The problems of number theory
The third question is a famous conjecture of Goldbach, and though it has been around for more than 200 years, no one has yet managed to prove it. Question (4) is known as Catalan s conjecture and is also an open question. However, it is known that if the difference between a cube and a square is ±1 , then the cube is 8 and the square is 9. This was shown by Euler, and the proof is not very difficult. For more details on Catalan’s conjecture see Ribenboim 1994.