THE NUMBER 13

 Two interesting facts about the number 13: 

(i)  (1 / 13 )   =     0.076923076923076923076923...

                 (rational periodic infinite decimal)

On multiplying the periodic number 76923 by successive multiples of 13, the following curious numerical pattern is obtained.

76923 * 013 = 0999999
76923 * 026 = 1999998 
76923 * 039 = 2999997 
76923 * 052 = 3999996
76923 * 065 = 4999995

76923 * 078 = 5999994

76923 * 091 = 6999993

76923 * 104 = 7999992

76923 * 117 = 8999991

76923 * 130 = 9999990

 (ii) The numerical pattern abcabc is divisible by 13.

VAMPIRE NUMBER

The vampire numbers were introduced by Clifford A. Pickover in 1994.

A Vampire number z  is a number which can be written as a product of two numbers x and y  containing the same digits the same number of times as the vampire number. Here x and y are called FANGS.

FOR EXAMPLE:

1827 can be written as a product of two numbers 21 and 87.
i.e., 1827 = 21 * 87
Here the digits 1,8, 2,7 are repeated  the same number of times but in different order. 21 and 87 are called fangs.

A true Vampire number isa number which can be written with 2 fangs having the same number of digits not both ending in zero.

FOR EXAMPLE :

1260 is a vampire number, with 21 and 60 as fangs, since 21 × 60 = 1260. However, 126000 – 210 × 600 – is not, as both 210 and 600 have trailing zeroes.

All vampire numbers must clearly have an no. of digits.


Some vampire numbers:
 117067 = 167*701
124483 = 281 * 443
536539 = 563 · 953
23287176 = 2673 * 8712.




Vampire number having two distinct pairs of fangs :


125460 = 204 *615 = 246 * 510

Vampire number having three distinct pairs of fangs :

13078260 = 1620*8073 = 1863*7020 = 2070*6318

NUMBER PATTERN USING 1, 2, 8 AND 9

                      9 x 2 = 18

              99 x 2 = 198

            999 x 2 = 1998

          9999 x 2 = 19998

        99999 x 2 = 199998

      999999 x 2 = 1999998

    9999999 x 2 = 19999998

  99999999 x 2 = 199999998

999999999 x 2 = 1999999998

PALINDROMIC NUMBER PATTERN WITH 1

111111111
               * 111111111
--------------------------
 12345678987654321

A Strange Prime Number - 73,939,133

The prime number 73,939,133 has a very strange property.

If you keep removing a digit from the right hand end of the number, each of the remaining numbers is also prime.

It's the largest number known with this property.

Take a look: 73,939,133 and 73,939,13 and 73,939,1 and 73,939 and 7,393 and 739 and 73 and 7 are all prime!

142857 - CYCLIC NUMBER

142857 is a cyclic number, the numbers of which always appear in the same order but rotated around when multiplied by any number from 1 to 6.

142857 * 2 = 285714

142857 * 3 = 428571

142857 * 4 = 571428

142857 * 5 = 714285

142857 * 6 = 857142

1729 - TAXICAB NUMBER

1729 is smallest taxicab number , i.e., the smallest number representable in two ways as a sum of cubes. It is given by

1729 = 12³ + 1³

1729 = 10³ + 9³

The number derives its name from the following story:

G. H. Hardy told about Ramanujan. I remember once going to see him when he was ill . I had ridden in taxi cab number 1729 and remarked that the number seemed to me rather dull one, and that I hoped it was not an unfavorable omen. "No," he replied, "it is a very interesting number; it is the smallest number expressible as the sum of two cubes in two different ways."

1729 is the second taxicab number (the first is 2= 1^3 + 1^3). The number was also found in one of Ramanujan's notebooks dated years before the incident.

"Every positive integer is one of Ramanujan's personal friends."—J.E. Littlewood, on hearing of the taxicab incident

More details are available on the attached link.
Source(s):
http://en.wikipedia.org/wiki/1729_(number)