37
votes
Accepted
Assuming the Collatz conjecture is false, what is known about the size of the false set?
Krasikov and Lagarias prove the following result (Theorem 6.1):
For any $a\neq 0\pmod 3$, and for all sufficiently large $x$ (depending on $a$), there are at least $x^{0.84}$ integers below $x$ such ...
34
votes
Accepted
Can $9xy$ divide $1+x^2+x^3+y^2$?
The equation is solvable in integers. Take, for example,
$$
x = -19578556686240310295378317903565, \\
y = -101658411567714319887, \\
z = 418962851513108789978912616277180591709694.
$$
Verification can ...
30
votes
Accepted
Powers of $3$ close to powers of $2$
Ellison (1971), see MR0417051, proved that
$$|3^n-2^m|>1.8^m,\qquad m>27.$$
From this bound and the SAGE program below, we see that the only solution of $|3^n-2^m|\leq 13$ with $n\geq 5$ is $(m,...
18
votes
Are there mutually independent undecidable statements?
Here is an easy way to see it.
Let $A$ assert that if PA is inconsistent, the smallest $k$ for which $\Sigma_k$ induction is inconsistent is a multiple of $3$.
Let $B$ make the similar assertion ...
17
votes
Accepted
Using the Eichler-Selberg Trace formula to compute class numbers?
Generally speaking, formulas like the trace formula admit an uncertainty principle: To obtain an identity where one side is highly concentrated (e.g. a sum over a small number of class numbers, or ...
15
votes
Accepted
On the equation $9x^3+y^3=z^2+3$
This equation is unsolvable. Modulo 9 analysis shows that $z \ne 0 \pmod{3}$ and $z \ne \pm 1 \pmod{9}$. Let $p$ be a prime divisor of $z^2 + 3$ for which 3 is not a cubic residue. From
$$z^2 + 3 = 1/...
13
votes
Accepted
Do there exist positive integers $m$, $n$, $p$, $q$ such that $m>1$, $p\neq q$, $p$ and $q$ divide $mn^2 - 1$, and $mn$ divides $p - q$?
I will show below that there are no solutions. I will use the ideas of Denis Shatrov from the MSE page along with Keith Conrad's notes about the Pell equation. I will also use some ideas from comments ...
12
votes
Accepted
Prove positivity of a binomial sum
Let $N$ be a positive integer and $c_1,\ldots,c_N$ non-negative real numbers. Denote $f(x)=((x+c_1)\ldots (x+c_N))^{-1}$.
Lemma 1. For all integer $d\geqslant 0$ and all $x>0$ we have $(-1)^df^{(d)}...
12
votes
Accepted
Lindelöf hypotheses for derivatives of zeta
The Lindelöf hypothesis yields the same bound for each derivative of $\zeta(s)$ via Cauchy's formula. Indeed, let $n\in\mathbb{N}$, $\sigma\in\mathbb{R}$, $T\in(1,\infty)$, $\varepsilon\in(0,1/2)$. ...
12
votes
Sum of three square is a square and sum of their product taken two at a time is also a square
According to Tito Piezas's website $9. (x^2+y^2+z^2, x^2y^2+x^2z^2+y^2z^2)$,
one of the solutions is
$(x,y,z) = (p^2+q^2-r^2, 2pr, 2qr)$ where $(p,q,r) = (8n(n^2-1), n^4-10n^2+5, n(n^2+1)(n^2-3)).$...
11
votes
Accepted
Original proof of Hilbert irreducibility theorem
See Hilbert's Proof of His Irreducibility Theorem by Mark B. Villarino, Bill Gasarch, and Kenneth Regan. (Also available as an arXiv preprint.)
10
votes
Using the Eichler-Selberg Trace formula to compute class numbers?
A similar approach, using the closely related Selberg Trace Formula, is used by Ce Bian, Andrew R. Booker, Austin Docherty, Michael J. Jacobson, Jr.,
and Andrei Seymour-Howell in their paper ...
10
votes
Accepted
Jacobi symbols for two-square sums of primes
The first observation follows from the law of quadratic reciprocity. Indeed, assume that $p\equiv 1\pmod{8}$ and $p=A^2+B^2$. Let $A'$ denote the odd part of $A$. Then $p\equiv B^2\pmod{A'}$, and ...
9
votes
Prove positivity of a binomial sum
I think one can simplify Fedor's proof using the point of view of Bernstein's Theorem for completely monotone functions, together with Bonferroni's Inequalities from probability.
The given sum is
$$
S=...
