This document outlines the research conducted to solve the remaining values of K in the Diophantine equation x^3+y^3+z^3=k. Researchers faced challenges using previous algorithms to find the last two values. They utilized a supercomputer and altered algorithms to factor large numbers faster. This new approach found the solutions for K=33 as 8866128975287528^3+(-8778405442862239)^3+(-2736111468807040)^3 and K=795 as (-14219049725358227)^3+14197965759741571^3+2337348783323923^3, achieving the goal of solving all values of K

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The Diophantine Equation
Case Analysis

2. OUTLINE
INTRODUCTION
• What is The Diophantine Equation?
• What’s the current status of K’s value?
CHALLENGES
• What are the challenges faced by mathematicians to find the
last two values of K?
METHODS
• What method the researchers followed?
RESULTS
• What results were found by the researchers at the end?
EVALUATIONS
• Findings after the final assessment
REFERENCES
• Important references that were used

3. Area 1: Introduction
What is
The Diophantine Equation?
 A polynomial equation, usually involving two or more unknowns, such
that the only solutions of interest are the integer ones.
 Also known as The Sum Of Three Cubes Puzzle x^3+y^3+z^3=k
Current status of
K’s value?
 Use of new algorithms and 64-bit computers to find
more values of K
 Found solution of every value of K except the last two.
3

4. 4
Area 2: Challenges
Use of a
complete new
approach
Following are the challenges faced to find the values of K:
No use of
previously used
algorithms
Elkies & Euclidean
algorithm functioned
separately
Use of different
algorithms and
scientific computers

5. 5
Area 3: Methods
Use of a smart computer called the cyber 205 vector (ibid) to find a range of values
for x, y and z for k near 33
An alteration of the vector computer and the algorithms were used to factorize large
numbers
Rework of different algorithms helped sped up the process and produced the
values of k=33
Methods that were used to find the values of K are as follows:

6. 6
Area 4: Results
In about three weeks, the
super computer produced
answers of two unsolved
equation i.e. k=33 and
k=795 (ibid)
k = 33 =
8866128975287528^3
+(−8778405442862239)^3 +
(−2736111468807040)^3
(ibid)
k= 795 =
(−14219049725358227)^3
+ 14197965759741571^3
+ 2337348783323923^3
(ibid)
The below-mentioned results were found after an intensive analysis:

7. 7
Area 5: Evaluations
Opportunities
Achieved the goal what was sought after
Direct contribution of new advancements in mathematics and technology
Learning of new techniques that could be used on other equations in the future.
Limitations
Unable to uncover the solution for x, y and z when k=3
Modern technologies are yet to perform 100% in many equations

8. Area 6: References
Beck, M., Pine, E., Tarrant, W., Yarbrough Jensen, K. (2007), New integer representations
as the sum of three cubes, https://www.ams.org/journals/mcom/2007-76-259/S0025-
5718-07-01947-3/S0025-5718-07-01947-3.pdf [Accessed 18 November 2021]
Booker, Andrew R (2019), Cracking the problem with 33, Research in Number Theory ,
https://arxiv.org/pdf/1903.04284v2.pdf [Accessed 18 November 2021]
Maina, K, (2019) Math wizards and supercomputers solve
’42’,https://www.englishforums.com/news/math-wizards-and-supercomputer-solve-42/
[Accessed 18 November 2021].
Palvus, John (2019), Sum-of-Three-Cubes Problem Solved for ‘Stubborn ’Number 33 ,
https://www.quantamagazine.org/sum-of-three-cubes-problem-solved-for-stubborn-
number-33-20190326/ [Accessed 18 November 2021]
Weisstein, Eric (2003), Diophantine Equation,
https://mathworld.wolfram.com/DiophantineEquation.html [Accessed 18 November
2021]

9. Thank You