Already at a young age Sergei Prokofiev was a cocky sod. Several years younger, he stil managed to annoy his classmates by counting their errors. At eleven years he insisted his teacher allow him to write his own symphony.
Around ten years later he wrote his first piano concerto. Three years later he managed to win a competition with it, even though the rules stipulated the music was actually published. He published it, sure. Just twenty copies of it, and just in time for the event.
The story of the second piano concerto is a bit of a strange one. It was written in 1912, so around the same time as the first. Prokofiev finished it in one year. However, nothing of it remains, as it got lost in the turmoil of the Russian Revolution. The composer reconstructed the score in 1923, but admitted himself he produced a completely new work, and could just as well have called it his fourth.