All numbers (here meaning, all integers) can be represented as the product of two numbers. Thus
6 = (2)(3)
6 = (1)(6)
Such representation, or resolution into factors, is called factorization. A prime number p is one that can be factored in only one way, into the numbers (1) and (p). All other numbers, which can be factored in more than one way, like 6 in the case above, are called composite.Theory
Prime numbers have long fascinated Western mathematicians, and the attempt to investigate their properties has generated much of Western number theory as a sort of byproduct. Primes are difficult subjects for investigation. They have few qualities of their own, and appear instead to be the residue that is left after all composite numbers have been eliminated from the sequence of natural numbers. (The elimination of composites, first the multiples of 2, then the multiples of 3, and so on, leaving at the end only the primes still standing, is called "the sieve of Eratosthenes").
Some facts about primes are known. A few of them are listed here as a matter of curious interest.
- There is no highest prime (Euclid). Here is why. Suppose there were, and call it h. Then we form the number consisting of all primes between 1 and h multiplied together, and add 1 to it. Either the result (call it r) is prime or it is not. (1) If r is prime, then it is larger than h, refuting the claim that h is the largest prime. (2) If r is not prime, then it has a prime divisor other than any of the primes between 1 and h (none of which can divide r, given the way r was formed), and thus that prime divisor must be larger than h, again refuting the claim that h is the largest prime.
- There is also no limit to the number of double primes (primes which differ by 2, such as 29 and 31), eg 1,000,000,000,000,061 and 1,000,000,000,000,063.
- Primes do grow less dense as we go higher in the natural number sequence. We would have expected this a priori, and the tendency can be observed directly by counting primes per hundred numbers in the table of primes from 1 to 3000.
- The density of primes can be approximated by the formula n / log n (Legendre; proved by Hadamard).
- If 2*n - 1 is prime, then n itself is prime (Cataldi-Fermat).
- Every prime of the form (4n + 1) is the sum of two squares, and is such a sum in only one way (Fermat).
- If from q (not necessarily a prime) we form the number (q - 1)! + 1, calling it r, then r is divisible by q if and only if q is a prime (Wilson, proved by Lagrange).
- There is no algebraic formula which yields only primes.
Primes as such play no role in statistics. There is no harm in thinking about them, with the warning that all the easy facts about them have probably been picked up by now, and that thinking about them is not really contributing to your statistical education.
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