We define recursively the Ulam numbers by setting u1 = 1, u2 = 2, and for each
subsequent integer n, we set n equal to the next Ulam number if it can be written
uniquely as the sum of two different Ulam numbers; e.g.: u3 = 3, u4 = 4, u5 = 6,
etc.

Prove that there are infinitely many Ulam numbers.

Now a D4 BONUS.

3 (=1+2).
Find another Ulam number is that is the sum of two consecutive Ulam numbers.

How about a 3rd one?

Assume the Ulam numbers are finite. Then there must exist a largest and a next-to-largest Ulam number. But the sum of those two is also a Ulam number, one which is larger than either one. This is a contradiction, so our assumption is wrong and the Ulam numbers are infinite.