This title space is too small! The topic is Hardy-Ramanujan numbers. #3

Number Theory Level 3

This problem's question: 1729 is first sum of paired cubes that can be done two different ways. What is the sum of the positive integers that comprise those two pairs (four positive integers altogether)?

See A011541 .

This is a pencil-and-paper problem.

See Taxicab number :

"Godfrey Hardy was a professor of mathematics at Cambridge University. One day he went to visit a friend, the brilliant young Indian mathematician Srinivasa Ramanujan, who was ill. Both men were mathematicians and liked to think about numbers.

When Ramanujan heard that Hardy had come in a taxi he asked him what the number of the taxi was. Hardy said that it was just a boring number: 1729. Ramanujan replied that 1729 was not a boring number at all: it was a very interesting one. He explained that it was the smallest number that could be expressed by the sum of two cubes in two different ways.

This story is very famous among mathematicians. 1729 is sometimes called the 'Hardy-Ramanujan number'."

The answer is 32.

1 solution

A Former Brilliant Member
Jun 16, 2019

Limit on positive integers to be checked: $13$ .

The pairs: $\left( \begin{array}{cc} 1 & 12 \\ 9 & 10 \\ \end{array} \right)$ .

The sum: $32$ .

s = 1729;
q = Timing[n = Ceiling[N[Power[s, (3)^-1]]];
   Print[n];
   c = Table[i^3, {i, n}];
   Flatten[
    ParallelTable[
     If[c[[i]] + c[[j]] == s, {i, j}, Nothing], {i, n}, {j, i, n}], 1]
   ];
Print[q];
Print[Plus @@ Flatten[q[[2]]]]

This title space is too small! The topic is Hardy-Ramanujan numbers. #3

The answer is 32.

1 solution

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