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Can every positive integer eventually be expressed in this form?

Based on John Omielan's outline, I will prove below that every integer $a\geq 122$ has a representation with exponents $s_k\in\{1,2\}$. Let us use the fact that every integer exceeding $33$ is a sum ...
GH from MO's user avatar
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2 votes

Can every positive integer eventually be expressed in this form?

Using almost exclusively just the first and second power terms, all positive integers (apart from the 8 exceptions you found) can be expressed in your specified form. First, with $s_k = 1$ for all $0 \...
John Omielan's user avatar
2 votes

On a efficient algorithm for factoring bivariate polynomials modulo composite modulus assuming the solution is unique

We wrote a 2 page preprint on researchgate Abstract: We give efficient probabilistic algorithm for factoring bivariate polynomials modulo composite integers with unknown factorization assuming the ...
joro's user avatar
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2 votes

About the units in $\mathbb{Z}[\frac{1+\sqrt{d}}{2}]$

The page https://mathworld.wolfram.com/FundamentalUnit.html has a table of fundamental units, with $D$ there in the role of your $d$. When $D = 37$ the table has David Speyer's example $6+\sqrt{37}$. ...
KConrad's user avatar
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0 votes

Is there a bicyclic irregular pentagon in integers?

Can we use degenerate pentagons? Draw an equilateral triangle, but when you draw one of its sides retrace it backwards and then go forwards again so that in effect you have made it three out of the ...
Oscar Lanzi's user avatar
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6 votes

A question on hybrid subconvexity for individual L-functions

There are several confusions in your post. 1. Automorphic forms for the group $\mathrm{SL}_2(\mathbb{Z})$ have level $q=1$ by definition. You probably wanted to talk about newforms for the Hecke ...
GH from MO's user avatar
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5 votes

About the units in $\mathbb{Z}[\frac{1+\sqrt{d}}{2}]$

At the 2022 Western Number Theory meeting, Michael Beeson asked the question, when $d\equiv5\bmod8$, what are conditions for the fundamental unit to be in ${\bf Z}[\sqrt d]$. https://oeis.org/A107997 ...
Gerry Myerson's user avatar
10 votes
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Solving $2^{x+1} m -1=p^y$ for prime $p$ and natural $x,y$

No, a counterexample (possibly not the smallest) is given by Riesel number $m=509203$. We know that every number of the form $2^{x+1}m-1$ has a prime factor from the set $\{3,5,7,13,17,241\}$ and it ...
Max Alekseyev's user avatar
5 votes
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About the units in $\mathbb{Z}[\frac{1+\sqrt{d}}{2}]$

Regarding (i): No, not always. $37$ is $5 \bmod 8$, and the primitive unit in $\mathbb{Z}[(1+\sqrt{37})/2]$ is $6+\sqrt{37}$. In this example, $x^2-d y^2=-1$; the first case I can find with $x^2-d y^2 ...
David E Speyer's user avatar
7 votes
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Bounding the sum $f(x)=-\frac{x}{2}+\sum_{p\le x}\log(p)-\frac{1}{x}\sum_{p\le x}p\cdot \log(p)$

Just an $\epsilon$-expansion of my comment on the question. Recall the usual notation for the Chebyshev $\vartheta$-function: $$\vartheta(x) = \sum_{p\leqslant x} \log p. $$ Then, using the notation ...
Anurag Sahay's user avatar
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10 votes
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Bounds for Dirichlet L-functions

As Peter Humphries said in a comment, the best known bound for $\sigma=1/2$ (applying to all $\chi$) is due to Petrow and Young: $$L(1/2 + it,\chi) \ll_{\varepsilon} (q(|t| + 1))^{1/6 + \varepsilon}.$$...
GH from MO's user avatar
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2 votes

1-1 map on the $\{0,1\}^k$

I assume words “more” and “less” to be non-strict, otherwise shift $k_0$ by 1. Assume that the weight (=number of 1s) of $x$ is $k/2-k_0$. There are plenty of $a$s which agree with $x$ only at $2k_0$ ...
Ilya Bogdanov's user avatar
4 votes
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Bounding $H^4_{\text{ėt}}$ of a surface

