Prime Numbers

[LeedsRhinos]

What is it about prime numbers that deems them so important?

[Peter Pedant]

their primeness?

they're so simple in idea and spring up all over the place?

They're simple in idea, but their distribution has not been adequately explained?

Primes are interesting and prime pairs (11 n 13,  17 n 19, 29 n 31) are even more incredibly interesting!

[bernardo]

The number of prime numbers (equal to or below a certain limit) of the form 4n+1 becomes more than the number of prime numbers (equal to or below the same limit) of the form 4n+3 for the first time when the limit is 26861, although it has been proved that the number of primes of the form 4n+3 subsequently outnumbers those of the form 4n+1 (and vice versa) an infinite number of times.

[bernardo]

The largest known region of primes for which the number of primes (up to and including said prime) of the form 4n+1 outnumbers the number of primes of the form 4n+3 is the region from 18,465,126,293 to 19,033,534,538.

[bernardo]

For all values of n greater than 48, there is always at least one prime muber between n and 9n/8.

[bernardo]

For all values of n equal to or greater than 118, the region of numbers from n to 4n/3 always contains primes of the forms 4n+1, 4n-1, 6n+1 and 6n-1.

[bernardo]

There are 455,052,511 prime numbers less than 10,000 million.

[bernardo]

15 is the smallest product of two odd primes.

[bernardo]

1234567891

and

12345678901234567891

and

1234567891234567891234567891

are prime numbers.

[bernardo]

Every compound number can be factorised uniquely as a product of a set of prime numbers.

[bernardo]

396,733 and 396,833 is the smallest pair of consecutive primes differing by 100.

[Hippy]

Many years ago Euclid of Alexandria (possibly 325 BC- 265 BC) proposed several number, arithmetical and geometrical theories in his famous book, "The Elementals". One of these translates as follows.

 

Any integer can be represented as a product of primes. Since, by definition, a number is composite if its factors are other than 1 and itself wherein it would be prime, and it must follow that these factors will be smaller than the original number. It is possible to extract factors until only prime factors remain. Therefore: N = pqr..., where all p, q, r,... are prime. To prove uniqueness, assume there are two representations: N = pqr... = fgh... It is obvious that p divides fgh... It also follows that it divides one of the factors f,g,h,... So they cancel out by division. It is possible to continue chipping away at the factors left and right until no factors remain.

 

Representation of a number as the product of primes is called prime number decomposition. The Fundamental Theorem of Arithmetic asserts that each integer has an unique prime number decomposition.

 

If you are still with me and accept that an infinite number of primes exist, then there is scope for using prime factors as a basis for cryptography. Now that is really useful, and we all need security, locks, and codes. Primes provide a source of "interesting" arithmetical puzzles and number surprises. A simple one is 27 (product of prime 3 cubed than is raised to power 3) multiplied by 37 (prime formed from two prime digits) produces 999, three more prime digits, divisible by 3 (prime) and 111 (product of 3 and 37) and so on ...

 

[mr. piper]

They have an important job, although the are scarcely recognised as so. remember they are elected officials are worthy of our patience and trust, i have myself seen one at a distance ( Mrs. Thatcher) and i particlarly admired her twin set and pearls, but it might have been john major, whatever, they came out of no. 10 and drove to parliament.