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Prove that

p (n)= 2+ \left (p (1) + \cdots + p\left ( \left [\frac {n}{2} \right ] + \chi_1 (n)\right ) + \left (p'_2(n) + \cdots + p' _{ \left [\frac {n}{2} \right ] - 1}(n)\right )\right )

for every

n \in \mathbb {N}

with

n>2

where

\chi

denotes the principal character Dirichlet modulo 2, i.e.

\chi _1 (n) = \begin{cases} 1 & \text{if } (n,2)=1 \\ 0 &\text{if } (n,2)>1 \end{cases}

with

p (n)

we denote number of possible partitions of

n

and

p' _m(n)

we denote the number of partitions of

n

in exactly

m

sumands.

Problem Statement