Number Theory: Lecture 13

In which we think about which numbers are represented by the forms x^2 + y^2 and x^2 + xy + y^2 (amongst others).

  • Theorem 38: Let n be a natural number.

    1. Suppose that n is properly represented by a form of discriminant d.  Then there is a solution to the congruence w^2 \equiv d \pmod{4n}.
    2. Suppose that there is a solution to the congruence w^2 \equiv d \pmod{4n}.  Then there is a form of discriminant d that properly represents n.

    We proved this with the help of Lemma 37.

  • Proposition 39 (Hensel‘s Lemma): Let f be a polynomial with integer coefficients, and let p be an odd prime.  Suppose that there is an integer x_1 such that f(x_1) \equiv 0 \pmod{p} and f'(x_1) \not\equiv 0 \pmod{p}.  Then for each r \geq 1 there is some x_r such that f(x_r) \equiv 0 \pmod{p^r}.  We proved this inductively, showing that for each r there is an x_r such that f(x_r) \equiv 0 \pmod{p^r} and x_r \equiv x_1 \pmod{p} (so f'(x_r) \not\equiv 0 \pmod{p}).

Understanding today’s lecture

Pick a form f.  Can you find a necessary and sufficient condition for f to represent a prime p?  (How does the class number of the discriminant of f make a difference?)

Further reading

I mentioned that Hensel’s Lemma is reminiscent of Newton’s method.

Jones and Jones (Elementary number theory) has another explicit example of an application of Hensel’s lemma.

Preparation for Lecture 14

Can you finish classifying the numbers that can be written as a sum of two squares?

We’re about to move on to a new topic: the distribution of the primes.  Here are some questions to consider.

  • Show (if you haven’t done so previously) that there are infinitely many primes congruent to 1 modulo 4, and that there are infinitely many primes congruent to 3 modulo 4.  What conditions would you need to put on a and n to ensure that there are infinitely many primes congruent to a modulo n?
  • You hopefully know that \sum_n \frac{1}{n} diverges, but that \sum_n \frac{1}{n^2} converges.  What about \sum_p \frac{1}{p} (where the sum is over all primes p)?  Can you estimate \sum_{p \leq x} \frac{1}{p} as a function of x?
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