Many questions regarding prime numbers remain open, such as Goldbach's conjecture (that every even integer greater than 2 can be expressed as the sum of two primes), and the twin prime conjecture (that there are infinitely many pairs of primes whose difference is 2). Such questions spurred the development of various branches of number theory, focusing on analytic or algebraic aspects of numbers. Primes are used in several routines in information technology, such as public-key cryptography, which makes use of properties such as the difficulty of factoring large numbers into their prime factors. Prime numbers give rise to various generalizations in other mathematical domains, mainly algebra, such as prime elements and prime ideals.
What is the name of the supposition that any number larger than 2 can be represented as the sum of two primes? What is the name of the supposition that there are infinite pairs of primes whose difference is 2? Besides the analytic property of numbers, what other property of numbers does number theory focus on? What is the application of prime numbers used in information technology which utilizes the fact that factoring very large prime numbers is very challenging? What is the name of one algebraic generalization prime numbers have inspired?