Exercise 5.6.11 (Size of the solutions of $x^3 + y^3 = 7$)

Question

Let the triple $(X_n, Y_n, Z_n)$ of integers be defined as in the proof of Theorem 5.16. Show that for $n=1$ this is $(20, -17, 7)$, and that for $n=2$ this is $(-36520, 188479, 90391)$. Show also that $X_3$ is a $21$-digit number and that $X_4$ is a $85$-digit number.

Richard Ganaye · Accepted Answer

\begin{proof} 
This is an Exercise for computer: with Sagemath,
\begin{verbatim}
sage: def triple(n):
....:     x, y, z = 2, 1, 1
....:     for i in range(n):
....:         x, y, z = x * (x^3 + 2 * y^3), -y * (2 * x^3 + y^3), z * (x^3 - y^3)
....:     return x, y, z
....: 
sage: list_triples = [triple(n) for n in range(5)]
....: list_triples[:3]
....: 
[(2, 1, 1), (20, -17, 7), (-36520, 188479, 90391)]
sage: x3, x4 = list_triples[3][0],  list_triples[4][0]
....: len(str(abs(x3))), len(str(abs(x4)))
....: 
(21, 85)
\end{verbatim}
This confirms the results of the statement.

\end{proof}

Exercise 5.6.11 (Size of the solutions of $x^3 + y^3 = 7$)

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Exercise 5.6.11 (Size of the solutions of $x^3 + y^3 = 7$)

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