Diophantine equations are hard

There's a solution to x^3 + y^3 +z^3 =33 for x, y, z all integers. 29 is a pretty trivial target. 32 is impossible (as are "all whole numbers that leave behind a remainder of 4 or 5 when divided by 9").

"Until Booker found his solution, it was one of only two integers left below 100 (excluding the ones for which solutions definitely don’t exist) that still couldn’t be expressed as a sum of three cubes. With 33 out of the way, the only one left is 42."

What is “sufficiently interesting,” Booker explained, is that each newfound solution is “a tool for helping you decide what’s true” about the sum-of-three-cubes problem itself. That problem, stated as k = x³ + y³ + z³, is what number theorists call a Diophantine equation — a kind of algebraic structure whose properties have fascinated mathematicians for millennia. “They’re rich enough to encode [other mathematical] statements that have nothing to do with Diophantine equations,” said Browning. “They’re rich enough to simulate computers.”