What is so interesting about 13?

• Reverse of the square of 13 is the same as the square of the reverse of 13 i.e. 13²= 169; on reversing 169, we get 961 which is the same as the square of reverse of 13 i.e. 31² = 961.
  
  13 x 13 = 169
  31 x 31 = 961

• Number 13 is the smallest prime number which can be expressed as the sum of the squares of two prime numbers i.e.
  
  13 = 2² + 3²

Where 2, 3, 13 are prime numbers.

• From number 13 if we subtract the sum of its digits, we get a perfect square i.e.
  
  13 - (1 + 3) = 9 = 3²

Also if the product of digits of 13 is added to it, we again a perfect square i.e.

13 + (1 x 3) = 16 = 4²

• Consider the equations:
  
  13²= 169
  31²= 961

If we insert the + sign between all the digits of above equation, the equation still holds good i.e.

(1+3)² = 1+6 +9
(3+1)² = 9 +6+1

• The smallest number whose sum of digits is 13 is a perfect square i.e. 49 = 7² and 4 + 9 = 13.
• The smallest square which contains last three digits alike is 1444. The sum of digits of this number is also 13 i.e.
  
  1+4+ 4+ 4 = 13

• If we place the number 13 before its reverse i.e. 31 we get the number 1331 which is a cubic number i.e. 1331= 11³
• If we take the cube of number 13 i.e. 13³ = 2197 and rearrange the digits of the cube 2197 we get the famous Ramanujan's number 1729.
• 13 is the average number of the prime factors of the famous Ramanujan's number 1729. i.e.
  
  1729 = 7 x 13 x 19

  (7 + 13 + 19) ÷ 3 =13

• Take the square of the number 13 i.e. 13² = 169. Now we see that 169 contains many perfect squares in it:
  
  The sum of the digits of the number 169 is a perfect squares and is the square of the sum of the digits of the original number i.e.
  
  1+6+9 = 16 = 4² =(1+3)²

Divide the number in two blocks i.e. 16 and 9. We see that both are perfect square numbers and their sum and product is a perfect square:

16 + 9 = 25 = 5²
16 x 9 = 144 = 12²

The number 169 which is perfect square can be expressed as sum of two square i.e.

169 = 13² = 5² + 12²

The number 169 can be expressed by number of its two block i.e. 16 and 9 which are perfect squares as follows:

169 = (16 + 9) + (16 x 9)

• Sum of the numbers from 1 to 13 gives 91 which is the smallest number which can be expressed as the sum of two cubes and also as the difference of two cubes i.e.
  
  91 = 3³ + 4³
  91 = 6³ - 5³

• The reciprocal of number 13 gives

1/13 = 0.076923076923076923...

i.e. the number 076923 repeating indefinitely. This number 76923 is one of the most curious numbers in mathematics.On multiplying the number 76923 by successive multiples of 13, the following beautiful pattern is obtained.

76923	x	13	=	0999999
76923	x	26	=	1999998
76923	x	39	=	2999997
76923	x	52	=	3999996
76923	x	65	=	4999995
76923	x	78	=	5999994
76923	x	91	=	6999993
76923	x	104	=	7999992
76923	x	117	=	8999991
76923	x	130	=	9999990

I published the above properties in reference [1] and [2]. Some more fascinating Prime Curios about 13 can be found at PRIME CURIOS.

Some New Fascinating Properties

• The Sum of all Prime Numbers up to 13 is equal to the 13^th Prime Number and this is the largest such number.
• The number 13 is a Fibonacci prime, so also the 13^th Fibonacci number is Fibonacci Prime and this is also the largest known such number.
• The sum of first 13 Prime numbers is 238, whose sum of digits is 13.

Update on 1-12-2002

• The number 13 appears for the first time at 111^th place in decimal expansion of Pi as shown below:

3.141592653589793238462643383279502884197169399375105820974944
   592307816406286208998628034825342117067982148086513282306647
   093844609550582231725359408128481117450284102701938521105559
   6446229489549303820......

Interestingly sum of digits of prime factors of 111 ( i.e. 3*37) is 13.

• The concatenation of cubes of numbers from 13 to 1 i.e.

2197172813311000729512343216125642781  is prime.

• The only known prime in the Smarandache Mirror sequence i.e.

1, 212, 32123, 4321234, 543212345, 65432123456, 7654321234567,
876543212345678, 98765432123456789,109876543212345678910,
1110987654321234567891011, 12111098765432123456789101112,
131211109876543212345678910111213,
1413121110987654321234567891011121314,.....
is the 13^thterm in the sequence.

• A beautiful anagrammatical equation involving 13 is:

ELEVEN + TWO = TWELVE + ONE.

• The Smallest composite titanic number whose none of the prime factor is known is 10⁹⁹⁹ + 13.

Update on 13-06-2003

• 13 is the Smallest Prime which can be represented as sum of two composite numbers and also sum of two prime numbers in only one way.

Update on 01-01-2004

• Sum of digits of Smallest 13 digit Prime (i.e. 1000000000039) is equal to 13.
• 31^st Prime is preceded by 13 consecutive composite numbers.

Some interesting facts

• The premier of many countries is designated as PRIME MINISTER containing exactly 13 letters.
• Some other common designations containing 13 letters are CHIEF MINISTER, CHIEF ENGINEER, VICE PRINCIPAL etc.
• Some of the mathematical terms containing 13 letters are PERPENDICULAR, CIRCUMFERENCE, ANTILOGARITHM, APPROXIMATION, QUADRILATERAL and INDETERMINATE.
• The standard length of Broad Gauge Rail in India is 13 meters.
• Teen-hood starts from Thirteen.
• 13 is also called a moon number because the moon moves approx. 13 degrees a day so there are 13 moon months in a year.

Update on 1-12-2002

• In Japan, the numbers 4 and 9 are considered unlucky, not because 13 can be represented as sum of these two perfect squares but because of their pronunciation. In Japanese, four is pronounced as shi which is the same pronunciation as death. Nine is pronounced ku which has the same pronunciation as torture.
• There is always atleast one friday 13^th in each year. In some years, there are two and rarely three. Most recently in 1998 there were three friday 13^th. Next such year will be 2009.