Timeline for Nice diophantine equations with large smallest solutions
Current License: CC BY-SA 4.0
17 events
| when toggle format | what | by | license | comment | |
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| Mar 21 at 2:32 | comment | added | Gerry Myerson | Some very large numbers come up in smallest solutions to problems in the thread math.stackexchange.com/questions/514/… I don't know what those problems would look like if you turned them into diophantine equations. Likewise, mathoverflow.net/questions/15444/… | |
| Mar 21 at 1:46 | comment | added | D.W. | $(x_1-2)^2 + \sum_{i=1}^{k-1} (x_i^2 - x_{i+1})^2$ doesn't meet your requirements, but arguably it gets somewhat "close". The smallest solution has size $2^{2^k}$, and $v(P) = 32 \times 48^{k-1}$. | |
| Mar 21 at 1:34 | comment | added | D.W. | When you write "has such solution", what does "such" refer to? Do you mean "has a solution"? | |
| Mar 20 at 23:35 | history | became hot network question | |||
| Mar 20 at 20:13 | comment | added | Stefan Kohl♦ | @DenisShatrov If the smallest solution would be large enough, that equation would make an excellent answer to the question. | |
| Mar 20 at 20:08 | history | edited | Stefan Kohl♦ | CC BY-SA 4.0 |
Added further explanation of what is meant by an equation being "nice".
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| Mar 20 at 19:57 | comment | added | Denis Shatrov | @WillSawin Yes, I calculated $v(P)$ wrongly. | |
| Mar 20 at 19:12 | comment | added | Will Sawin | @DenisShatrov If I counted right that has $v(P)= 324$ but the solution in that question is merely of size $\approx 2^{140}$, so not large enough. | |
| Mar 20 at 16:58 | comment | added | Denis Shatrov | mathoverflow.net/questions/466362/… This is a nice equation with large minimal solutions. | |
| Mar 20 at 16:52 | answer | added | Will Sawin | timeline score: 7 | |
| Mar 20 at 16:09 | comment | added | Will Sawin | (By Theorem 3.1 of Hilbert's Tenth problem is unsolvable by Martin Davis, this equation, if I typed it right, forces $x$ to be exponentially large as a function of $a+C$ and $x'$ to be exponentially large as a function of $x$, so if $a$ is positive then $x'$ is double-exponential as a function of $C$, but $v$ is merely polynomial in $C$.) | |
| Mar 20 at 16:06 | comment | added | Will Sawin | If I typed it right, $(x^2-((a+C)^2-1)y^2-1)^2+(u^2-((a+C)^2-1)v^2-1)^2+(s^2-(b^2-1)t^2-1)^2+(v-ry^2)^2+(b-1-4py)^2+(b-a-C+qu)^2+(s-x-cu)^2+(t-k-4(d-1)y)^2+(y-k+e-1)^2 + (x^{'2}-(x^2-1)y^{'2}-1)^2 + (u^{'2} - (x^2-1)v^{'2}-1)^2 + (s^{'2}-(b^{'2}-1)t^{'2}-1)^2+ (v'-r'y^{'2})^2 + (b'-1-rp'y')^2 + (b'-x-q'u')^2 +(s'-x' - c'u')^2 +(t'-k'-4(d'-1)y')^2$ with the restriction that $a$ is positive does the trick for all sufficiently large $C$. | |
| Mar 20 at 16:02 | comment | added | Stefan Kohl♦ | @WillSawin Nice examples are welcome in any case. Just the smallest solution (respectively, smallest non-zero or smallest positive solution, if applicable) should exceed the given bound. | |
| Mar 20 at 15:54 | comment | added | Will Sawin | Is the intention in the question to allow equations where we consider only positive solutions or only nonzero solutions, as you do in some of the examples? | |
| Mar 20 at 15:40 | history | edited | Stefan Kohl♦ | CC BY-SA 4.0 |
added 1 character in body
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| Mar 20 at 15:33 | history | made wiki | Post Made Community Wiki by Stefan Kohl♦ | ||
| Mar 20 at 15:33 | history | asked | Stefan Kohl♦ | CC BY-SA 4.0 |