Questions tagged [number-theory]
Questions on advanced topics - beyond those in typical introductory courses: higher degree algebraic number and function fields, Diophantine equations, geometry of numbers / lattices, quadratic forms, discontinuous groups and and automorphic forms, Diophantine approximation, transcendental numbers, elliptic curves and arithmetic algebraic geometry, exponential and character sums, Zeta and L-functions, multiplicative and additive number theory, etc.
39,534
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Theorem 1.27 a) in Diamond, Darmon, Taylor, "Fermat's Last Theorem"
I am looking for a reference for part a) of Theorem 1.27 here regarding the proof of the growth of coefficients of cusp forms. Theorem 1.24 gives a very sketchy argument of a less specific fact, but I ...
- 3,991
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0
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7
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Problem about Elliptic regulator on a set of points in an elliptic curve
I am a bit on the following problem:
Let $\{P_1,\ldots,P_r\}$ be a set of rational points on an elliptic curve $E/\mathbb{Q}$ and $\mathfrak{H}:=(\langle P_i,P_j\rangle)_{1\le i,j,\le r}$. Then, I ...
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39
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Similarity of Lifting the Exponent Lemma in Pell numbers
Pell number is a term of the sequence $\{P_n\}$ determined by a recurrence relation
$$P_{n+2}=2P_{n+1}+P_n, P_0=0, P_1=1.$$
Let $v_p(x)$ be the $p$-adic valuation of an integer $x$ (the number of ...
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29
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Help me solve this diophantine equation- $x^5+x^4+1=7^y$ [duplicate]
An obvious soloution is $(x,y) = (2,2)$, but I can't figure out which module should I look for to show there are no more solutions...
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1
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1
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51
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What is the equation for the number of integer solutions of $n$ for $n^2\equiv -1\pmod m$ where $0≤n<m$, given $m$?
I can’t work out how to find the equation for the number of integer solutions of $n$ for $n^2\equiv -1\pmod m$ for any $m$ where $0≤n < m$.
The number of solutions, when non-zero, seems to be ...
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0
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24
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What Irreducible cubic curve and Siegel's theorem
Let $c(x, y)$ be a cubic polynomial with rational coefficients which is irreducible over $\mathbb{Q}$. I'm trying to understand when the equation $c(x) = 0$ has infinitely many integral points. If $c$ ...
- 2,969
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3
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40
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Order of Odd Elements in $\mathbb{Z}_{2^n}$
I'm wondering if there is a way to calculate the order of odd numbers in the cyclic ring $\mathbb{Z}_{2^n}$. I found a paper that shows that for any odd $x$, $x^{2^{n-2}} \equiv 1 \mod{2^n}$, and that ...
0
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2
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71
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What is a sufficient and necessary condition for a system of linear equations to have a unique integer solution?
Bézout's theorem states that a linear diophantine equation $ax+by=c$ has integer solutions if and only if $\gcd(a,b)|c$. Then is there a theorem regarding the existence of a unique integer solution ...
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22
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Maximizing a Sum over Partitions of an Interval: An Optimization Problem
Suppose $N$ is a fixed natural number. We split the interval $[1,N]$ into $L-1$ parts, where $L$ is some natural number less than $N$. This gives us an ordered set of points $\{n_i\}_{i=1}^L$, where $...
3
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36
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Strassmann's thoerem and irrationality measure of certain number
In this note from Keith Conrad, he explains an interesting application of Strassmann's theorem to the divergence of certain linear recurrence integer sequence. More precisely, the sequence defined as $...
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Prove that is always an integer $\frac{p_n!}{p_n(p_n+1)(p_n+2)(p_n+3)\dots(p_{n+1}-1)}$
Let $p_1<p_2<\dots<p_n<\dots$ be the consecutive primes . Prove that $$\frac{p_n!}{p_n(p_n+1)(p_n+2)(p_n+3)\dots(p_{n+1}-1)}$$ is always an integer except when $p_n=3$ .
$\textbf{Attempt:}...
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Another proof of e is irrational?
I find some proof here, but can I prove it in the following way?
Assume $e$ is rational, then $e=\frac{p}q$, both $p$ and $q$ are positive integers. By Lagrange's Remainder Theorem,
$$e^x=1+x+\frac{1}{...
