*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?)

#### 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 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 ?

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