Timeline for Reduce over integers - why does MMA not give ~impossible?
Current License: CC BY-SA 4.0
13 events
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| Oct 22, 2022 at 17:26 | comment | added | Daniel Lichtblau |
Not buying this convexity argument. With a slight change: In[15]:= FunctionConvexity[(2^x - 3 k - 2)^2 + (2^(x - 1) - 3 m - 1)^2, {k, m}, Assumptions -> x \[Element] Reals] Out[15]= 1 In[13]:= NMinimize[{(2^x - 3 k - 2)^2 + (2^(x - 1) - 3 m - 1)^2, x >= 1, k >= 0, m >= 0, {x, k, m} \[Element] Integers}, {x, k, m}] Out[13]= {0., {x -> 1, k -> 0, m -> 0}} But notice the solution is not unique. In[14]:= (2^x - 3 k - 2)^2 + (2^(x - 1) - 3 m - 1)^2 /. {x -> 3, m -> 1, k -> 2} Out[14]= 0
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| Oct 22, 2022 at 3:50 | comment | added | user64494 | @user293787: Thank you for your valuable comment. See Addition 3. | |
| Oct 22, 2022 at 3:49 | history | edited | user64494 | CC BY-SA 4.0 |
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| Oct 22, 2022 at 1:36 | comment | added | user293787 | One more thing: $2^x$ is not an integer if $x$ is a negative integer. | |
| Oct 21, 2022 at 21:07 | comment | added | Julian Moore | Just to clarify, the question is fundamentally about how Reduce works; the non-existence of a solution can be established by various means. This minimisation trick is new to me - duly noted! | |
| Oct 21, 2022 at 19:49 | comment | added | user64494 | @user293787: Please, see Addition 2, Don't hesitate to ask for further explanation in need. | |
| Oct 21, 2022 at 19:47 | history | edited | user64494 | CC BY-SA 4.0 |
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| Oct 21, 2022 at 19:37 | history | edited | user64494 | CC BY-SA 4.0 |
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| Oct 21, 2022 at 19:22 | comment | added | user293787 |
The original function is only convex in $k,m$ but not in $k,m,x$. Convexity of your new function in $k,m,y$ does not imply convexity of the original function, since $y=2^x$ is not affine linear. Nevertheless, I agree that you will somehow be able to conclude in this way that there is no solution. I think that by introducing $y$ you are doing part of what OP probably had in mind when they wrote "By inspection there are no solutions". I did not know about FunctionConvexity. +1
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| Oct 21, 2022 at 18:38 | history | edited | user64494 | CC BY-SA 4.0 |
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| Oct 21, 2022 at 18:27 | comment | added | user64494 |
@user293787: Thank you for your valuable comment. At least, the command FunctionConvexity[(2^x - 3 k - 1)^2 + (2^(x - 1) - 3 m - 1)^2, {k, m}, Assumptions -> x \[Element] Reals] results in 1.
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| Oct 21, 2022 at 18:14 | comment | added | user293787 |
But... NMinimize[{(x^3+y^3-z^3)^2,Element[x|y|z,PositiveIntegers]},{x,y,z}] returns {1.,{x->1,y->1,z->1}}. Do you consider that a proof of the $n=3$ case of a famous theorem? The documentation says: "If f and cons are linear or convex, the result given by NMinimize will be the global minimum, over both real and integer values; otherwise, the result may sometimes only be a local minimum. "
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| Oct 21, 2022 at 18:04 | history | answered | user64494 | CC BY-SA 4.0 |