Good evening, I was self-studying Discrete Mathematics from a book and found the following problem: 1. The problem statement, all variables and given/known data We define the Ulam numbers by setting u1 = 1 and u2 = 2. Furthermore, after determining whether the integers less than n are Ulam numbers, we set n equal to the next Ulam number if it can be written uniquely as the sum of two different Ulam numbers. Note that u3 = 3, u4 = 4, u5 = 6, and u6 = 8. Prove that there are infinitely many Ulam numbers. 2. The attempt at a solution I imagine that I have to show that, if there are n Ulam numbers, for example, there is always an un+1. To show this, I have to show somehow that there will always be two numbers among these n numbers whose sum is unique and greater than un. If it were the case that un is the greatest Ulam number, it would mean that no sum between two of the numbers is unique and greater than un. I had no clue on how to continue, so I searched on the Internet and found the following proof method: Assume that there are only finitely many Ulam numbers. Let the two largest Ulam numbers be un-1 and un. Then the integer un-1 + un is an Ulam number larger than un. It is the unique sum of two Ulam numbers, since it is the largest sum between two of the n Ulam numbers. While trying to understand this proof, I found that there seems to be something wrong, because it is not always true that, for n Ulam numbers, un-1 + un will be an Ulam number. For example, consider the n Ulam numbers with n = 3: 1, 2, 3. The sum 3 + 2 = 5 is not an Ulam number, because 4 is an Ulam number and 4 + 1 = 5. Thank you in advance.