Prime (or prime number) (from Latin primum, "the first"): a natural number with exactly two distinct divisors. This means that is has to be greater than 1 and that it is divisible only by 1 and by itself.

Together with the number 1, the prime numbers can be used to generate all natural numbers, for each natural number can be generated in a unique manner through multiplication of prime numbers. The Greek mathematician ►Euclid¬†proved that there is an infinite number of primes. The proof is that if there were not, a number N could be derived from the product of all existing primes plus 1. This number, however, would be a further prime, for whenever it is divided by a prime there will be a remainder of 1. In short, the product of any finite number of primes will be yet another prime. Therefore, the number of primes cannot be finite.

Prime numbers seem to be a simple matter. Nonetheless, their distribution on the ►number ray is one of the great puzzles of mathematics. It appears to obey some rules whose precise nature mathematicians have yet to discover. If you are interested in doing your own research in this area, try figuring out the distribution of primes among the first 150 numbers:

0 1 2 3 4 5 6 7 8 9
10 11 12 13 14 15 16 17 18 19
20 21 22 23 24 25 26 27 28 29
30 31 32 33 34 35 36 37 38 39
40 41 42 43 44 45 46 47 48 49
50 51 52 53 54 55 56 57 58 59
60 61 62 63 64 65 66 67 68 69
70 71 72 73 74 75 76 77 78 79
80 81 82 83 84 85 86 87 88 89
90 91 92 93 94 95 96 97 98 99
100 101 102 103 104 105 106 107 108 109
110 111 112 113 114 115 116 117 118 119
120 121 122 123 124 125 126 127 128 129
130 131 132 133 134 135 136 137 138 139
140 141 142 143 144 145 146 147 148 149

Except for the first row, primes must always end with 1, 3, 7 or 9; otherwise they would be divisible by 2 or 5. If you look at the patterns in one or the other column, regularities seem to emerge, only to disappear again at some later point. It can be proven that primes become less and less frequent as the numbers get larger. The distribution shows that many primes occur in pairs with an interval of 2 (for example, 5 and 7, 11 and 13, 17 and 19, 29 and 31.). These prime twins exist even among very large prime numbers. However, no one to date (2008) has found out whether the number of prime twins is infinite.

Another possibility awaiting proof — this one for over 400 years — is Goldbach's conjecture that every even integer is the sum of two primes (counting 1 as a prime, as Goldbach did; since 1 is no longer considered a prime, the conjecture is now formulated as "Every even integer greater than 2 is the sum of two primes"). No one has found a counterexample to this conjecture, but its proof has yet to be delivered.

There is no mathematical formula to calculate, from any given number, the nearest prime greater than it. This makes it difficult to identify primes. It is more or less a matter of systematic hunting and requires sophisticated computer programs. As of 2008, the largest prime yet identified is

243112609-1

This is a prime with 12978189 digits discovered in September 2008 by the GIMPS project at the University of California.

By the way, there is some money in the search for primes, for they are needed in cryptography. The record prime number just mentioned earned the organizers of the project a cash award of $100,000 from the ►Electronic Frontier Foundation in San Francisco.


Links Related to the Topic

■ Largest Known Primes
■ List of the first 10000 primes

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