Some Interesting Integers

The following are some integers and the relationships between them, that I have found interesting. I suppose that you could argue that all integers are interesting in some way, and undoubtedly you know others that interest you, but I will focus on only a few that have fascinated me.

4 = 2 + 2 = 2 x 2

Four is the only non-prime integer whose prime factors add to create the number.
It is also the first composite number.

8 = 2³ and 9 = 3²

These happen to be the only consecutive integers that are powers of primes.

13 & 31 and 17 & 71

These two digit reversals create the following powers of 3:
(31 - 13)/2 = 3²
(71 - 17)/2 = 3³

24 and 70

sum( n² ) = 70² where n goes from 1 to 24. This has been proven to be the only sum of squares of consecutive integers (starting at 1), that creates a square.

31 and 8191

31 and 8191 are the only two known numbers (N) that have 3 different cases of the following relation. All the rest have only one or two such cases. Notice that 31 and 8191 are the 3rd and 5th Mersenne primes respectively.
N = (bⁿ - 1)/(b - 1) for 1 < b < N and n > 1
Aside: N is the n-digit number in base b with all digits equal to 1.

31 = (2⁵ - 1) = (5³ - 1)/4 = (30² - 1)/29
8191 = (2¹³ - 1) = (90³ - 1)/89 = (8190² - 1)/8189

Also N = (31³¹ - 1)/30 is prime, and so is N = (19¹⁹ - 1)/18

55 = 5 x 11

55 = 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10
55 = 1² + 2² + 3² + 4² + 5²
55 is also the tenth Fibonacci number

89 = the 24th prime

89 interests me since it is the 11th Fibonacci number, and all terms of the Fibonacci sequence are embedded in the decimal expansion of 1/89:

1/89 = 0.01123595506
     =    112358
               13
                21
                 34
                  55
                   89 
                   144 
                    233

Add the Fibonacci numbers, and you get the decimal digits of 1/89. Start
adding from the bottom right, and don't forget to carry. You will need more
terms to obtain more decimal digits.

The prime 2521

(2521-1) x 2 = 7! = 10!/6!
2521 = 35² + 36²
2521 x 2 - 1 = 71²
2521 - 1 = 2 x 2 x 2 x 3 x 3 x 5 x 7

21 = 3 x 7

And last, but not least, twenty-one just happens to be my favorite number, so I have included it also.
21 is the 8th Fibonacci number. It is also the palindrome 1221 in base 2 of the Alternate Number System and the palindrome 10101 in base 2 of our Existing Number System. Coincidently, I spent my youth at a house which has the number 21, and was born on the 21st of June. The number 21 seems to follow me! ;-)

Enjoy

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