19.7.17 ** Find the smallest integer greater than 2 that can be expressed both as the sum of the squares of two positive integers and as the sum of the cubes of two positive integers. Also prove that there are infinitely many positive integers that can be expressed both as the sum of the squares of two coprime positive integers and as the sum of the cubes of two coprime positive integers.
The first 6 integers greater than 2 that can be expressed as the sum of the cubes of two positive integers are: $1^{3}+2^{3}=9$, $2^{3}+2^{3}=16$, $1^{3}+3^{3}=28$, $2^{3}+3^{3}=35$, $3^{3}+3^{3}=54$, $1^{3}+4^{3}=65$. The numbers 9, 16, $28, 35, 54$ cannot be expressed as the sum of the squares of two integers, while $65=1^{2}+8^{2}$. Therefore, the smallest integer greater than 2 that can be expressed both as the sum of the squares of two positive integers and as the sum of the cubes of two positive integers is 65. For each positive integer $k$, we have $1+2^{6 k}=1^{2}+\left(2^{3 k}\right)^{2}=1^{3}+\left(2^{2 k}\right)^{3}$.