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Years ago, I watched this YouTube video. At 17:30, the professor said that the smallest integer solution to $313(x^3+y^3)=z^3$ has more than $1000$ digits!

I remembered this today and I was curious to see the $1000$-digit solution and how it was even possible to find such a thing. However, after hours of searching, I found nothing.

I would like to ask for a reference about this theorem. Who discovered it, how did they discover it, and what is the $1000$-digit solution?

asked one of my friends who has a master's degree in mathematics and he was also surprised by this fact. So I decided to ask this question here.

pie
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  • Curiosity... Is that over $1000$ total digits between the three variables, or around $1000$ digits each? – abiessu Dec 31 '23 at 02:17
  • @abiessu I think he meant around $1000$ digits each – pie Dec 31 '23 at 02:18
  • probably computer bashing... although $313$ is prime so the RHS is a multiple of $30664297$ – Aaa Lol_dude Dec 31 '23 at 02:21
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    Also, if we include negative numbers, I'm pretty sure the smallest solution is less than $1000$ digits, since we can have $x=355507307842882624593086325021133856149447336710120844428552934573043094018915289363$ $y=−354602746692986709129018423204648314355484458881941451025238387384142099383045862152$ $z=1517122651849438712721950935044230084378368307868200665761294465082177989014675811$ – Aaa Lol_dude Dec 31 '23 at 02:23
  • @AaaLol_dude Wow, how did you find that ? and what is the smallest possible positive solution – pie Dec 31 '23 at 02:38
  • See https://math.stackexchange.com/questions/1613119/a-diophantine-equation-with-only-titanic-solutions for a discussion of this equation. – Gerry Myerson Dec 31 '23 at 02:40
  • @GerryMyerson is there some site that has the 21000 digit solution ? and who is the first one to discover it – pie Dec 31 '23 at 02:46
  • Related: https://people.math.harvard.edu/~elkies/sel_p2.html – Travis Willse Dec 31 '23 at 02:52
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    To be precise, the triplet given by @AaaLol_dude is a solution to the equation $x^3+y^3=313^2 z^3$. But this is equivalent to the given equation upon substituting $z\mapsto 313z$, so just take the given $z$ and multiply by $313$. – Semiclassical Dec 31 '23 at 03:06
  • @Prem it answers 90% of it , the remaining part is how is the first to discover that ?, Is thgere is a website or pdf that have the 21000 digit solution ? – pie Dec 31 '23 at 06:06
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    @pie The 21000-digit solution I found is not the smallest in positive integers, but uses only elementary methods. Using elliptic curves, MacLeod found a smaller one with around 6700 digits. Both are discussed in an old MSE whose link is given above. – Tito Piezas III Dec 31 '23 at 06:35

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