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If seen as an algorithm question, the taxicab numbers can be found in $O(N^2)$. However, I am wondering are there any improvement to get this numbers with less time complexity considering number theory results.

Given an upper bound, i.e. $10^6 = N^3$(in this case N = $10^2$), My target here is to find all the taxicab numbers which are smaller than $N^3$ through less time complexity than $O(N^2)$. I think it must requires some number theory ideas.

Peppep
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  • define taxicab number. what is $n$? – kodlu Nov 10 '18 at 04:55
  • n here is a upper bound of the taxicab number. For exmaple, the target is to find all the taxicab which are smaller than $10^6$. I will edit the problem to make it more clear – Peppep Nov 10 '18 at 04:59
  • your question is answered elsewhere: https://math.stackexchange.com/questions/2815/find-taxicab-numbers-in-on-time?rq=1 – kodlu Nov 10 '18 at 04:59
  • Possible duplicate of Find taxicab numbers in $O(n)$ time – kodlu Nov 10 '18 at 04:59
  • thank you. But I think this link treats the problem as an algorithm question. I am wondering can the time complexity be even smaller if some number theory observations are included? – Peppep Nov 10 '18 at 05:03
  • These MIT notes suggest an algorithmic approach on page 9 that you might find interesting. See https://ocw.mit.edu/courses/mathematics/18-704-seminar-in-algebra-and-number-theory-rational-points-on-elliptic-curves-fall-2004/projects/lugo.pdf – kodlu Nov 10 '18 at 05:15
  • Thank you. But I do not think this paper can guarantee to find all the taxicab number. Do I understand it in a wrong way? – Peppep Nov 10 '18 at 16:20
  • Actually, I think even with the general solutions of Ramanujan' works. The time complexity can not be reduced less than $O(N^2)$. Do you have some better ideas? – Peppep Nov 10 '18 at 17:34

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