A magic square
M
M
over an integral domain
D
D
is a
3
×
3
3\times 3
matrix with entries from
D
D
such that the elements from each row, column, and diagonal add to the same sum. If all the entries in
M
M
are perfect squares in
D
D
, we call
M
M
a magic square of squares over
D
D
. In 1984, Martin LaBar raised an open question: “Is there a magic square of squares over the ring
Z
\mathbb {Z}
of the integers which has all the nine entries distinct?” We approach to answering a similar question when
D
D
is a finite field. We claim that for any odd prime
p
p
, a magic square over
Z
p
\mathbb Z_p
can only hold an odd number of distinct entries. Corresponding to LaBar’s question, we show that there are infinitely many prime numbers
p
p
such that, over
Z
p
\mathbb Z_p
, magic squares of squares with nine distinct elements exist. In addition, if
p
≡
1
(
mod
120
)
p\equiv 1\pmod {120}
, there exist magic squares of squares over
Z
p
\mathbb Z_p
that have exactly 3, 5, 7, or 9 distinct entries respectively. We construct magic squares of squares using triples of consecutive quadratic residues derived from twin primes.
Note on the cardinality difference between primes and twin primes and its impact on function x/ln(x) in prime number theorem
