Thursday, January 12, 2006

Prime Numbers and laws

The first few primes are 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37...

Euler commented "Mathematicians have tried in vain to this day to discover some order in the sequence of prime numbers, and we have reason to believe that it is a mystery into which the mind will never penetrate" (Havil 2003, p. 163).

In a 1975 lecture, D. Zagier commented "There are two facts about the distribution of prime numbers of which I hope to convince you so overwhelmingly that they will be permanently engraved in your hearts. The first is that, despite their simple definition and role as the building blocks of the natural numbers, the prime numbers grow like weeds among the natural numbers, seeming to obey no other law than that of chance, and nobody can predict where the next one will sprout. The second fact is even more astonishing, for it states just the opposite: that the prime numbers exhibit stunning regularity, that there are laws governing their behavior, and that they obey these laws with almost military precision" (Havil 2003, p. 171).

From Prime number - Wikipedia, the free encyclopedia

4 = 2 times 2

23244 = 2 times 2 times 3 times 13 times 149

The fundamental theorem of arithmetic states that every positive integer larger than 1 can be written as a product of primes in a unique way, i.e. unique except for the order. Primes are thus the "basic building blocks" of the natural numbers. For example, we can write 23244 = 2 times 2 times 3 times 13 times 149 and any other such factorization of 23244 will be identical except for the order of the factors. See prime factorization algorithm for details for how to do this in practice for larger numbers.

The importance of this theorem is one of the reasons for the exclusion of 1 from the set of prime numbers. If 1 were admitted as a prime, the precise statement of the theorem would require additional qualifications.


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