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Results tagged with nt.number-theory
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user 10365
Prime numbers, diophantine equations, diophantine approximations, analytic or algebraic number theory, arithmetic geometry, Galois theory, transcendental number theory, continued fractions
16
votes
1
answer
2k
views
On J. T. Condict's Senior Thesis on Odd Perfect Numbers
I am trying to locate a copy of J. T. Condict's senior thesis on odd perfect numbers:
J. Condict, On an odd perfect number's largest prime divisor, Senior Thesis, Middlebury College (1978).
I am sur …
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 …
9
votes
2
answers
759
views
Is there an odd integer $x < 105$ for which it is known that $x \nmid N$, if $N$ is an odd p...
Is there an odd integer $x < 105$ for which it is known that $x \nmid N$, if $N$ is an odd perfect number?
I have asked the same question in MSE, but did not get any answers. I was wondering if anyb …
6
votes
0
answers
503
views
Does the equation $\sigma(\sigma(x^2))=2x\sigma(x)$ have any odd solutions?
This question was posted in MSE in early August 2020. It did garner several upvotes, but did not receive any responses. I have therefore cross-posted it here, hoping that it gets answered.
Let $\sig …
6
votes
3
answers
4k
views
Re: Mordell's equation $y^2 = x^3 + k$ and perfect numbers
I have already tried a somewhat exhaustive search of the literature, but couldn't find anything close to the problem that I am working on.
My question is: When does Mordell's equation
$$Y^2 = X^3 + K$ …
6
votes
1
answer
2k
views
If $N = qn^2$ is an odd perfect number with $\gcd(q,n)=1$, is it possible to have $q + 1 = \...
The title says it all.
Question
If $N = qn^2$ is an odd perfect number with Euler prime $q$ and $\gcd(q,n)=1$, is it possible to have $q + 1 = \sigma(n)$?
Heuristic
From the Descartes spoof, with qu …
5
votes
Algebraic Attacks on the Odd Perfect Number Problem
I agree with Pace - the correct function to consider would be the abundancy index instead of the sigma function itself. In a certain sense, the abundancy index value of 2 for perfect numbers (odd or …
4
votes
1
answer
1k
views
On Sorli's Conjecture Re: OPNs (Circa 2003)
In the PhD dissertation titled "Algorithms in the Study of Multiperfect and Odd Perfect Numbers" (hyperlinked here) and completed in 2003, Ronald Sorli conjectured that the exponent $k$ on the Euler p …
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 …
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 …
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 …
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 …
3
votes
0
answers
179
views
Does this Theorem 2 from Dandapat et al. imply that $\gcd(\sigma(p^k),\sigma(a^2)) > 1$?
Write the odd perfect number $m=p^k a^2$ as a product of primes
$$m = p^k {p_1}^{2a_1} \cdots {p_v}^{2a_v}.$$
(Note that it is known that $v \geq 9$ by work of Nielsen.) Let $N(m)$ be the number of s …
3
votes
Generalized quasi-perfect numbers
The related equation
$$\sigma(n) = An + B(n)$$
where $B(n)$ "is a function that may depend on properties of $n$" is considered in the paper Variations on Euclid’s Formula for Perfect Numbers by Faride …
3
votes
4
answers
1k
views
A conjecture regarding odd perfect numbers
(Note: I asked this question in MSE this June 2018 but did not receive any responses there. I have therefore cross-posted it here, hoping that it gets answered.)
Let $\sigma(z)$ denote the sum of t …