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21h

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What notions are used but not clearly defined in modern mathematics?
I posted "quantum group" as an answer, Feb 26 '11 at 5:30. You may be interested in the comments that were made. 
21h

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Graphs from which two vertices can be exchanged
"For the last claim...." But no claims are made. What do you mean? 
21h

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Fermat's proof for $x^3y^2=2$
$x^3=(a^36ab^2)^2+2(3a^2b2b^3)^2$, and also $x^3=y^2+2(1)^2$, but does that imply $(3a^2b2b^3)^2=1$? 
22h

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Are there infinitely many $k$ for which $\frac{\sigma(k)}{k}=n^p$ and $p$ is an odd prime?
I checked it. It's still not terribly interesting (to me). Noam's question is more interesting. 
23h

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Are there infinitely many $k$ for which $\frac{\sigma(k)}{k}=n^p$ and $p$ is an odd prime?
@Noam, "Numbers $n$ such that sum of divisors of $n^2$ is a square" is oeis.org/A008847 which has a reference to a preprint of Kaplansky. I can't find the Kaplansky work anywhere, so I don't know whether he proved anything in it. 
1d

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Are there infinitely many $k$ for which $\frac{\sigma(k)}{k}=n^p$ and $p$ is an odd prime?
@zeraouliarafik, hard, yes; interesting, that's in the eye of the beholder. In two short paragraphs, GH has reduced it to some wellknown hopeless problems. There is no difficulty in Number Theory in tweaking wellknown hard problems to come up with any number of further hard problems. 
1d

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Probability of correlated residues
13 edits in 21 hours. 
1d

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Examples of common false beliefs in mathematics
$$\pmatrix{0&0\cr0&1\cr}$$ 
2d

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An equality for the dimension of the sum of subspaces (in the nondegenerate case)
Thanks. So, what does it mean for a bunch of things to be "2 by 2 nonequal"? Is it different from just saying the things are distinct? 
2d

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An equality for the dimension of the sum of subspaces (in the nondegenerate case)
What does "22 distinct" mean? 
Jul 27 
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Recent progress on the busy beaver problem?
Mota et al., Sophistication as randomness deficiency, Lecture Notes in Comput. Sci., 8031 (2013) 172181, "derive an alternative formulation for busy beaver computational depth.'' Bienvenu and Shen, Random semicomputable reals revisited, Lecture Notes in Comput. Sci., 7160 (2012) 3145, make "several simple observations relating lower semicomputable random reals and busy beaver functions.'' 
Jul 27 
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Eigenvalues of the sum of two matrices, where one is $B=\operatorname{diag}(1, 0,\dots,0)$
@Gerardo, note that one of the eigenvalues of $A$ is zero. 
Jul 27 
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Minimal number of intersections in a convex $n$gon?
The numbers are tabulated at oeis.org/A230281  but only up to $n=8$. It seems $f(7)=29$ and $f(8)=49$. There are links to diagrams, and to articles (in Russian). $f(7)=29$ is illustrated at oeis.org/A230281/a230281.pdf 
Jul 27 
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Eigenvalues of the sum of two matrices, where one is $B=\operatorname{diag}(1, 0,\dots,0)$
I doubt it. If $$A=\pmatrix{1&1\cr1&1\cr}$$ then one eigenvalues of $A+B$ is positive, and one is negative. 
Jul 27 
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Vector inequation problem
@Dongryul, evidently, it was put on hold because 5 people agreed that it was not about researchlevel mathematics. If you and/or Bob can convince 5 people that it is about researchlevel mathematics, you can get it reopened. 
Jul 27 
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The error in Petrovski and Landis' proof of the 16th Hilbert problem
@Ali, sorry, I don't have the letter, and I don't read Russian. I'm sure all I had was the item in Math Reviews. 
Jul 26 
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Elementary Proof of Infinitely many primes $\mathfrak{p} \in \mathbb{Z}[i]$ in the sector $\theta < \arg \mathfrak{p} <\phi $
Ah. What I had in mind was an edit to the body of the question (not so much the title). 
Jul 26 
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Straight line complexity of $k!a$ where $(a,p)=1$
I think I'll just wait until you make the 20th edit. 
Jul 25 
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Straight line complexity of $k!a$ where $(a,p)=1$
You are asking for a probability. I don't see how you will get one, if you can't first tell me the probability that $a=17$. 
Jul 25 
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Straight line complexity of $k!a$ where $(a,p)=1$
What does it mean for $a$ to be a random natural number? 