Computational number theory is the branch of number theory concerned with finding and implementing efficient computer algorithms for solving various problems in number theory. Much progress has been made in this field in recent years, both in terms of improved computer speed and in terms of finding more efficient algorithms. Two important applications of computational number theory are primality testing and prime factorization of large integers.Primality testing is considered easy in the sense that very large general numbers (currently up to 4000 digits or so) can be tested reliably for primality. In fact, on August 6, 2002, Agrawal, Saxena, and Kayal found a polynomial time algorithm for testing and proving the primality of general numbers. Although this algorithm is still impractical, it was a landmark discovery, since polynomial time algorithms are considered easy. On the other hand, factoring is considered hard in the sense that..
Analytic number theory is the branch of number theory which uses real and complex analysis to investigate various properties of integers and prime numbers. Examples of topics falling under analytic number theory include Dirichlet L-series, the Riemann zeta function , the totient function , and the prime number theorem.
Number theory is a vast and fascinating field of mathematics, sometimes called "higher arithmetic," consisting of the study of the properties of whole numbers. Primes and prime factorization are especially important in number theory, as are a number of functions such as the divisor function, Riemann zeta function, and totient function. Excellent introductions to number theory may be found in Ore (1988) and Beiler (1966). The classic history on the subject (now slightly dated) is that of Dickson (2005abc).The great difficulty in proving relatively simple results in number theory prompted no less an authority than Gauss to remark that "it is just this which gives the higher arithmetic that magical charm which has made it the favorite science of the greatest mathematicians, not to mention its inexhaustible wealth, wherein it so greatly surpasses other parts of mathematics." Gauss, often known as the "prince of..