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.

Posted by Steve Herman on 2017-06-03 08:33:53