Are you absolutely sure you want to compute $p$-adic etale cohomology for a smooth proper $\mathbb{Z}[1/S]$-scheme with $p \notin S$, so $p$ is not invertible on $X$? This will be painful, and I ...
David Loeffler's user avatar
2 votes

Bloch–Beilinson conjecture for varieties over function fields of positive characteristic

This may not be precisely what you want, but a function field analogue of Beilinson's conjectures is formulated in R. Sreekantan, Non-Archimedean regulator maps and special values of $L$-functions, ...
Oli Gregory's user avatar
  • 1,369
0 votes

Is it possible to have square-free order(s) in $\mathbb{Z}^\times_N$?

It is impossible. The proof goes here. Denote $d$ as the order of $p_0$, so $d$ divides the order of $\mathbb{Z}^\times_N$ that is $4p'q'$. Let, $e=p_1\times p_2\times\cdots\times p_m$ be the exponent....
Somudro Gupto's user avatar
1 vote

Automorphism groups in class sets of ternary lattices

I put lots of references at http://zakuski.math.utsa.edu/~kap/ I've got an early version working. At first I thought it would be just class number one or two. The six coefficients $a,b,c,d,e,f$ ...
Will Jagy's user avatar
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1 vote
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Divisors on product abelian fourfolds

Unfortunately I do not really know a reference, but the proof is simple enough, so let me reproduce it here. For this answer, I denote by $\operatorname{NS}(X)$ the image of $c_1 \colon \operatorname{...
R. van Dobben de Bruyn's user avatar
16 votes
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What are $L$-functions?

You seem to be looking for a single unifying idea that explains why $L$-functions are the way they are. I don't think this is possible. $L$-functions are more like an elephant, with different ...
Will Sawin's user avatar
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0 votes

Applications of the Chinese remainder theorem

I didn't read this obvious application yet in the other answers: if you want to solve a polynomial equation in $\Bbb{Z}/n\Bbb{Z}$ and you factorize $n=q_1\dots q_m$ via CRT, you can calculate the ...
3 votes
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Difficulties in the proof of finiteness of n-Selmer group using cohomology

(Not sure any of these questions are at the right level for this forum, but here the comments that may help.) question : Inflation-restriction sequence. question : The target can be identified with ...
Chris Wuthrich's user avatar
3 votes
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Correctness of the algorithm for the A329369, A347205 and related sequences

Generalise to $$b(2^m(2k+1)) = \sum\limits_{j=0}^{m}C_{m+1,j} \, b(2^jk), \\ b(0) = 1$$ Consider the infinite matrices: $$M_0 = \begin{pmatrix} 0 & 1 & 0 & 0 & \cdots \\ 0 & 0 &...
Peter Taylor's user avatar
  • 6,776
0 votes

Applications of the Chinese remainder theorem

Didn't find this one above. I learned it in Silverman's A Friendly Introduction to Number Theory. Namely, CRT implies that Euler's totient function is multiplicative. That is, $$(a,b)=1\implies \...
6 votes
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Atkin-Lehner involution on the modular abelian varieties

Since an algebraic number is zero if and only if any of its conjugates is zero, $I_f J_1$ is stable under $W_N$ and so indeed $W_N$ descends to an automorphism of $A_f$. Now, the important thing to ...
Olivier's user avatar
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9 votes
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Field of definition of elliptic curves

The field of moduli of an elliptic curve is always a field of definition, a fact that is not true for abelian varieties of higher dimension. You can use the fact that two elliptic curves are ...
Joe Silverman's user avatar
2 votes
Accepted

Finiteness of Selmer group

Here's the explanation in the second edition of the work you cite.
anon's user avatar
  • 476
6 votes
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Lifting mod $p$ representations of arithmetic fundamental groups of a non-affine scheme over a finite field of characteristic $p$

I think the answer is no. There is a construction, due to Godeaux--Serre, which shows that (*) if $\Gamma$ is any finite group, then over any field $k$ there exists a smooth geometrically connected ...
SeanC's user avatar
  • 576
7 votes
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Let $X$ be a positive integer. Then $\pi{(X+\ln^2{X})}-\pi{(X-\ln^2{X})}>\ln{X}$?

