Question

Prove by strong induction that any nonzero natural number can be written as a sum of distinct powers of 2.

Answer 1

Proof. We will use strong induction to prove that Vn e Zt we can express n as a sum of distinct powers of 2. Base Case. For n=1 note that 2° = 1 and thus for n=1 our proposition holds. Inductive Hypothesis. Assume me Zt with 1 <m <k and that our proposition holds for m. Inductive Step. We consider two cases. Case 1. If k + 1 is even then observe that ku must be an integer. Now since 1 < kui <k we know by our inductive hypothesis that kyl is the sum of distinct powers of 2. But then multiplying by 2 gives k+1 2 k+1 2 .2=k+1 and since (by the distributive property of multiplication over addition) each distinct power of 2 in the sum *4 is multiplied by a factor of 2, each power of 2 is increased by 1 and thus remains distinct. Case 2. If k +1 is odd then we know k is even. Furthermore we know by our inductive hypothesis that k may expressed as the sum of distinct powers of 2. But if k is even we know k does not contain a 2° = 1 in its sum of distinct powers of 2.

Thus we see that k+1=k+2 and if k +1 is odd we may express k +1 as the sum of distinct powers of 2. It follows by strong mathematical induction that Vn ezt we can express n as a sum of distinct powers of 2.0