant $\Delta$ is a perfect square, say $\Delta = k^2$, then the quadratic has rational (in fact, integer) roots:

$$ m = \frac{(p+v)-k}{2}, \quad n = \frac{(p+v)+k}{2} $$

These satisfy:

$$ m + n = p + v, \quad mn = v(2p + v) $$

and are positive integers for suitable $v$. With these $m, n$ in hand, the nine square entries are defined as follows:

$$ \begin{aligned} a &= p^2 + n^2, &\quad b &= 2p^2 - v^2, &\quad c &= p^2 + 2pm - m^2,\\ d &= 2p^2 + 4pv + v^2, &\quad e &= p^2 + m^2, &\quad f &= 2pn + n^2 - p^2,\\ g &= 2pm + m^2 - p^2, &\quad h &= p^2 + 2pn - n^2, &\quad i &= (p+v)^2 + p^2. \end{aligned} $$

It can then be verified that the following matrix:

$$ \begin{pmatrix} a^2 & b^2 & c^2\\ d^2 & e^2 & f^2\\ g^2 & h^2 & i^2 \end{pmatrix} $$

has the property that every row, column, and the main diagonal sums to $9p^4$. (Note: the anti-diagonal does not necessarily sum to this value.)

By permuting entries while preserving these sums, one obtains 12 distinct PMSS. Allowing the anti-diagonal $(c,e,g)$ to sum instead of the main diagonal yields 12 additional symmetric arrangements. Thus, each $(p,v)$ pair gives rise to 24 distinct PMSS if we allow the sum to match along either diagonal.

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Summary:

• There are infinitely many primes $p \equiv 1 \pmod 6$.
• For each such $p$, whenever a suitable $v$ exists such that $\Delta = p^2 - 6pv - 3v^2$ is a square, we obtain a PMSS.
• Each PMSS yields 12 permutations (or 24 if allowing either diagonal to sum correctly).

Hence, this parametrization produces infinitely many pseudo-magic squares of squares with distinct magic sums $9p^4$, as $p$ ranges over primes congruent to 1 mod 6.

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Further information:

It may be possible to refine the bounds on $v$ or identify a deeper structural pattern relating $v$ to $p$, but I wasn't able to derive a universal rule. Interestingly, while the construction works cleanly for all primes $p \equiv 1 \pmod{6}$, the full sequence of valid $p$ values corresponds exactly to OEIS A004611 >1, which includes some composite integers such as $p = 49$ and $p = 91$.

These composite cases are particularly notable:

• Each still yields at least one valid PMSS using the same construction.
• Some even admit multiple distinct solutions.
• In several cases, one of these solutions appears to be a scaled version of an earlier PMSS corresponding to a smaller prime.

In contrast, each prime $p \equiv 1 \pmod{6}$ produces exactly one PMSS (up to symmetry) using this method.