Timeline for Are these equations solvable in positive integers?
Current License: CC BY-SA 4.0
13 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Aug 22, 2023 at 18:55 | answer | added | Max Alekseyev | timeline score: 4 | |
| Aug 22, 2023 at 16:09 | comment | added | kodlu | Fixed the exponent of 2 in $H(P)$ please undo if I misinterpreted | |
| Aug 22, 2023 at 16:08 | history | edited | kodlu | CC BY-SA 4.0 |
fixed the exponent of 2 in $H(P)$
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| Aug 22, 2023 at 14:57 | comment | added | Bogdan Grechuk | Thank you! I had a feeling that I am missing something obvious with (b). Equation (a) looks more interesting (but should be doable), while equations (c) and (d) look difficult. | |
| Aug 22, 2023 at 14:55 | history | edited | Bogdan Grechuk | CC BY-SA 4.0 |
Question updated after equation (b) has been solved.
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| Aug 22, 2023 at 13:38 | comment | added | Denis Shatrov | The equation (b) is unsolvable, because $y(xy - 1) = x^3 + 2z^2$ and Jacobi symbol $\left(\frac{-2x}{xy - 1} \right)$ is equal to $-1$ (note that modulo 8 we see that $x, y$ are both even). | |
| Aug 22, 2023 at 13:27 | comment | added | mathworker21 | @BogdanGrechuk Thank you. And thanks for your work and your questions! | |
| Aug 22, 2023 at 12:59 | comment | added | Bogdan Grechuk | For this reason, almost all equations in the short list happens to have no positive integer solutions. The only exception so far is the equation $x^3+y^3=z^3+2$ for which my program did not find any positive integer solutions, but there is a solution $1214928^3+3480205^3=3528875^3+2$ discovered in (Heath-Brown, D. R., Walter M. Lioen, and H. J. J. Te Riele. "On solving the Diophantine equation $x^3+y^3+z^3=k$ on a vector computer." Mathematics of computation 61.203 (1993): 235-244.) | |
| Aug 22, 2023 at 12:51 | comment | added | Bogdan Grechuk | Yes. As you may guess, there are hundreds of thousands of equations of size $H\leq 26$. My computer program excludes all families of equations that are easy to solve (e.g. linear ones), all equations that have no solutions for obvious reasons (like no solutions modulo some m), and then of course excludes all equations for which it can find a positive integer solution. Then it outputs a list of equations that it cannot exclude and I work with them manually. Equations with small solutions are therefore excluded and not go to the short list. | |
| Aug 22, 2023 at 12:33 | comment | added | mathworker21 | @BogdanGrechuk Did you test numerically that each of these $4$ has no solutions? | |
| Aug 22, 2023 at 10:19 | comment | added | Bogdan Grechuk | I have 4 open equations of the same size, and I though it is more convenient to have it all in one place rather than creating 4 questions. When I answer a question, I care about sharing knowledge not about whether "accept" button will be pressed. But I am not sure about other users. If formal "accept" is important for them, then this is indeed an issue. | |
| Aug 22, 2023 at 9:59 | comment | added | Gerry Myerson | We generally frown on having more than one question in a question. Which will you accept if four different users solve your four equations? | |
| Aug 22, 2023 at 9:28 | history | asked | Bogdan Grechuk | CC BY-SA 4.0 |