Sunday, December 5, 2010

creative thoughts-17

Creative thought -17

Equal product of two numbers with sum in one pair equals to the difference
 in the other pair.

It is very simple to everyone to show that a sum of two numbers is equal to a
sum of two other numbers and a product of two numbers is equal to a product
of two other numbers separately. It can be proved that same two pairs of numbers
will not satisfy both the conditions simultaneously. Mathematically it can be
 stated if a + b = c +d ,then ab =/= cd and if ab = cd then a+b =/= c+d.
If we assume supernaturally that they coexist, then the equality would become
true for all powers of the numbers in the pairs and it leads to an impossible relation

  n    n       n       n
a + b   = c +   d

where n = 1,2,3,,4…….. In fact such relation will be true under trivial condition
where the numbers in the pairs are equal.

However, a sum and product of two numbers may be made equal
with a difference and product of two other numbers respectively. e.g.,

2+3 = 5 = 6 – 1
2x3 = 6 = 6x1
3+10 = 13 = 15-2
3x10 = 30 = 15x2

For a same sum and difference, one can have two or more pairs of numbers
to give different products. e.g.,

4+21 = 25 = 28-3 ; 4 x 21 = 84 = 28x3
10+15 = 25 = 30-5 ; 10x15 = 150 = 30x5

As the pair of such numbers has substantial influence on the equal sum of
like powers , it is worth to investigate a method of identifying such pairs. For
a given sum (or difference) S and product P , the pairs (a,b) and (c,d) are
related by the following relations

a+b = S = c-d
and axb = P = cxd

Substitute for d (= P/c) in S = c-d ,we get,

(cxc) –Sc – P = 0

The roots of the quadratic equation give the value of c and d

c = [S+ √ (SxS) +4P]/2 and d = [S- √ (SxS)+4P]/2

We derived the solutions for a and b ( See-Creative thought-15)

a = [S + √ (SxS)-4P]/2 and b = [ S- √ (SxS)- 4P]/2

To have integral solutions for c and d , (SxS)+4P must be a square number
and for a and b (SxS)-4P must be a square number .

If (SXS)+4P = (ZxZ) and (SXS) -4P = (YxY),where Z and Y are integral
numbers ,then its sum and difference give

2(SxS) = (ZxZ)+ (YxY) and 8P = (ZxZ)- (YxY)

It implies that the pairs of numbers whose sum and difference are equal
and their products are equal are closely related with the

numeral relation representing the sum of two squares is equal to twice that
of an another square. e.g.,

(1x1) + (7x7) = 2 x (5x5)

It gives S = 5 and P = 6 and therefore we can derive the pairs as (2,3)and (1,6).

(7x7)+ (17x17) = 2 x(13x13)

It gives S=13 and P =30 and the pairs are (10,3) and (15,2).

The numbers expressed by the relation 2(nxn)+2n +1 ,where n is a number ,
generate such numeral relation where the sum of two squares is equal to twice
 the square of that number 2(nxn)+2n + 1

It can be further derived,

(cxc)+(dxd) = (SxS)+2P
(axa) + (bxb) = (SxS) – 2P

(axa)+(bxb)+(cxc)+(dxd) = 2 (SxS)= 2 [(a+b)(a+b)]=2[(c-d)(c-d)] Few solutions are

(2,3),(1,6); (2x2) + (3x3) + (1x1) + (6x6) = 2 (5x5)
(3,10),(2,15); (3x3) +(10x10) + (2x2) + (15x15) = 2 (13x13)
(5,12),(3,20); (5x5)+(12x12)+(3x3)+(20x20)= 2x(17x17)
(14,15)(6,35); (14x14)+(15x15)+(6x6)+(35x35)= 2(29x29)

From (ZxZ) +(YxY) = 2 (SxS), one can derive a,b,c and d,the members
of the pairs

a = √ [(ZxZ)+(YxY)]/8 + Y/2 and b = √ [(ZxZ)+(YxY)]/8 – y/2
c = Z/2 + √ [(ZxZ)+(YxY)]/8 and d = Z/2 - √ [(ZxZ)+(YxY)]/8

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