- 7,007
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vote
2
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32
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Gaussian Sums of a Dirichlet's Character
In Davenport's chapter 9, They defined
$$\tau(\chi)=\sum_{m=1}^{q} \chi(m) e_q(m)$$
Further if $(n,q)=1$, then we have that $$\chi(n)\tau(\overline{\chi})=\sum_{h=1}^{q} \overline{\chi}(m) e_q(nh)$$ ...
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$2^n$ contains 9 in base 10 if $n \geq 109$?
While working on https://www.reddit.com/r/math/comments/11wds6n/repeated_doubling_conjecture/ I observed that the decimal expansion of $2^n$ appears to contain a $9$ when $n \geq 109$.
Can/has this ...
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0
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31
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Product in ideal class group
let $R$ be a Dedekind domain, $K$ its field of fractions. In the class group $Cl(K),$ is it true that the product of classes of two ideals equals to the class of their product?
- 27
3
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0
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46
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$(-1,0,1)$-square matrix has different line sums?
Let $A$ be a $n\times n$ matrix with coefficients from the set $\{-1,0,1\}$. Let $r_i$ and $c_i$ denote the sum of the elements of the $i$-th row and column of $A$ respectively. For which $n$ is it ...
0
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1
answer
26
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Radius of convergence in p-adics
Let $f(x)=\displaystyle{\sum_n a_nx^n}$ be a power series with coefficients in the field of p-adic rationals, $\mathbb{Q}_p$. Let $R_f$ be its radius of convergence and let $f'$ denote its term-by-...
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set all values of n such that the sum $\sum_{k=1}^{n}k^{2}$ is a perfect square [closed]
n is a non-null natural number ($n\in \mathrm{N^{*}})$
set all values of n such that the sum $\displaystyle\sum_{k=1}^{n}k^{2}$ is a perfect square
$(\displaystyle\sum_{k=1}^{n}k^{2}=a^{2}:a\in \...
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1
answer
36
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Finding the value of the square of the Gauss sum
While finding the value of the square of the Gauss sum $G^2$, at some point we make a substitution as follows:
$$G^2 = \sum_{m_1=1}^{q-1}\sum_{m_2=1}^{q-1} \left(\frac{m_1 m_2}{q}\right) e_q(m_1+m_2) =...
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44
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Which general methods are there for determining if $t$ and $e^t$, where $t$ is any given transcendental number, are algebraically independent?
Which general methods are there for determining if $t$ and $e^t$, where $t$ is any given transcendental number, are algebraically independent?
Can Schanuel's conjecture be used for this?
We have ...
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2
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69
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Find the continued fraction for $\sqrt{k^2+k},$ where $k$ is a natural number
Find the continued fraction for $\sqrt{k^2+k},$ where $k$ is a natural number.
I know the algorithm how to find it for a specific number, let's say $\sqrt5,$ but any idea and any understanding how it ...
1
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1
answer
52
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Does there exists any positive integer solution to the equation $x^3+y^3=2z^3$.
Does there exists any positive integer solution to the equation $x^3+y^3=2z^3$ where $x\neq y\neq z$. I tried to find but could not get such triple. We all know that $x^3+y^3=z^3$ don't have a integer ...
2
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1
answer
86
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Finding rational points on a circle such that $X^2+Y^2=r^2=k \in \mathbb{Z}$
I am interested in finding rational points on a circle with radius $r$, such that $r^2=k$ is an arbitrary integer. I tried reducing the problem to the unit circle, and maybe use pythagorean triples as ...
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The sum of each digit of a polynomial is not a Fibonacci numbers. [closed]
P is an integer coefficient function. For the natural number n, S(|P(n)|) is not a Fibonacci number. Then, is P bound to be a constant? However, S(|P(n)|) is the sum of each digit of P(n).
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small clarification about S-class ideal
in this question about the S-ideal class group https://math.stackexchange.com/q/4310256, we have this answer
enter image description here
i didn't understand the last result, how he concludes from $(a)...
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0
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Example of the real numbers $x$ appearing in the Jarnik's theorem: $||nx||\leq n^{-\beta}$ for infinitely many $n\in\mathbb{N}$ for fixed $\beta>1$
According to Falconer (Falconer, '85, The Geometry of Fractal Sets, theorem 8.16, p. 134), a part of the Jarnik's theorem is the following:
Take $\beta>1$. The set of real numbers $x$ for which ...
