Timeline for Extending rational Diophantine triples to sextuples
Current License: CC BY-SA 4.0
11 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Aug 14, 2019 at 15:46 | history | edited | Tito Piezas III | CC BY-SA 4.0 |
Made clearer
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| Mar 22, 2016 at 19:07 | history | edited | Tito Piezas III | CC BY-SA 3.0 |
Improved.
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| Mar 21, 2016 at 6:28 | comment | added | Tito Piezas III | @JesperPetersen: Since $x_1x_2+1$ is a square, then $a,b,c,d$ obey, $$2(a^2+b^2+c^2+d^2)-(a+b+c+d)^2-3-6abcd+(abcd)^2 = y^2$$ as in Section III of this post, while $a,b,c,d,x_k$ satisfy, $$(\alpha_1-\alpha_5)^2=4(\alpha_2+\alpha_4+1)$$ where the $\alpha_i$ are in the elementary symmetric polynomials in this answer. However, there is no known general relation where all six variables appear at once. | |
| Mar 20, 2016 at 8:45 | comment | added | Jesper Petersen | Great finding indeed! How does this relate to the previous post linked at the top? I.e. what I am asking is if $a, b, c, d, x_1, x_2$ satisfy a relation in terms of elementary symmetric polynomials as in that post? | |
| Mar 20, 2016 at 6:45 | comment | added | Tito Piezas III | @duje: Ah, nice! Letting $n=p/q$, the denominator $D$ of $b$ or $c$ becomes, $$D=p^2\pm pq-q^2$$ and to solve $D=\pm1$ in integer $p,q$ would entail the Pell equation $x^2-2y^2=\pm1$. | |
| Mar 19, 2016 at 8:54 | comment | added | duje | You may take also $n=\pm \frac{P_k - P_{k-1}}{P_{k+1} - P_k}$ to get $b$ to be an integer. | |
| Mar 19, 2016 at 8:47 | comment | added | duje | By taking $n=\pm\frac{P_k}{P_{k+1}}$, where $P_k$ is $k$-th Pell number, you obtain infinitely many sextuples with one non-zero integer element. | |
| Mar 18, 2016 at 22:08 | comment | added | duje | Excellent finding! Congratulations! | |
| Mar 18, 2016 at 20:57 | history | edited | Tito Piezas III | CC BY-SA 3.0 |
Better format.
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| Mar 18, 2016 at 20:28 | history | edited | Tito Piezas III | CC BY-SA 3.0 |
added 56 characters in body
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| Mar 18, 2016 at 20:18 | history | answered | Tito Piezas III | CC BY-SA 3.0 |