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We will prove Dirichlet’s Theorem on Primes in Arithmetic Progressions.
Euclid proved that there are infinitely many primes. This, however, says nothing about the distribution of those primes. Dirichlet proved a stronger statement: there are infinitely many primes in every arithmetic progression; in this sense, the primes are “uniformly distributed”.
Theorem 1 (Dirichlet’s Theorem) If , then there are infinitely many primes satisfying .
Many ideas of the proof of Dirichlet’s Theorem were first seen in Euler’s proof of the infinitude of primes. Here, Euler actually showed that diverges, and that is the approach that we will take for Dirichlet’s Theorem.
Happy Pigeonhole Day!
In case you haven’t heard of Pigeonhole Day, here’s an explanation:
Since 365 is congruent to 1 (mod 7), there is one day of the week that appears once more than the others. This year, it is Friday. Each month has four or five Fridays, so there are seven months with four Fridays. This means that there is one quarter of the year with only one such month. This is the current quarter, and the month is November. Additionally, since November has only four Fridays, the Fridays cannot be on November 1st or 2nd. This means that there can only be one Friday with a single-digit date, and that is today, November 5.
For a (better) explanation of Pigeonhole Day, see this mathoverflow post.
In fact, in every non-leap year, Pigeonhole Day can be defined in the same way; simply relabel the days of the week and observe that every year has isomorphic calendars. Pigeonhole Day is on November 5 in every year except for leap years, when it isn’t well-defined; in those years, it is probably best to celebrate leap day on February 29 instead.
Why do we care about Pigeonhole Day? It is a mathematical holiday, and it’s always nice to have an excuse to celebrate. Yes, people often think of Pi Day as the canonical mathematical holiday, but Pi Day is inherently numerological; there is no good reason for the decimal digits of pi to have any deep mathematical meaning, and mathematics should celebrate important principles instead of devolving into a subfield of numerology. So we can think of Pigeonhole as a purer holiday. That’s not to say that I’m not going to celebrate Pi Day as well — I can’t resist the temptation. However, Pi Day is for the masses, while Pigeonhole Day is for the truly enlightened.