Cryptography relies heavily on number theory, particularly on the behavior of prime numbers, to create encryption methods that transform readable text into something indistinguishable from random noise, making it nearly impossible to crack mathematically.

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 Riemann hypothesis, which concerns the distribution of prime numbers, is a fundamental problem in mathematics that remains unsolved, with significant implications for number theory.

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

Quantum computers pose a risk to current encryption schemes, particularly public key cryptography, which relies on the difficulty of certain computational problems.

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.

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.

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