A NEW CLASSIFICATION OF NUMBERS

By
P. G. RAJESWARA RAO
p_grrao@yahoo.co.nz

We can’t live without numbers. We need them in our daily activities – big and small.
I would like to share with you some special classification that I have observed with natural numbers

Take any natural number and find its digit sum square it and find the digit sum of the square. If we continue the procedure the outcome is as follows:

Take 22

Sum of digits = 4
Square of 4 = 16
Sum of digits = 7
Square of 7 = 49
Sum of digits = 13
Square of 13 = 169
Sum of digits = 16
Square of 16 = 256
Sum of digits = 13
Square of 13 = 169

22 , 4 , 16 , 7 , 49 , 13 , 169 , 16 , 256 , 13 , 169
The series repeats from 13 to 256 continuously

Take 35

35 , 8 , 64 , 10 , 100 , 1 , 1 , 1 , 1 , ……………………………….
It continues with one only

Take 75

75 , 12 , 144 , 9 , 81 , 9 , 81 , 9 , 81……………………………….
It continues with 9 and 81 alternatively

According to my observation all natural numbers came under any of these three types

So the numbers can be classified into three categories based on the special property

1) C TYPE : Cyclic type which will have repeating series of 13 – 256
2) N TYPE : These are the series of numbers which have 9 , 81 as repeating numbers
3) O TYPE : these are the series if numbers which will have only one recurring

From the above examples

22 is C TYPE
35 is N TYPE
75 is O TYPE


SOME SPECIAL OBSERVATIONS

1) Among the single digit numbers O type are 2, N type are 3, C type are 4.
2) Among the double digit numbers O type are 20, N type are 30, C type are 40.
3) N TYPE has no primes.
4) O TYPE has 7 primes among which 2 pairs are twin primes.
5) C TYPE numbers have 14 primes
6) Multiples of 3 , 6 , 9 belong to N TYPE
7) Multiples of 2 , 4 , 5 , 7 , 8 are distributed in the three categories
8) If a number belongs to one category then the sum of that number and nine multiples belongs to same category
9) This type of classification can also be done with cubes and digit sum there also three categories will exist.