If we solve the Riemann hypothesis, it could lead to new mathematics that millions of mathematicians would be eager to understand.

The parity barrier in number theory limits the density of primes, making it a significant obstacle in proving conjectures like the twin prime conjecture.

In undergraduate education, you learn about really hard problems like the Riemann Hypothesis and the Twin Primes Conjecture, but what's interesting are the problems that exist on the boundary between solvable and unsolvable.

If our core belief about the randomness of prime numbers is proven wrong, it could reveal patterns that undermine many cryptographic systems, leading to significant doubts about their security.

The twin prime conjecture posits that there are infinitely many pairs of prime numbers that differ by two, but proving it remains elusive due to the sparsity of twin primes as numbers increase.

The twin prime conjecture is difficult because while there are infinitely many primes, twin primes become increasingly rare as numbers grow larger, complicating proof strategies.

Turing's work on uncomputable numbers and Gödel's incompleteness theorems reveal that not all mathematical truths can be proven, challenging the belief that every mathematical problem is solvable.

Peter Shor's suggestion that quantum computers could solve problems related to numbers, like finding the prime factors of large integers, sparked significant excitement due to its implications for cryptography.

The Collatz conjecture, which involves iterating a simple process on natural numbers, is easy to state but remains unsolved, illustrating the complexity of seemingly simple mathematical problems.