9
votes
Counting points on elliptic curves
Corresponding sums of Legendre symbols are know as Jacobsthal sums $$J(u)=\sum_{x\mod p}\left(\frac {x^3+ux}p\right).$$
They are equal to $0$ for prime $p\equiv 3\pmod4$. If $p\equiv 1\pmod4
$, $\...
8
votes
Accepted
Trying to understand the topology of the Weil group for the local Langlands conjecture
Note that in a topological group $G$, any subgroup $H$ containing an open subgroup $U$ is itself open: we can write $H$ as a disjoint union of $hU$ for a set of coset representatives for $H/U$, and ...
8
votes
Do there exist positive integers $m$, $n$, $p$, $q$ such that $m>1$, $p\neq q$, $p$ and $q$ divide $mn^2 - 1$, and $mn$ divides $p - q$?
I'll first give a quick summary of this solution, and then fill in the details of each step below.
It suffices to show that if $a>1$, $b$, and $c$ are positive integers, then there are no integral ...
8
votes
Accepted
Prime differences and zero multiplicity
This problem is connected with the L^2 average of primes in short intervals, see Selberg (1942 paper entitled “on the normal density…”). In particular, results on the integral of $\psi(x+h)-\psi(x)-h$ ...
7
votes
Lower bounding a partition-related sum
I have no idea why Lucia removed her nice answer, but here goes a short elementary argument.
We want to bound from below the expectation of $f(\pi)$, where $\pi$ is a random permutation and $f(\pi)=\...
7
votes
What is the difference between Hida and Coleman families?
The difference is the generality of the setting: Hida families (first introduced by Hida in the early 80s) apply only to eigencuspforms which are so-called ordinary at $p$ (roughly speaking, the $p$-...
6
votes
Conditional convergence of exponential sums related to a Hecke modular form
Since you allow $r \in \mathbf Q$, what is the point of writing $12$ in $e^{i\pi nr/12}$? It might as well be written as $e^{i \pi nr}$ by replacing $r$ with $12r$.
In any case, your question has a ...
6
votes
A basic conjecture/observation on the Riemann $\xi$-function
Consider any analytic function $f$ and a zero $s_0 \ne 0$ say simple, though things do not really change if there is a multiplicity as below. Then for some $c \ne 0$ we have $f(s)=f(s_0)+c(s-s_0)+O(|s-...
6
votes
Representing natural numbers as sums of distinct prime powers
Brüdern (2021) proved that every sufficiently large even integer $n$ can be written as
$$n=\sum_{k=1}^{20}x_k^{k+1},$$
where each $x_k$ is a prime number. A similar result with a few more summands was ...
6
votes
Prime differences and zero multiplicity
It is not know that RH implies EH, or that EH implies RH. Let us denote
$$S(x):=\sum_{p_n < x} (p_n -p_{n-1})^2.$$
Assuming the Lindelöf hypothesis, Yu (1996) proved that $S(x)\ll_\varepsilon x^{1+\...
5
votes
Accepted
Generators of the ideal class group
Let $G$ be the ideal class group, and let $H$ be the subgroup generated by the prime ideals of norm at most $3\log^2(d^2)=12\log^2(d)$. Assume that $H$ is a proper subgroup of $G$. Then there is a ...
5
votes
Accepted
Difficult elliptic curves for $a^4+b^4+c^4+d^4 = (a+b+c+d)^4$?
Regarding your first question, there are no solutions $v$ to your equation for the three values of $u$ given (at least assuming GRH). For $u = 97/72$ and $u = 103/78$, the $D^{2} = \text{ quartic in } ...
5
votes
Accepted
On shortest vector problem
Yes you can. I'll write $\lambda_1(L) = \min_{l\in L\setminus \{0\}} \lVert l\rVert_2$.
I will also identify a lattice with any of its basis, e.g. I will write $\lambda_1(B)$ where $B$ is a lattice ...
5
votes
Accepted
Lower bounding a partition-related sum
The sum you want is the coefficient of $z^N$ in the generating function
$$
F(z) = \prod_{k=1}^{\infty} \Big( \sum_{m=0}^{\infty} \frac{2^m}{(m+1)!} \frac{z^{km}}{k^m m!}\Big).
$$
The function $F(z)$ ...
5
votes
Accepted
Stabilizing conjugacy classes of integer matrices
${\rm Conj}(A)\to{\rm Conj}(A\oplus I_m)$ is not surjective, because
$$I_m\oplus A\in{\rm Conj}(A\oplus I_m) \quad\hbox{but}\quad\not\in R({\rm Conj}(A)).$$
5
votes
Representing natural numbers as sums of distinct prime powers
There's a lot of literature on the topic; the magic words are: "Waring-Goldbach" problem.
Starting point: https://en.wikipedia.org/wiki/Waring%E2%80%93Goldbach_problem
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