# Number Theory and the Partition function

I have a question with a problem from the book "An Introduction to The Theory of Numbers" by Niven (5th edition). This question is for anyone who has the book. In Chapter $10$ section $\S 3$ Formal Power Series, in exercise $1$, it mentions $p'(n)$ at the end of the problem. I can't seem to find any reference as to what that means in the book. If anyone can shed some light on this problem I am having I would be grateful. I also check the list of errors posted on previous and current editions for this book and haven't found anything regarding this. Also, before the book begins, it has a list of symbols introduced in the book along with the page number, but I do not see the symbol $p'(n)$ in there. Unless the symbol $p'(n)$ is used for the derivative for $p(n)$.

Thanks!

EDIT: Exercise $1$: Show that the infinite product

$(1+x_1)(1+x_1x_2)(1+x_1x_2x_3)\cdots = 1 + \sum x_1^{a_1}x_2^{a_2}\cdots x_k^{a_k}$

where $a_i - a_{i+1}$ is $0$ or $1$, and $a_k = 1$. Count the number of terms in the expansion that are of degree $n$. Set $x_1 = x_2 = x_3 = \cdots = x$ to show that $(1+x)(1+x^2)(1+x^3)\cdots$ is the generating function for $p'(n)$ of Problem $1, \S 10.2$.

EDIT: I had initially said $\S 10.4$ but I had made a mistake. It is actually $\S 10.3$.

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It may mean $p(n+1)-p(n)$ or $p(n)-p(n-1)$ in analogy with the derivative. I have seen that elsewhere, but don't have the book. – Ross Millikan Nov 14 '12 at 21:48
could you post an image or copy the text? – picakhu Nov 14 '12 at 21:48
Sure. I'll copy the problem verbatim. – MathNewbie Nov 14 '12 at 21:51
Unless I am missing something, I've looked at problem $1, \S 10.2$ and it makes no reference to $p'(n)$. – MathNewbie Nov 14 '12 at 21:57

The problem you have written out is actually from 10.3, Formal power series, generating functions, and Euler's Identity, not from 10.4. I think $p'(n)$ is a typo for $p^d(n)$, the number of partitions of $n$ into distinct parts, which is defined in Definition 10.2 and referred to in 10.2 problem 3 (which, in turn, refers to 10.2 problem 1).

EDIT: Actually, it's a carryover from the 4th edition, in which 10.2 problem 1 was,

Let $p'(n)$ denote the number of partitions, $$n=a_1+a_2+\cdots+a_r,$$ of $n$ into summands $$a_1\ge a_2\ge a_3\ge\cdots\ge a_r=1$$ such that consecutive $a_i$ differ by at most $1$. Read the graphs of such partitions vertically in order to prove that, $$p'(n)=p^d(n).$$

Note by the way that the book is not by Niven but by Niven and Zuckerman (first 4 editions), Niven, Zuckerman, and Montgomery (5th edition).

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Thanks. I misread this problem as in section 10.4 in lieu of 10.3. – MathNewbie Nov 14 '12 at 23:10
Ah I see, thanks for the clarification. This is really helpful! – MathNewbie Nov 14 '12 at 23:12

Looking up the book in Amazon, I was able to "look inside".

The p'(n), according to the problem, is in problem 1, section 10.2.

Amazon stopped me from looking further, but you have the book.

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Thanks for the response. I have the book in front of me and when I did the exercise $1, \S 10.2$, they only talked about partitions $\pi$ of $n$ and $\pi '$ being the conjugate partition of $\pi$ of $n$. At least for me, I do not see any connection with the function $p'(n)$. – MathNewbie Nov 14 '12 at 22:00