## Waring’s Problem: Lecture 8

In which we study the local factors $\beta_p(N)$ in more detail.

• Lemma 16: The sum $S(a,q)$ is multiplicative, in the sense that if $q_1$ and $q_2$ are coprime natural numbers then
$S(a_1,q_1)S(a_2,q_2) = S(a_1 q_2 + a_2 q_1, q_1 q_2)$
for any $a_1$ coprime to $q_1$ and any $a_2$ coprime to $q_2$.
We quickly proved this, essentially using the Chinese Remainder Theorem.
• Lemma 17: If $q_1$ and $q_2$ are coprime, then $A(q_1) A(q_2) = A(q_1 q_2)$.  (That is, $A$ is multiplicative.)  This was easy using Lemma 16.
• Corollary 18: The $\beta_p(N)$ are the local factors for this problem, in the sense that $\beta(N) = \prod_p \beta_p(N)$.  This follows immediately from Lemma 17 and the absolute convergence of the series for $\beta_p(N)$.
• Lemma 19: (i) Let $p$ be an odd prime, and write $k = 2^v k_0$ where $p$ and $k_0$ are coprime.  Let $c$ be an integer not divisible by $p$.  Suppose that there is some $m$ with $m^k \equiv c \pmod{p^{v+1}}$.  Then for each $r \geq v + 1$ there is a solution to $n^k \equiv c \pmod{p^r}$.
(ii) Write $k = 2^v k_0$ where $k_0$ is odd.  Let $c$ be an odd integer.  Suppose that there is some $m$ with $m^k \equiv c \pmod{2^{v+2}}$.  Then for each $r \geq v + 2$ there is a solution to $n^k \equiv c \pmod{2^r}$.
We proved this by considering the structure of the multiplicative group modulo $p^r$.  One could also use a suitable form of Hensel’s Lemma.
• Lemma 20: (i) Let $p$ be an odd prime, and write $k = p^v k_0$ where $p$ and $k_0$ are coprime.  Suppose that we have $a_1, \dotsc, a_s$ such that $a_1^k + \dotsb + a_s^k \equiv N \pmod{p^{v+1}}$.  Suppose moreover that $a_1 \not\equiv 0 \pmod{p}$.  Then for any $r \geq v+1$ there are at least $p^{(r-v-1)(s-1)}$ solutions to the congruence $n_1^k + \dotsb + n_s^k \equiv N \pmod{p^r}$.
(ii) Write $k = 2^v k_0$ where $k_0$ is odd.  Suppose that we have $a_1, \dotsc, a_s$ such that $a_1^k + \dotsb + a_s^k \equiv N \pmod{2^{v+2}}$.  Suppose moreover that $a_1$ is odd.  Then for any $r \geq v+2$ there are at least $2^{(r-v-2)(s-1)}$ solutions to the congruence $n_1^k + \dotsb + n_s^k \equiv N \pmod{2^r}$.
This was easy with the help of Lemma 19.

Wikipedia (inevitably) has some discussion of Hensel’s lemma.  You might like to look at Ben Green’s lecture notes, which give a slightly different way of tackling the material from today’s lecture.

#### Preparation for Lecture 9

Our next task is to show that there is a solution to $a_1^k + \dotsb + a_s^k \equiv N \pmod{p^R}$ (for suitable $R$), for large enough $s$.  You could try to prove this yourself ahead of the lecture on Wednesday.