Abstract:

In this report, we want to know whether there is a magic square whose entries are distinct perfect squares.

Firstly, we analyze the basic properties of a magic square and find that the magic sum of a magic square is equal to 3 times of the central entry and the 9 entries of a magic square contain 8 arithmetic progressions.

Secondly, we focus on our main target, magic square of squares. Investigating the properties of the prime factors of those 9 entries, we find that if the greatest common divisor of all entries is equal to 1, the prime factors of central entry are of the form \(p \equiv 1\) (mod 4), the central entry must not be a square of a prime number and the common prime factors of any two adjacent entries (if exist) are not of the form \(p \equiv 3\) (mod 4).

Thirdly, we find that this problem is equivalent to a system of Diophantine equations with ten variables. We provide a construction method of the solution to these partial equations:

$$a^2 + b^2 = c^2 + d^2 = e^2 + f^2 = g^2 + h^2 = 2M^2$$ ,

where these nine perfect squares are distinct.

Finally, based on the theorems obtained, we find that given a positive integer \(N\), there exists a positive integer \(M\) such that it has \(N\) essentially different representations of a sum of two perfect squares.

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