All Questions
12 questions
6
votes
2
answers
804
views
Must Mersenne numbers be divisible by arbitrary large primes with exponent one?
Let $M_n$ denote the Mersenne numbers $M_n=2^n-1$.
As $n$ varies, must $M_n$ be divisible by arbitrary large prime $p$
with exponent one, i.e. $p \mid M_n, p^2 \nmid M_n$?
In other words, must the ...
- 44.4k
2
votes
0
answers
221
views
Squares whose differences are squares
EDIT. I've just noticed a thread from 2011 in the "Related" column on the right (click me), where a closely related question is being discussed (the main difference seems to be that, in ...
- 10.5k
38
votes
5
answers
10k
views
Are nontrivial integer solutions known for $x^3+y^3+z^3=3$?
The Diophantine equation
$$x^3+y^3+z^3=3$$
has four easy integer solutions: $(1,1,1)$ and the three permutations of $(4,4,-5)$. Elsenhans and Jahnel wrote in 2007 that these were all the solutions ...
2
votes
1
answer
656
views
Reference Request - Sharp Estimates for a Logarithmic Sum
Can anybody suggest a good (e.g. "non-technical") introduction to estimating bounds for logarithmic sums of the form
$$\sum_{i=1}^{r}{{\alpha_i}{\log(q_i)}}$$
where the $$\alpha_i$$ are positive ...
1
vote
1
answer
446
views
Existence of Solutions to an Equation Involving the Sum-of-Divisors Function [Reference Request]
Let $\sigma(x) = \sigma_1(x)$ denote the sum of all the positive divisors of $x$.
If $n \in \mathbb{N}$ is odd and $\gcd(n, \sigma(n)) = 1$, then do there exist any solutions to the following ...
2
votes
0
answers
487
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 $\...
CommunityBot
- 1
2
votes
1
answer
322
views
Reference request: Research done on whether the Euler prime can be the largest factor of an odd perfect number
(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 ...
CommunityBot
- 1
2
votes
0
answers
398
views
Counting factors: is this approach in the literature on multiperfect numbers?
Does the following approach (or something near it) exist in the number theory
literature?
I will provide some motivation for $\omega(p^n - 1)$ as $n \rightarrow \infty$
and for this question. First, ...
CommunityBot
- 1
7
votes
4
answers
2k
views
Interactions of number theoretic conjectures and other fields of mathematics
There are many interesting open conjectures in number theory. My question is not about partial results or possible ways to prove them. It is about their interactions with the other fields of ...
- 11.2k
3
votes
1
answer
1k
views
Collatz conjecture— finite state machine transducer construction, origination?
wikipedia has an entry on the Collatz conjecture with a section on As an abstract machine that computes in base two. this apparently describes a construction of a FSM transducer computing sequential ...
CommunityBot
- 1
11
votes
1
answer
2k
views
The Class Number One Problem for Real Quadratic Fields
An approach to the Gauß class number one problem for imaginary quadratic fields is to determine the integral points on the modular curve $Y_{nonsplit}(n)$ for a suitable $n$. Here follows a quick ...
- 5,670
1
vote
0
answers
452
views
Reference Request - Jakob Weisblat's "The Search for the Odd Perfect Number" [closed]
Hi All!
I am currently trying to locate an online copy of Jakob Weisblat's paper titled "The Search for the Odd Perfect Number". I could only get hold of the abstract:
"A perfect number is a number ...