# Cracking the problem with 33

@article{Booker2019CrackingTP, title={Cracking the problem with 33}, author={Andrew R. Booker}, journal={Research in Number Theory}, year={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 solution for $k=33$.

## 6 Citations

Diophantine equations: a systematic approach

- Mathematics
- 2021

By combining computer assistance with human reasoning, we have solved the Hilbert’s tenth problem for all polynomial Diophantine equations of size at most 28, where the size is defined in [37]. In…

Sum of Three Cubes via Optimisation.

- Mathematics
- 2020

By first solving the equation $x^3+y^3+z^3=k$ with fixed $k$ for $z$ and then considering the distance to the nearest integer function of the result, we turn the sum of three cubes problem into an…

On a question of Mordell

- Mathematics, MedicineProceedings of the National Academy of Sciences
- 2021

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.

## References

SHOWING 1-10 OF 39 REFERENCES

Newer sums of three cubes

- Mathematics
- 2016

The search of solutions of the Diophantine equation $x^3 + y^3 + z^3 = k$ for $k<1000$ has been extended with bounds of $|x|$, $|y|$ and $|z|$ up to $10^{15}$. The first solution for $k=74$ is…

Solutions of the diophantine equation

- Mathematics
- 1964

with the help of the EDSAC computer at Cambridge University; their results are tabulated in [1]. Mordell's original interest in this equation centered on the case d = 3 ; in particular, he wanted to…

New sums of three cubes

- Mathematics, Computer ScienceMath. Comput.
- 2009

The search for solutions of the Diophantine equation x 3 + y 3 + z 3 = n for n < 1000 and |x|, |y |, |z| < 10 14 is reported on.

Speeding the Pollard and elliptic curve methods of factorization

- Mathematics
- 1987

Since 1974, several algorithms have been developed that attempt to factor a large number N by doing extensive computations module N and occasionally taking GCDs with N. These began with Pollard's p 1…

On searching for solutions of the Diophantine equation x3 + y3 +2z3 = n

- Computer ScienceMath. Comput.
- 2000

An efficient search algorithm to solve the equation x3 + y3 + 2z3 = n for a fixed value of n > 0 by parametrizing |z| and obtains |x| and |y| (if they exist) by solving a quadratic equation derived from divisors of 2|z|3±n.

Rational Points Near Curves and Small Nonzero |x3-y2| via Lattice Reduction

- Mathematics, Computer ScienceANTS
- 2000

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.

A Note on the Diophantine Equation x 3 + y 3 + z 3 = 3

- Mathematics
- 1985

Any integral solution of the title equation has x =y z (9). The report of Scarowsky and Boyarsky [3] that an extensive computer search has failed to turn up any further integral solutions of the…

On searching for solutions of the Diophantine equation x3 + y3 + z3 = n

- Computer Science, MathematicsMath. Comput.
- 1997

A new search algorithm to solve the equation x 3 + y 3 + z 3 = n for a fixed value of n > 0.5 is proposed, using several efficient number-theoretic sieves and much faster on average than previous straightforward algorithms.

New integer representations as the sum of three cubes

- Mathematics, Computer ScienceMath. Comput.
- 2007

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.

The density of zeros of forms for which weak approximation fails

- Mathematics
- 1992

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…