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More about 1729 (Posted on 2012-01-04) Difficulty: 3 of 5
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.
 

See The Solution Submitted by Ady TZIDON    
Rating: 5.0000 (3 votes)

Comments: ( Back to comment list | You must be logged in to post comments.)
re(2): computer solution | Comment 4 of 8 |
(In reply to re: computer solution by Jer)

well I believe there is room to argue how one defines a distinct fibonacci number, is it distinct by its index or its value.  The former allows for 1 and 1 to be differently as they occur twice in the sequence, it is the latter that forces us to use only a single 1.  In the case of the latter, we get all the solutions that have been previously shown.  The former allows for another 30 solutions,  20 of which are duplicates of the others (as you are simply using the other 1 value instead).  However, there are 10 additional solutions that are of more interest in which both 1's are used.

1: {1,1,2,5,34,89,1597}
2: {1,1,2,5,13,21,89,1597}
3: {1,1,2,5,34,89,610,987}
4: {1,1,2,5,13,21,34,55,1597}
5: {1,1,2,5,13,21,89,610,987}
6: {1,1,2,5,34,89,233,377,987}
7: {1,1,2,5,13,21,34,55,610,987}
8: {1,1,2,5,13,21,89,233,377,987}
9: {1,1,2,5,13,21,34,55,233,377,987}
10: {1,1,2,5,13,21,34,55,89,144,377,987}

  Posted by Daniel on 2012-01-04 14:31:42
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