singularity

singularity 🔎

Propositions

Definition. A proposition is a statement that is either true or false.

\[p(n) \mathrel{::=} n^2 + n + 41\]

• The symbol WWD means “equal by definition.” It’s always ok simply to write “=” instead of ::=, but reminding the reader that an equality holds by definition can be helpful.

In fact, it’s not hard to show that no polynomial with integer coefficients can map all nonnegative numbers into prime numbers, unless it’s a constant

• This statement is saying that there’s no magic formula using addition, subtraction, and multiplication of whole numbers that can turn every non-negative number (0, 1, 2, 3…) into a prime number.

• Think of it this way: A polynomial is like a formula where you plug in a number and get another number out. For example, x² + 2x + 1 is a polynomial. If we wanted this polynomial to always give prime numbers, then no matter what whole number we put in for x, the result would need to be prime.

• The statement tells us this is impossible unless the polynomial is just a constant (like “5” all by itself).

• Here’s why: If a polynomial isn’t constant, it eventually grows larger and larger. As it grows, it will eventually produce numbers that are divisible by other numbers - meaning they can’t be prime. For example, if f(n) = n² + n + 41, this gives primes for n = 0 to 39, but f(40) = 40² + 40 + 41 = 1681 = 41², which isn’t prime.

• No matter how clever you are with your polynomial formula, you can’t escape this limitation - there will always be some input that produces a non-prime output.

You can’t check a claim about an infinite set by checking a finite set of its elements, no matter how large the finite set.

\[\forall n \in \mathbb{N}. \, p(n) \text{ is prime}\]

for all n where n is a member of nonnegative integers, p(n) is prime