Tidbits

Pick two pairs of numbers at random. The probability that the pair with the larger sum also has the larger product is 11/12.

Formally, given $N \in \mathbb{N}$, choose integers $a,b,c,d$ uniformly at random from $[1,N]$. Then $P[ad>bc \mid a+d>b+c] \to 11/12$ as $N \to \infty$. (John D. Cook’s blog)

A peculiar interaction between blanket trait implementations and type inference in Rust:

trait Foo {
  fn abs(&self) -> i64 { 42 }
}
impl<T> Foo for T {}

fn main() {
    let x = 5;
    println!("{}", x.abs()); // outputs 42
    let _: i64 = x;
    println!("{}", x.abs()); // outputs 5
}

(r/rust comment)

Given a number $x_1>0$, consider the sequence defined recursively by

It turns out that there exists a unique starting value $x_1 = \alpha$ such that the displayed sequence diverges to $\infty$; this value $\alpha=1.187\dots$ is called the Foias constant. (Wikipedia)

In Go, type inference of type parameters doesn’t see through parentheses.

func id[T any](x T) T { return x }
var _ func(int) int = id    // infers T=int
var _ func(int) int = (id)  // type error

(I found the example above in this GitHub issue.)

A fun construction of the reals: define a function $f : \mathbb{Z}\to\mathbb{Z}$ to be an almost homomorphism if the set $\{f(a+b)-f(a)-f(b) :a,b \in \mathbb{Z}\}$ is finite. (This set is exactly $\{0\}$ when $f$ is a homomorphism, hence the name.) Let $S \subseteq \mathbb{Z}^\mathbb{Z}$ be the set of almost homomorphisms, and let $B \subseteq S$ be the set of bounded functions.

Then the quotient $S/B$, equipped with pointwise addition and composition as multiplication, is isomorphic to the real numbers. (“The Eudoxus Real Numbers”)

The Fibonacci sequence was originally used to model the growth of (idealized) rabbit populations (Wikipedia):

A newly born breeding pair of rabbits are put in a field; each breeding pair mates at the age of one month, and at the end of their second month they always produce another pair of rabbits. How many pairs of rabbits are there after a year?

There exists a complex sequence $(a_n)_n$ such that $\sum_{n \ge 0} a_n^k$ converges if and only if $k$ is prime.

More generally, for any fixed subset $A \subseteq \mathbb{Z}^+$, there exists a complex sequence $(a_n)_n$ such that $\sum_{n \ge 0} a_n^k$ converges if and only if $k \in A$. (Math Stack Exchange)

If $a(b+1)(ab+1)$ is a perfect square, then $(b+1)$ divides $a(ab+1)$.

(I originally posed this as a conjecture on Math Stack Exchange based on computational evidence, and user Mastrem provided a proof using a clever descent argument.)

In Go, the following is not an error:

type A struct { Field string }
type B struct { Field int }
type C struct { A; B }

even though C embeds A and B, both of which declare a field called Field.

If $f : \mathbb{Z}^\infty \to \mathbb{Z}$ is an additive function that vanishes on all sequences of the form $(1, 0, 0, \dots)$, $(0, 1, 0, \dots)$, $(0, 0, 1, \dots)$, and so on, then $f$ is the zero function. (Math Stack Exchange)

If this seems obvious, note that the statement is false for $f : \mathbb{Z}^\infty \to \mathbb{Q}$ under AC.