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.
Divisibility on the sequence of perfect squares minus one: The gap principle
