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31

Science Discussion Forum / even perfect number

« on: March 05, 2012, 03:23:41 PM »

Over 2300 years ago Euclid proved that If 2^(k)-1 is a prime number (it would be a Mersenne prime), then 2^(k-1)(2^(k)-1) is a perfect number. A few hundred years ago Euler proved the converse (that every even perfect number has this form).  It is still unknown if there are any odd perfect numbers (but if there are, they are large and have many prime factors).

32

Science Discussion Forum / What is an angle measure of regular 12 sided polygon?

« on: March 05, 2012, 03:18:05 PM »

Answer:

Each exterior angle measures 30 degrees

Each interior angle measures 150 degrees

33

Science Discussion Forum / Cross product is not define in two space why?

« on: March 05, 2012, 03:15:15 PM »

why Cross product is not define in two space? 

Answer:
When performing the cross product of two vectors (vector A and vector B), one of the properties of the resultant vector C is that it is perpendicular to both vectors A & B. In two dimensional space, this is not possible, because the resultant vector will be perpendicular to the plane that A & B reside in. Using the (i,j,k) unit vector notation, you could add a 0*k to each vector when doing the cross product, and the resultant vector will have zeros for the i & j components, and only have k components.

34

Science Discussion Forum / perfect numbers

« on: March 04, 2012, 04:08:36 PM »

In number theory, a perfect number is a positive integer that is equal to the sum of its proper positive divisors, that is, the sum of its positive divisors excluding the number itself (also known as its aliquot sum). Equivalently, a perfect number is a number that is half the sum of all of its positive divisors (including itself) i.e. σ1(n) = 2n.
The list of the perfect numbers are as follows:
6
28
496
8128
33550336
8589869056
137438691328
.......................


The four perfect numbers 6, 28, 496 and 8128 seem to have been known from ancient times and there is no record of these discoveries.

    6 = 1 + 2 + 3,
    28 = 1 + 2 + 4 + 7 + 14,
    496 = 1 + 2 + 4 + 8 + 16 + 31 + 62 + 124 + 248
    8128 = 1 + 2 + 4 + 8 + 16 + 32 + 64 + 127 + 254 + 508 + 1016 + 2032 + 4064