The answer is No. According to Mathematica, for $X=1693182318746937$, we have $$ \pi{(X+\ln^2{X})}-\pi{(X-\ln^2{X})} = 34 < 35.065... = \ln{X} $$ Why this $X$? The Cramér–Shanks–Granville ratio is ...
Bogdan Grechuk's user avatar
2 votes

Quick proof of the fact that the ring of integers of $\mathbb Q(\zeta_n)$ is $\mathbb Z[\zeta_n]$?

I recently found a very short proof of the prime power case in the incredibly enlightening notes of Peter Stevenhagen: Number rings. The proof follows almost entirely from the Dedekind–Kummer theorem ...
JZweifler's user avatar
  • 121
7 votes

Let $X$ be a positive integer. Then $\pi{(X+\ln^2{X})}-\pi{(X-\ln^2{X})}>\ln{X}$?

Just to flesh out Lucia's predicted negative answer a little bit using the older heuristic arguments from Granville, Andrew, Harald Cramér and the distribution of prime numbers, Scand. Actuarial J. ...
Terry Tao's user avatar
  • 110k
4 votes

Increasing sequences and Wieferich primes

I will focus on Q3. Let us evaluate $J(2^{2^n}+1)$. Denote $N=2^{2^n}+1$. Then $N-1=2^{2^n}$. Notice that $2^{2^{n+1}}\equiv 1 \pmod N$. Let $M=\frac{N-1}{2^{n+1}}=2^{2^n-n-1}$, which is an integer. ...
Alexander Kalmynin's user avatar
12 votes

Probability of finding a prime number between $x-\ln(x)$ and $x+\ln(x)$

It is known that the probability in question exceeds a positive constant for $N>N_0$, and it is conjectured that it tends to $1-e^{-2}\approx 0.864665$ as $N\to\infty$. I suggest to read the ...
GH from MO's user avatar
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9 votes

Let $X$ be a positive integer. Then $\pi{(X+\ln^2{X})}-\pi{(X-\ln^2{X})}>\ln{X}$?

It is a subtle matter even to make a guess on how many primes must exist in such short intervals. A recent study of this, and related problems, was made in the paper of Granville and Lumley (the ...
Lucia's user avatar
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11 votes
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If $(a,b,c)$ are the sides of a triangle, then the probability $P(ax + by \ge c) = \frac{4}{\pi^2}\chi_2(x) + \frac{4}{\pi^2}\chi_2(y)$

Both conjectures are true. The proof below uses fedja's method in a simplified form, and also a nice observation by Zacky (see the comments below this post). Since $ax+by\geq c$ is equivalent to $b\...
GH from MO's user avatar
  • 100k
6 votes
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Test for pair of odd primes $(p, 2p^2-1)$

Below I will prove that the proposed test is necessary, that is, if $k\in\text{A106483}$ then $b(2k+1)=6k$. Following the simplification proposed by Will Sawin in the comments, the test for a given ...
Max Alekseyev's user avatar
15 votes

Let $X$ be a positive integer. Then $\pi{(X+\ln^2{X})}-\pi{(X-\ln^2{X})}>\ln{X}$?

Concerning lower bounds, we don't even know that the left-hand side is positive for every sufficiently large $X$. The best result of this kind is that, for every sufficiently large $X$, $$\pi(X+X^{0....
GH from MO's user avatar
  • 100k
11 votes

Given an irreducible polynomial over $\mathbb{Z}$, how often is it irreducible modulo a prime?

