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 it 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 u 6 = 8.

a) Find the first 20 Ulam numbers.

b) Prove that there are infinitely many Ulam numbers.

SOLUTION

Step 1

We have to find the first 20 Ulam numbers.

We have .

Number 5 and 7 are not Ulam numbers because their addition is not distinct.

We have 5 = 1+4 , 2+3 where 1,2,3,4 are Ulam numbers.

And 7 = 3+4, 1+6 where 1,6,3,4 are Ulam numbers.

8 = 6+2 , Ulam number

9 = 3+6 , 1+8 not an Ulam number

10 = 2+8, 4+6 not an Ulam number.

Proceeding like this we have to find the first 20 Ulam numbers.