Much was said about 1729 a.k.a. Ramanujan number.or a taxicab number.
1729 is also a Carmichael number and the first absolute Euler pseudoprime, a sphenic number, a Zeisel number etc etc

  Please note the following (and solve):
  1. 1729 is one of four positive integers (with the others being A, B and the trivial case 1) which, when its digits are added together, produces a sum which, when multiplied by its reversal, yields the original number:
1 + 7 + 2 + 9 = 19 ; 19*91 = 1729
Find A and B.

2. 1729= xyz, where x,y and z are integer members of an arithmetic progression.
Find these members.

3. 1729 can be expressed (in more than one way) as a sum of distinct Fibonacci numbers
List all the expressions.
 

'1729= xyz, where x,y and z are integer members of an arithmetic progression. Find these members'.

Incidentally, 1729 is the third member of an infinite series of such numbers:

Let (x-d)x(x+d)=y^3+1

Set x=(3a^2+1), y=3a^2, d=3a

Then ((3a^2+1)-3a)(3a^2+1)((3a^2+1)+3a)=(3a^2)^3+1

x={1,4,13,28,49,76..} see A056107 in Sloane.

 

Posted by broll on 2012-01-05 02:16:11