This is a supplement to David E Speyer's excellent answer. It was already proved by Frobenius (1880) that the fraction of $p$ for which $f(x)\bmod p$ has a given decomposition type $(n_1,\dotsc,n_t)$ ...
GH from MO's user avatar
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3 votes
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Finding a rational point of large height on an elliptic curve knowing a real approximation

This answer does the same, best approximations using rationals, but with a small intermezzo that shows in the particular tested cases the difference. The best rational approximation is done not for $x$...
dan_fulea's user avatar
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26 votes
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Given an irreducible polynomial over $\mathbb{Z}$, how often is it irreducible modulo a prime?

Let $K$ be a splitting field of $f$ over $\mathbb{Q}$. Since $f$ is irreducible, it is in particular separable; let $\theta_1$, $\theta_2$, ..., $\theta_n$ be the roots of $f$ in $K$. Then $G:=\text{...
David E Speyer's user avatar
1 vote
Accepted

"Infinity": A card game based on prime factorization and a question

Here is a rephrasing that may help: the graph $G$ has vertices $[c]$ and an edge between two vertices if one is a prime multiple of the other. Then the game is for each player to take some vertices, ...
Zack Wolske's user avatar
  • 1,877
10 votes

On A057985 and A287066

This is something that can be proved (in principle) with Walnut, a system for automatically proving results like this. You would have to use a special Pisot numeration system, the P4 system, as ...
Jeffrey Shallit's user avatar
2 votes

Number fields with finite maximal unramified $p$-extensions

One can do this without any recent theorems. Let $\ell$ be a prime congruent to $1$ mod $p$. Then there is a Galois extension $F$ of $\mathbb Q$ with Galois group $\mathbb Z/p$ ramified only at $\ell$,...
Will Sawin's user avatar
  • 138k
5 votes

What do theta functions have to do with quadratic reciprocity?

I think you're looking for the work of Tomio Kubota. The square of the theta function is a modular form. For a while, and still today, the theta function itself is sometimes considered a modular form ...
LeSserafimRuleTheKPOPSky's user avatar
0 votes

A comprehensive overview of finite fields

The "Handbook of Finite Fields" is probably the most comprehensive book, treating all aspects of finite fields and their applications. Mullen, G.L., & Panario, D. (2013). Handbook of ...
Reyx_0's user avatar
  • 149
5 votes
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Counting lattice points inside a parallelepiped

I have not filled in absolutely all the details, but hopefully this is enough to be convincing. Let's let $M$ take positive integer values, and let's consider the parallelepiped: $$P'=\{x+Mtv\mid x\in ...
Hugh Thomas's user avatar
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6 votes

Integer Points on an Elliptic Curve

The sage code is an one-liner: EllipticCurve(QQ, [-4, 9]).integral_points(both_signs=True, verbose=True) This gives some few details on the computations and lists ...
dan_fulea's user avatar
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0 votes

Finding rational points on intersection of quadrics in affine 3-space

If you want to solve a particular pair of these equations locally, and you know the intersection isn't singular, then you can always create a genus one model for the intersection on Magma and use the ...
Aditya Khurmi's user avatar
6 votes
Accepted

Quantum probabilistic method?

The Hilbert-Polya approach to the Riemann hypothesis follows this path, by attempting to relate the zeroes of the Riemann zeta function to a quantum mechanical scattering problem. The probability ...
Carlo Beenakker's user avatar
2 votes

Sequence derived from transform of a given vector (with Fibonacci as partial sums)

Not a complete answer, but too long for a comment and addressing the conjecture which I take to be the most important part of the question. The double-loop transformation process seems familiar to me ...
Peter Taylor's user avatar
  • 6,776
9 votes

Convergence of a product in $\mathbb Q_2[[X]]$

The coefficients you use are all in $\mathbf Z_2$, so I advise working in $\mathbf Z_2[[x]]$ rather than in $\mathbf Q_2[[x]]$. You are using the wrong topology on the power series, as mentioned ...
KConrad's user avatar
  • 49.8k
4 votes

Reference Request: Test vectors for local Rankin-Selberg L-factors in ramified cases

Are you asking for a proof of existence, or an explicit construction? These are very different things! It is immediate from the definition that there exists a finite family $(W_i, W_i')_{i \in I}$ ...
David Loeffler's user avatar

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