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 ...
Wojowu's user avatar
  • 27.3k
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 ...
Bogdan Grechuk's user avatar
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,...
GH from MO's user avatar
  • 97.8k
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 ...
Joel David Hamkins's user avatar
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 ...
Will Sawin's user avatar
  • 135k
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/...
Denis Shatrov's user avatar
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 ...
GH from MO's user avatar
  • 97.8k
12 votes
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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)}...
Fedor Petrov's user avatar
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 $n\in\mathbb{N}$, $\sigma\in\mathbb{R}$, $T\in(1,\infty)$, $\varepsilon\in(0,1/2)$. ...
GH from MO's user avatar
  • 97.8k
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 $(p,q,r) = (8n(n^2-1), n^4-10n^2+5, n(n^2+1)(n^2-3)).$...
Tomita's user avatar
  • 1,427
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.)
Peter Mueller's user avatar
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 ...
davidlowryduda's user avatar
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 ...
GH from MO's user avatar
  • 97.8k
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=...
Abdelmalek Abdesselam's user avatar
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 $, $\...
Alexey Ustinov's user avatar
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 ...
R. van Dobben de Bruyn's user avatar
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 ...
Thomas Browning's user avatar
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 $\psi(x+h)-\psi(x)-h$ ...
Alessandro Languasco's user avatar
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)=\...
Fedor Petrov's user avatar
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 $p$-...
Olivier's user avatar
  • 10.2k
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 ...
KConrad's user avatar
  • 49.5k
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-...
Conrad's user avatar
  • 1,854
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 ...
GH from MO's user avatar
  • 97.8k
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+\...
GH from MO's user avatar
  • 97.8k
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 ...
GH from MO's user avatar
  • 97.8k
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 } ...
Jeremy Rouse's user avatar
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 ...
Mark Schultz-Wu's user avatar
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 $F(z)$ ...
Lucia's user avatar
  • 43.3k
5 votes
Accepted

Stabilizing conjugacy classes of integer matrices

${\rm Conj}(A)\to{\rm Conj}(A\oplus I_m)$ is not surjective, because $$I_m\oplus A\in{\rm Conj}(A\oplus I_m) \quad\hbox{but}\quad\not\in R({\rm Conj}(A)).$$
Denis Serre's user avatar
  • 51.5k
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
Alessandro Languasco's user avatar

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