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Oct
5 |
answered | Did Erdős prove there are two primes $4a+1, 4b+3$ between between $n$ and $2n$? |
Sep
28 |
comment |
Does the equation $241+2^{2s+1}=m^2$ have a solution?
If anyone is wondering, the argument Mike Bennett refers to is worked out in math.dartmouth.edu/~carlp/2tokmmminus1v8.pdf . See Theorem 1. |
Aug
15 |
answered | Prime divisors of values of a polynomial on an infinite set |
Jul
31 |
answered | Lower bounding the probability that $\gcd(t,N)≤B$, for a random $t$ and fixed (large) $N$ |
Jul
28 |
comment |
Upper density of the set of $n$'s such that $p(n)$ is prime, where $p$ is polynomial
An alternative is to use some of the theory of Beurling primes. See alpha.math.uga.edu/~pollack/beurling.pdf |
Jul
28 |
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Upper density of the set of $n$'s such that $p(n)$ is prime, where $p$ is polynomial
A quick comment: One can derive the estimate $\sum_{p \le x} \nu_F(p)/p = \log\log{x} + O(1)$ without worrying about zeros of $\zeta_K(s)$ on $\Re(s)=1$; e.g., apply the Tauberian theorem mentioned as Proposition 5 in alpha.math.uga.edu/~pollack/eulerprime.pdf to $\log \zeta_K(s)$. Everything needed to check the hypotheses is in Hecke's algebraic number theory book (and goes back to Dedekind, I think). |
Jul
17 |
awarded | Yearling |
Jun
25 |
comment |
Number of primes one larger than divisors of a fixed number, which is LCM of 1,2,3,…,L
I don't have an account on Math Stackexchange at the moment, but the conjecture of Wythagoras that motivated your question follows quickly from Theorem 1 in this paper of Erdos--Pomerance--Schmutz: math.drexel.edu/~eschmutz/PAPERS/lambda.pdf |
Jun
6 |
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When is $(-1+\sqrt[3]{2})^n$ of the form $a+b\sqrt[3]{2}$?
Have you seen Proposition 3.8 and Example 3.16 in esc.fnwi.uva.nl/thesis/centraal/files/f310232185.pdf ? |
May
25 |
comment |
On the number of consecutive divisors of an integer
Here's a PDF: renyi.hu/~p_erdos/1978-26.pdf |
May
25 |
answered | On the number of consecutive divisors of an integer |
May
14 |
answered | On the natural density of almost perfect numbers |
May
10 |
revised |
Are There Infinitly Many $n$ Which $a\times n!+1$ Be Composite?
deleted extraneous plus sign |
May
9 |
answered | Are There Infinitly Many $n$ Which $a\times n!+1$ Be Composite? |
May
4 |
answered | Powers modulo a fixed integer |
Apr
18 |
comment |
When has the Borel-Cantelli heuristic been wrong?
Another example of (probable) failure in the presence algebraic structure: Pomerance has a heuristic, described at oddperfect.org/pomerance.html, predicting no odd perfect numbers. The same heuristic could be adapted to suggest there are only finitely many even perfect numbers --- but of course we "know better" here, by Euclid. |
Mar
26 |
awarded | Nice Answer |
Mar
26 |
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Numbers $n$ such that the sum of the divisors of $n$ is a nontrivial power
Let me supplement my previous comment by saying that it's not realistic at this time to expect definitive results for fixed $a$. For example, consider the equation $\sigma(n) = 6^b$. Probably there are infinitely many solutions $n,b$, since if $p=2\cdot 6^m-1$ is prime, then $n=2p$ gives rise to a solution. On the other hand, it seems hopeless to prove at this time that there are infinitely many solutions. For then there are infinitely many $p$ with $p+1$ a $3$-smooth number. We don't know anything nearly this strong! The ``smoothest'' we can get $p+1$ is $p^{0.2931}$ (Baker-Harman)! |
Mar
25 |
answered | Does there exist an integer that is both solitary and almost perfect? |
Mar
25 |
comment |
Numbers $n$ such that the sum of the divisors of $n$ is a nontrivial power
Here is a comment on the fixed $a$ problem: One can show that there are only finitely many proper prime powers $p^e$ for which $\sigma(p^e)$ is supported on a given finite set of primes. See, e.g., the book of Shorey and Tijdeman on Exponential Diophantine Equations. This means that the problem in some sense reduces to understanding the shifted primes $\sigma(p)=p+1$. For instance, $\sigma(n) = 2^b$ has infinitely many solutions $(n,b)$ if and only if there are infinitely many Mersenne primes. |