What is the smallest palindromic prime whose cube can be expressed as the sum of three odd cubes ?

My seven year-old has become interested in math, so I told her what prime numbers were. I then asked her if even numbers could be prime, to which she replied "only 2". Elated, I decided to really encourage her, "I will give you the number of pennies equal to the largest prime you can name, but if it is not prime, you get nothing and must help me cook dinner tonight.

She said: "a billion seven." Aghast, I asked her how she picked that.

It turned out that it was because the character in the movie 'A Brilliant Young Mind', a maths movie we had recently watched, would only eat food orders that had prime pieces like 7 and 47. Additionally, she knew a billion was way big. I checked online. She had done it.

So now I owe her 10 million dollars and 7 cents. She tells me to get busy working and sings "I owe, I owe, so off to work I go!" around the house.