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Results tagged with reference-request
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user 10365
This tag is used if a reference is needed in a paper or textbook on a specific result.
1
vote
On odd perfect numbers and a GCD
Here is a conditional proof that
$$G = \gcd(\sigma(q^k),\sigma(n^2)) = i(q) = \gcd(n^2, \sigma(n^2)).$$
As derived in the OP, we have
$$G = \gcd\bigg(\frac{n^2}{i(q)}, i(q)\bigg).$$
This is e …
1
vote
On odd perfect numbers and a GCD
It turns out that
$$G \text{ is a square } \iff i(q) \text{ is a square.}$$
The proof is essentially contained in this answer to a closely related MSE question.
Thus, we have the implication
$$G \te …
3
votes
Has it been proved that odd perfect numbers cannot be triangular?
Not an answer, but I just want to point out some thoughts that recently occurred to me, which are related to this problem.
By this answer, we know that every odd perfect number $N = q^k n^2$ can be w …
0
votes
1
answer
101
views
Any results on $\gcd(N^2, D(N^2))$ where $N^2$ is deficient and $D(N^2)$ is the deficiency o...
Any results on $\gcd(N^2, D(N^2))$ where $N^2$ is deficient and
$$D(N^2)=2N^2 - \sigma(N^2)$$ is the deficiency of $N^2$?
I checked OEIS sequence A033879 and have so far been able to get hold of Kly …
3
votes
0
answers
175
views
Looking for an appropriate reference(s) for two conjectures on odd perfect numbers
(I apologize in advance if this question is unsuitable for MO. If so, please let me know and I will migrate it to MSE.)
Let $\sigma(M)$ be the sum of the divisors of the positive integer $M$. For ex …
2
votes
References for Yang-Mills Theory
You could start with Terence Tao's Local well-posedness of the Yang-Mills equation in the Temporal Gauge below the energy norm, available online via the arXiv.
A second paper by Tao and Gang Tian on …
13
votes
2
answers
1k
views
Has it been proved that odd perfect numbers cannot be triangular?
(Note: This question has been cross-posted from MSE.)
Euclid and Euler proved that every even perfect number is of the form $m = \frac{{M_p}\left(M_p + 1\right)}{2}$ where $M_p = 2^p - 1$ is a prime …
4
votes
1
answer
334
views
If $N = {q^k}{n^2}$ is an odd perfect number given in Eulerian form, is $n$ a square?
(I have asked a similar question in MSE four days ago, but did not receive any answers. I have therefore cross-posted it to this site, hoping to get some responses.)
An odd perfect number $N$ is sai …
1
vote
1
answer
235
views
If $N = {q^k}{n^2}$ is an odd perfect number given in Eulerian form, is $n$ squarefree?
(I have asked a similar question in MSE around a week ago, but did not receive any responses. I have therefore cross-posted it to this site, hoping to get some answers.)
An odd perfect number $N$ is …
0
votes
If $q^k n^2$ is an odd perfect number with Euler prime $q$, are the following statements kno...
If $k > 1$, we can do better if the following inequalities hold:
$$q < q^k < n.$$
The resulting lower bound is
$$\gcd(n^2, \sigma(n^2)) = \frac{D(n^2)}{\sigma(q^{k-1})} = i(q^k) = \frac{2n^2 …
1
vote
1
answer
230
views
If $q^k n^2$ is an odd perfect number with Euler prime $q$, are the following statements kno... [closed]
(Note: This has been cross-posted to MSE. However, I feel that it is more likely to receive a good answer here, because I believe that it is a research-level question. For the mathematicians who vo …
4
votes
2
answers
470
views
On the natural density of almost perfect numbers
This question is pretty basic, so I apologize in advance if it is unsuitable for MO. If so, please do let me know and I will migrate it over to MSE.
Essentially, by work of Kanold, we know that the …
1
vote
0
answers
460
views
A question on (odd) perfect numbers
I have asked this question in MSE a few weeks back, but did not receive any responses. I have cross-posted it to MO, hoping that it is appropriate for this site.
Let $\sigma(x)$ be the (classical) s …
2
votes
0
answers
485
views
On Descartes / spoof odd perfect numbers
Descartes, Frenicle, and subsequently Sorli, conjectured that $k = 1$, if $N = {q^k}{n^2}$ is an odd perfect number given in Eulerian form (i.e., $q$ is prime with $q \equiv k \equiv 1 \pmod 4$ and $\ …
2
votes
1
answer
319
views
Reference request: Research done on whether the Euler prime can be the largest factor of an ...
(Note: This was cross-posted from MSE.) I posted the following reference request in MSE three (3) days ago, but was unable to elicit any responses. I am cross-posting it to MO, hoping that it is app …