In which we think about which numbers are represented by the forms and (amongst others).
- Theorem 38: Let be a natural number.
- Suppose that is properly represented by a form of discriminant . Then there is a solution to the congruence .
- Suppose that there is a solution to the congruence . Then there is a form of discriminant that properly represents .
We proved this with the help of Lemma 37.
- Proposition 39 (Hensel‘s Lemma): Let be a polynomial with integer coefficients, and let be an odd prime. Suppose that there is an integer such that and . Then for each there is some such that . We proved this inductively, showing that for each there is an such that and (so ).
Understanding today’s lecture
Pick a form . Can you find a necessary and sufficient condition for to represent a prime ? (How does the class number of the discriminant of make a difference?)
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 modulo , and that there are infinitely many primes congruent to modulo . What conditions would you need to put on and to ensure that there are infinitely many primes congruent to modulo ?
- You hopefully know that diverges, but that converges. What about (where the sum is over all primes )? Can you estimate as a function of ?