Pierre de Fermat discovered that if $p$ is a prime and congruent to 1 mod 4, then it can be written as the sum of the square of two natural numbers.
Similarly, I was trying to find the list of those prime numbers which are sum of the cubes of three natural numbers.
My Attempt:
$\begin{align}\qquad
3 &= 1^3+1^3+1^3 \\
17 &= 1^3+2^3+2^3 \\
29 &= 3^3+1^3+1^3 \\
73 &= 4^3+2^3+1^3 \\
\end{align}$
My idea was to find at least 100 such prime numbers so that I can observe the common property hold by them.
So before observing such things, I need list of such prime number because I don't have any computer program to develop such a list.
Question:
Can anyone provide me wit a list of prime numbers which are the sum of the cubes of three natural numbers?
Best Answer
Here is a complete list up to $10\,000$ which are $159$ primes. The $100$-th entry is $5641 = 17^3 + 8^3 + 6^3$.
For reference, I attached the brute-force Sage script at the end.