A prime number is any number that cannot be divided by any other number except one and itself. The twenty-five prime numbers below 100 are 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, and 97. The only even prime number is 2 because all other even numbers are divisible by 2. The Fundamental Theorem of Arithmetic states that all numbers are divisible by only one product of prime numbers. Examples are 25=5×5, 35=5×7, 81=3^4, 63=3x3x7. It works because prime numbers do not have any divisors except 1 and themselves.
The Prime Number Theorem is an approximation of the number of primes up to and including any integer. It was discovered by the German mathematician Carl Friedrich Gauss (1777-1855). For example, the primes up to and including 25 are 2, 3, 5, 7, 11, 13, 17, 19, and 23. Since there are nine primes less than 25, the prime number theorem will equal about 9 when 25 is plugged into the formula pi(x)=x/ln(x)-1. The ln(x) means the natural logarithm of x. Pi(x) does not mean pi=3.141592653589793… . This pi is used to find the circumference of a circle and has nothing to do with the Prime Number Theorem. The natural logarithm of 25 is approximately 3.219 (actually it is an infinite decimal, but ln (25)=3.219 rounded to three decimal places. Now we have 25/3.219-1=25/2.219=11. The prime number theorem is +2 greater than the number of primes up to 25. There are many unsolved problems in mathematics that have to do with the prime numbers. The Goldbach Conjecture and The Riemann Hypothesis are two of the most important. The Goldbach Conjecture was discovered by the Prussian mathematician Christian Goldbach (1690-1764). It claims that any even integer greater than two can be expressed as the sum of two primes. Examples are 2+2=4, 7+3=10, 17+19=36, 31+37=68, and 29+97=126. One fact that suggests the conjecture is true is the number of sums of two primes that equal any even integer is very large. For example, 124=11+113, 17+107, 23+101, 41+83, and 53+71. The Riemann Hypothesis is a statement made by the German mathematician Bernhard Riemann (1826-1866). The hypothesis is a statement made by Riemann about the distribution of prime numbers. There is a one million dollar prize for a proof of the hypothesis.
The largest prime number that has been found was recently found on August 23, 2008 by computer. It has 12,978,189 digits. Computers were used to prove primality. It is a Mersenne Prime, which is a prime of the form (2^n)-1. It cannot be a prime number unless n is prime, but it is not necessarily prime if n is prime. As a matter of fact, it is usually not prime even if n is prime. The first Mersenne Primes are 2^2-1=3, 2^3-1=7, 2^5-1=31.
Another form of prime is the Fermat Prime, which is similar to Mersenne Primes. Fermat Primes have a plus sign instead of a minus sign: (2^n)+1. The only known Fermat Primes are (2^1)+1=3, (2^2)+1=5, (2^4)+1=17, (2^8)+1=257, and (2^16)+1=65537. Fermat Numbers are prime only if n is a power of two (except 3, the first Fermat Prime). There is also a computer project that is finding the factorization of large Fermat Numbers. If there are not any prime factors up to the square root of any integer, then that integer is prime. It is not known if either or both of the Mersenne Primes or Fermat Primes are infinite. It was proved by Euclid that the prime numbers are infinite many years ago. Other forms of primes are the Generalized Fermat Primes, which are (k*2^n)+1, and the Generalized Mersenne Primes (k*2^n)-1.