ramanujan numbers.?

Question

a^(n)-b^(n)=c^(2)-d^(2)
12^(3)-10^(3)=27^(2)-1^(2)
or
12^(3)-10^(3)=3^(6)-1^(6)
this is the smallest ramanujan number 1729 i.e
12^(3)+1^(3)=9^(3)+10^(3).
can u generate other ramanujam numbers from this information?

Answer 1

http://mathworld.wolfram.com/TaxicabNumber.html
There are a few there.

Answer 2

Ramanujam numbers are numbers that can be expressed as the sum of two cubes in two different ways.
1729 = 10^(3) + 9^(3) = 12^(3) + 1^(3)
Multiplying by n^(3) on both sides for any positive integer n is one easy way of generating the same family of Ramanujam numbers.
To get different families proceed as follows:
If .A^(3) + B^(3) = C^(3) + D^(3), take A= x+a, B x-a,C=(x-a)/b +c,
D=(x-a)/b -c. After simple calculations get x =a/6(2+b^(6)) & c=ab^(2).For fixed b & proper choice of a we get different Ramanujam numbers of the same family. For different b we get different families.
b=2, a=1 gives 1729.
b=3, a=6 gives 2211^3 + 563^3 =2175^3 + 725^3.....

Answer 3

87539319

Answer 4

I don't know