*In which we study the local factors in more detail.*

- Lemma 16:
*The sum is multiplicative, in the sense that if and are coprime natural numbers then*We quickly proved this, essentially using the Chinese Remainder Theorem.

for any coprime to and any coprime to . - Lemma 17:
*If and are coprime, then . (That is, is multiplicative.)*This was easy using Lemma 16. - Corollary 18:
*The are the local factors for this problem, in the sense that .*This follows immediately from Lemma 17 and the absolute convergence of the series for . - Lemma 19:
*(i) Let be an odd prime, and write where and are coprime. Let be an integer not divisible by . Suppose that there is some with . Then for each there is a solution to .*We proved this by considering the structure of the multiplicative group modulo . One could also use a suitable form of Hensel’s Lemma.

(ii) Write where is odd. Let be an odd integer. Suppose that there is some with . Then for each there is a solution to . - Lemma 20:
*(i) Let be an odd prime, and write where and are coprime. Suppose that we have such that . Suppose moreover that . Then for any there are at least solutions to the congruence .*This was easy with the help of Lemma 19.

(ii) Write where is odd. Suppose that we have such that . Suppose moreover that is odd. Then for any there are at least solutions to the congruence .

#### Further reading

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 (for suitable ), for large enough . You could try to prove this yourself ahead of the lecture on Wednesday.

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May 16, 2012 at 12:11 pm

[…] Theorem of the week Expositions of interesting mathematical results « Waring’s Problem: Lecture 8 […]

May 21, 2012 at 12:07 pm

[…] Proposition 28: Let be an integer and let be coprime to . Then . We defined and used the bounds from Lemma 26 and Lemma 27, together with the multiplicativity from Lemma 16. […]