Theorem: Bananas do not exist.
Proof: A banana, if it exists, must be a polynomial--it has finitely many roots in a field, and splits.
However, bananas do not split over fields, but in kitchens, because fields are too hot and would cause the ice cream to melt.
Thus, bananas cannot be polynomials, and so cannot exist.
Alternative proof: Consider the set of banana splitting field jokes. By commutativity of a field, this is equivalent to the set field of banana splitting jokes. However, the set of banana splitting jokes is not a field. For instance, when you add two jokes to each other, the result is often unfunny and thus not a joke.
Theorem: Peas do not exist.
Proof: If peas exist, split peas exist, and thus does split pea soup. In particular, you split peas in a number field.
However, by the Chebotarev density theorem, in any number field, infinitely many peas split completely. So split pea soup does not exist.
Thus, peas do not exist.
What is the difference between Évariste Galois and a vector space?
A vector space has a second dual.
The Prime Number Theorem states that as \( x \to \infty \), the number of primes \( \pi(x) \) up to \( x \) is asymptotically \( x/\log{x} \). More specifically, we have \[ \pi(x) = \frac{x}{\log{x}} + O\left(\frac{x}{\log^2{x}} \right). \] This is, of course, not the best statement of the Prime Number Theorem. We define the offset logarithmic integral \( \operatorname{Li}(x) \) by \[ \operatorname{Li}(x) := \int_2^x \! \frac{\mathrm{d}t}{\log{t}}. \] Then in fact we have that \[ \pi(x) = \operatorname{Li}(x) + O\left( x\exp\left(-c\sqrt{\log{x}}\right) \right) \] for some constant \( c > 0 \) which gives a much better error term. Under the Riemann Hypothesis, in fact, that error term can be improved to \( \sqrt{x}\log{x} \). In other words, sometimes, the closest thing to the truth is a Li.
From the prime number theorem to Dirichlet's theorem, now that's what you call character development. (Some would call it progression.)
Okay, I admit I'm not being entirely truthful here. Dirichlet's theorem, at least in terms of logarithmic density, predates the prime number theorem. So more accurately I should've said something about, say, the PNT for progressions for the former, or the infinitude of primes for the latter. But sometimes, as the PNT itself shows us, the closest thing to the truth is a Li.
Eventually Frobenius and Chebotarev realized the connection to splitting behavior of primes in Galois extensions and conjugacy classes in the Galois group. It was bound to happen; one might call it density.
Since then there have been a bunch of Tauberian theorems and the Landau-Selberg-Delange method, marking a regression to the mean.
What do you get when you combine two functions from the set of dyadic rationals to itself?
Compost bins.
While people tout walkability as one of the benefits of living in a college town, and I do find Athens quite nice to walk around, some places remain inaccesible without a car. For example, I find it mildly annoying that Epps Bridge, where Trader Joe's, Guitar Center, and Best Buy are located, is inaccesible from where I live by bus or by foot. Really, though, I should've expected this. You can't always commute in Georgia, because Georgia is not abelian: the Georgia Center is not all of Georgia.
How do number theorists do online research on additive functions securely?
By using \( v_p(n) \) for each \( \log{n} \).
Two thieves try to break into a metal safe in the shape of a cube.
The first thief picks at it and pulls at it and tries everything, but the door does not budge a bit.
"Ugh. It's strongly closed," he mutters before giving up.
The second thief simply smiles, walks up to the safe, and gently pulls the door open.
His companion looks at him in awe and asks, "How did you do that?"
To which he responds, "Since the safe was strongly closed and convex, it was also weakly closed."
High school principals often do not teach; the reason cited is that the administrative work is onerous enough.
However, principals should teach. In fact, they should teach the first class in the morning.
This is because class number one is a principal ideal domain.