**PRIME NUMBERS**

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One of the groups of numbers where a lot of time was spent by mathematicians of all ages was the subject of prime numbers.

Prime numbers are positive integers that have no other factors than itself and *1*. All even integers except *2* are not prime numbers because these numbers have at least *2 *as a factor other than themselves and *1.*

The search for a universal equation that covers all prime numbers have been going on for years, but so far none has been found. There are some popular equations of prime numbers but these are either of limited range and do not yield purely prime numbers.

**a. ****FERMAT PRIME**

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The French mathematician, FERMAT, gave the following equation for prime numbers;

*F*_{n} = 2^{2n} +1 (28)

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The equation however does not yield purely prime numbers as it can be readily seen that when *n = 3, 2*^{6} +1 is equal to *65*, which is a multiple of *5*, hence is not a prime number.

**b. ****MERSENNE PRIME**

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The monk, Father MERSENNE, provided this equation for prime numbers:

*P = 2*^{n} – 1 (29**)**

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Where *n* is a prime number.

However, when *n = 11, 23, 29,* and** ***37,** ***P** is not a prime number. Conversely however, if *2*^{n} –1 is a prime number, *n* is a prime number.

So far the largest verified prime number as of September 4, 2006 is the 44^{th} MERSENNE, which is equal to;

*2*^{3258267} – 1

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**c. ****OTHER EQUATIONS OF PRIME NUMBERS**

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Other equations (Equations supplied by this author) that yield prime numbers are the following:

* ** ***P = 2**^{(2n + 1)} –1 (30)

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* P = 2*^{(2+2n)} __+__ 3 (31*)*

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* P = 2*^{(2 +2n) }+ 7 (32)

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As in the cases FERMAT and MERSENNE primes, the above equations are not guaranteed to yield purely prime numbers.

**d. ****THE SIEVES OF ERASTOTHENES**

As early as 300 B.C., prime numbers have fascinated mathematicians. ERASTOTHENES came up with a method of finding prime numbers. The method is popularly known as the sieve of ERASTOTHENES. The method is done as follows:

For a given range of numbers, say *1* through *100*, make a table as follows;

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 |