Discipline in
digital world
Abstract: All numbers are having its own characteristics when mixed
together in a relation, they show strict discipline which is curious to know. The
disciplinary behavior of a group of numbers is quite amazing when exposed. Few inherent
disciplines in the digital world are described in this article.
Key words: Number theory, Numeral relations- Triangular numbers
Introduction:
In numeral relations, two or more numbers are connected together with
some mathematical operators When the numbers are varied under the same
condition, the disciplinary behaviors of the numbers strike our mind. The
inherent disciplines of a group of numbers in the digital world are exemplified
with few simple numeral relations.
1.Sum of three squares of two successive numbers and its product
Sum of two squares of any two successive numbers is added with the
square of its product always gives a square, the root number of which is one
excess over the product number. This in turn is a factor for the sum of fourth
powers of the same set of numbers.
12 + 22 +
22 = 32 ; 14 + 24 + 24 = 33 = 3 x 11
22 + 32 +
62 = 72 ; 24 +
34 + 64 = 1393 = 7 x 199
32 + 42 + 122 = 132 ; 34
+ 44 + 124 =
21073 = 13 x 1621
42 + 52 + 202 = 212 ; 44 + 54 +
204 = 160881 = 21 x 7661
For any number n, it can be expressed as n2 + (n+1)2 +[n(n+1]2 =[ n(n+1) +1]2 and it
is found that [n(n+1) + 1] is always a factor for n4 + (n+1)4 +[n(n+1]4
2. Sum of three squares of any two numbers and its sum
Sum of squares of any two numbers when added with the square of its sum
has a peculiar property . Half of the sum when added with the product of the
two numbers taken gives the square of its sum
12 +
122 + 132 =
2 x 157 ; 157 + 1 x 12 = 132
22 + 112 + 132 = 2 x 147 ; 147 + 2 x 11 = 132
32 + 102 + 132
= 2 x 139 ; 139 + 3 x 10 = 132
42 + 92 + 132
= 2 x 133 ; 133 + 4 x 9 = 132
For any two numbers m and n, we have
m2 + n2 +
(m+n)2 = 2( m2 + n2 + mn) ;
m2 + n2 + mn + (m
x n) = (m+n)2
3. Sum of four different powers of any four successive numbers
It is found that the sum of
four different powers of any four successive numbers where both the numbers and
the exponents are in ascending order has some curious properties
1 + 22 + 33 + 44 = 288 = 2 x 144 = 3 x 42 x 6
2 + 32 + 43 + 54 = 700 = 7 x 100 = 4 x
52 x 7
3 + 42 + 53 + 64 = 1440 = 10 x 144 = 5 x 62 x 8
4 + 52 + 63 + 74 = 2646
= 6 x 441 = 6 x 72 x 9
In general, n + (n+1)2
+ (n+2)3 + (n+3)4 = (n+2) (n+3)2 (n+5)
With any three successive numbers, we have
1 + 22 + 33
= 32 = 2 x 42
2 + 32 + 43 = 75 = 3 x 52
3 + 42 + 53 = 144 = 4 x 62
4 + 52 + 63 = 245 = 5 x 72
In general, n + (n+1)2 + (n+2)3 = (n+1) (n+3)2
4 The sum of any three successive numbers when added with its mean
gives the cube of that mean
1 x 2 x 3 + 2 = 23
2 x 3 x 4 + 3 = 33
3 x 4 x 5 + 4 = 43
In terms of n
n x (n+1) x (n+2) + (n+1) = (n+1)3
The square and
cube of successive numbers are related with successive triangular numbers Tn
which are nothing but the sum of natural
numbers from 1 to n, 1,3,6,10,15,21,28,36,45,55,66,78………. [n(n+1)]/2
12 = 1
22 = 2 + 2(1)
32 = 3 + 2(3)
42 = 4 + 2(6)
n2 = n + 2(Tn-1)
The cubes have different correlation with the triangle numbers
13 = 1
23 = 2 + 6(1)
33 = 3 + 6 (1+3)
43 = 4 + 6(1+3+6)
n3 = n + 6 x (∑ Tn-1)2
The fourth power of a number
14 = 1
24 = 22 + 12 (1)
34 = 32 + 12 (1+4)
44 = 42
+ 12( 1+4+ 9)
In general, n4 = n2
+ 12 n-1∑ x2
1
No comments:
Post a Comment