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.

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 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.

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2 Responses to “Waring’s Problem: Lecture 8”

  1. Waring’s Problem: Lecture 9 « Theorem of the week Says:

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

  2. Waring’s Problem: Lecture 11 « Theorem of the week Says:

    […] 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. […]

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