Which is the smallest integer that can be written as the sum of two different positive square numbers in two different ways?
Let $N$ be an integer with two different decompositions:

$$
a^{2}+b^{2}=c^{2}+d^{2}=N
$$

where $a, b, c, d$ are positive integers, and for example $a>c, d>b$.

Rearranging and factoring the equality, we get

$$
(a+c)(a-c)=(d+b)(d-b)
$$

The parity of the four factors is the same, because if $(a+c)$ is odd, then $(a-c)$ is also odd, so their product is odd, which means the two factors on the right side are also odd. Thus, in this case, $N$ has four different positive odd divisors. The same can be said for the even case. It is easy to find a small integer that has four different positive odd divisors. If we write down the integers in sequence and see how many divisors each has, we can immediately see that $N=15$ is such an example; its divisors are $1, 3, 5$, and 15. The product of the divisors satisfies the equality $1 \cdot 15=3 \cdot 5$.

Since $a+c > a-c$; $a+c=15, a-c=1$, from which $a=8, c=7$. Furthermore, since $d+b > d-b$, $d+b=5, d-b=3$, and from this $d=4, b=1$.

Indeed, $N=8^{2}+1^{2}=7^{2}+4^{2}=65$ is a solution.

Now we only need to prove that 65 is the smallest integer that satisfies the requirement.

For this, we construct the table of sums $x^{2}+y^{2}$ for $1<x, y \leq 8$. From the table, we can read that the sums of squares are all different, except for the 65 we have already obtained.

Remark. From the theory of Gaussian integers, we know that the number $N$ is of the form $p q$, where $p$ and $q$ are the two smallest positive primes of the form $4k+1$, i.e., $p=3, q=13$ and $N=5 \cdot 13=65$.

|  | 1 | 4 | 9 | 16 | 25 | 36 | 49 | 64 |
| :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: |
| 1 | 20 |  |  |  |  |  |  |  |
| 4 | 5 | 8 |  |  |  |  |  |  |
| 9 | 10 | 13 | 18 |  |  |  |  |  |
| 16 | 17 | 20 | 25 | 32 |  |  |  |  |
| 25 | 26 | 29 | 34 | 41 | 50 |  |  |  |
| 36 | 37 | 40 | 45 | 52 | 61 | 72 |  |  |
| 49 | 50 | 53 | 58 | 65 | 74 | 85 | 98 |  |
| 64 | 65 | 68 | 73 | 80 | 89 | 100 | 113 | 128 |