Ramanujan's number: finding sum of two positive cubes

Ideas
,
1

fi =
FindInstance[{a^3 + b^3 == c^3 + d^3, a != c, a != d}, {a, b, c, d},
PositiveIntegers]

{{a -> 10, b -> 9, c -> 12, d -> 1}}


{a^3 + b^3, c^3 + d^3} /. fi

 {{1729, 1729}}

Solve, NSolve and Reduce cannot do this calculation

2

It is known as the Hardy–Ramanujan number because of a famous conversation between mathematicians G. H. Hardy and Srinivasa Ramanujan.

Ramanujan pointed out that 1729 is the smallest positive integer that can be expressed as the sum of two positive cubes in two different ways:

Hardy had remarked that his taxi number, 1729, seemed rather dull. Ramanujan immediately replied that it was actually a very interesting number for precisely this reason.

3

Frank,
As we discussed via email yesterday, Minimize works:

In[7]:= Minimize[{(a^3 + b^3 - c^3 - d^3)^2, a >= 1, b >= 1, c >= 1,  d >= 1, a <= 20, b <= 20, c <= 20, d <= 20, a != c, a != d}, {a, b, c, d} \[Element] Integers]
Out[7]= {0, {a -> 1, b -> 12, c -> 9, d -> 10}}

The upper bounds were chosen somewhat subjectively (knowing the H-R solution).
Other reasonable bounds could also be chosen depending on how far one wants to go, in search of further H-R numbers.

Regards,
Janos

[Janos D. Pinter]

4

In[6]:= s =
Select[Tuples[Range[12],
4], #[[1]]^3 + #[[
2]]^3 == #[[3]]^3 + #[[4]]^3 && #[[1]] != #[[3]] && #[[1]] !=
#[[4]] &]

Out[6]= {{1, 12, 9, 10}, {1, 12, 10, 9}, {9, 10, 1, 12}, {9, 10, 12, 
  1}, {10, 9, 1, 12}, {10, 9, 12, 1}, {12, 1, 9, 10}, {12, 1, 10, 9}}

In[7]:= Table[{s[[i, 1]]^3 + s[[i, 2]]^3, 
  s[[i, 3]]^3 + s[[i, 4]]^3}, {i, Length[s]}]

Out[7]= {{1729, 1729}, {1729, 1729}, {1729, 1729}, {1729, 
  1729}, {1729, 1729}, {1729, 1729}, {1729, 1729}, {1729, 1729}}