Sums of integer cubes

  title={Sums of integer cubes},
  author={Samir Siksek},
  journal={Proceedings of the National Academy of Sciences},
  • S. Siksek
  • Published 30 March 2021
  • Medicine
  • Proceedings of the National Academy of Sciences


On a question of Mordell
This paper concludes a 65-y search with an affirmative answer to Mordell’s question and strongly supports a related conjecture of Heath-Brown and makes several improvements to methods for finding integer solutions to x3+y3+z3=k for small values of k.
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  • A. Booker
  • Mathematics
    Research in Number Theory
  • 2019
Inspired by the Numberphile video "The uncracked problem with 33" by Tim Browning and Brady Haran, we investigate solutions to $x^3+y^3+z^3=k$ for a few small values of $k$. We find the first known
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A new algorithm is described for finding integer solutions to x 3 + y 3 + z 3 = k for specific values of k and this is used to find representations forvalues of k for which no solution was previously known, including k = 30 and k = 52.
Rational Points Near Curves and Small Nonzero |x3-y2| via Lattice Reduction
A new algorithm using linear approximation and lattice reduction to efficiently calculate all rational points of small height near a given plane curve C is given, and its proof also yields new estimates on the distribution mod 1 of (cx)3/2 for any positive rational c.
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  • 1994
The density of zeros of forms for which weak approximation fails
The weak approximation principal fails for the forms x + y + z = kw, when k = 2 or 3. The question therefore arises as to what asymptotic density one should predict for the rational zeros of these
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An upper bound for the numberE(N) of numbers not exceedingN and not being the sum of four cubes is given; namelyE(N)〈N131/147+e.
Sums of Three Cubes