Assignment # 7 – Number Theory

(1) Find p(8) using the recursive property m(n) = m􀀀1(n)+m(n􀀀m) for n  m > 1.
(2) Prove that the number of partitions of n into at most m parts is the number of
partitions into parts whose largest part is at most m. I.e. pm(n) = m(n). (Hint:
consider Ferrers diagrams.)
(3) Use the method of proof for Property 4 to show that every positive integer can be
expressed uniquely as a sum of powers of 2 with each power appearing at most once
(i.e. there is a unique representation in base 2).

assignment 7