- 768
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1
answer
24
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Generators of coprime ideals perfect powers
Could you please help with this a practice exam problem?: Let $K=\mathbb{Q}(\sqrt{-22})$ with unit group $\mathcal{O}_k^\times=\{ \pm 1 \}.$ Suppose that $x,y,z\in\mathcal{O}_K$ are non-zero, satisfy ...
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1
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Subspace of newforms one-dimensional with CM $\implies$ unique newform a Poincare series.
Let $k \geq 2$. Say $S_{k}^{\text{new}}(\Gamma_{0}(N), \chi)$ is one-dimensional and spanned by a newform with CM, and $S_{k}^{\text{old}}(\Gamma_{0}(N), \chi)$ has positive dimension. Must it be true ...
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Period of Lehmer sequences
In his thesis (1930), D. Lehmer did not provide the general formula for the period of his sequences. And it does not appear in HC. Williams book about E. Lucas work. And, since Lehmer sequences are ...
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1
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46
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Orthogonal group of Lorentzian lattice $I_{1, n}$ is infinite for $n\geq 2$
I am looking for a reference or an elementary proof of the following fact:
The orthogonal group of the lattice $I_{1,2} = I^+ \oplus 2 I^-$ is ifinite.
Here the Lorentzian lattice $I_{1,n}$ is given ...
- 4,937
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1
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62
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What is the probability that the greatest prime factor of a sequence of uniformly distributed integers increases?
Let $f_k(n), n = 1,2,3...$ be a sequence of random integer uniformly distributed in $[2,k]$ for some fixed $k \ge 3$. Let $l_n$ be the largest prime factor of $f_k(n)$. What is the probability that $...
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Question about the order of apparition of prime gaps [closed]
Is there any consecutive prime gap that appears for the first time before a smaller gap appears for the first time?
i.e. there is two consecutive primes with a gap of 18 before there appear any two ...
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0
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125
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How to find this LCM sum function? $ \text{lcm}(m,n) +\text{lcm}(m+1,n) +\cdots+\text{lcm}(n,n)$
Problem:
$$S=\text{lcm}(m,n) + \text{lcm}(m+1,n) +\ldots+ \text{lcm}(n,n)$$
There is already a question on math.stackexchange for $[1,n]$ range . I am trying to generalise it further for $[m,n]$ range....
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Analytic Number Theory - distribution of $x^2$ vs. the distribution of $x^2 - 2y^2$
My question originates from Rational Points on Elliptic Curves, (Silverman & Tate), though has little to do with elliptic curves.
In chapter $V$: Integer Points on Cubic Curves, section $3$ it ...
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51
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Mazur's theorem for general number fields
Mazur's theorem completely classifies the possibilities of $E_{tors}(\mathbb{Q})$ for an elliptic curve $E/\mathbb{Q}$, including the fact that only finitely many groups occur. What happens with $E_{...
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Elliptic curves of the form $y^2=x^3+p^2$, $p$ a prime, has torsion subgroup isomorphic to $\mathbb{Z}/3\mathbb{Z}$.
Suppose that $p\ge 5$ is a prime and $P$ be a torsion point on the elliptic curve $E_5:y^2=x^3+p^2$. Using the Nagell-Lutz theorem we can write down all possible values of $y(P)$ (where $P=(x(P),y(P))$...
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70
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Write $7 \cdot 10^{100} + 7$ as a sum of four squares
How do you write $7 \cdot 10^{100} +7$ as a sum of four squares?
I know that you can write it as a sum of four squares by the Lagrange's Four Squares Theorem, but I don't know how to write such a big ...
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120
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An unusual equivalent form of Riemann hypothesis
Let $G(x)=\sum_{k\leq x}\frac{\mu(k)}{k}$, where $\mu$ is the Mobius function. From this question and its answer, its mention the Riemann hypothesis is equivalent to $G(x)=O(x^{-\frac{1}{2}+\epsilon})$...
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Does the polynomial $p(n) = (n^3 + 6n^2 + 17n + 6)/6$ record visible points in grids of size 7 and 11 for all dimensions n greater than 1?
The polynomial $p(n) = (n^3 + 6n^2 + 17n + 6)/6$ appears to coincide with the folowing: mark each point on a $7^n$ or $11^n$ grid with the number of points that are visible from the point; for $n > ...
1
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1
answer
60
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Infinite Series of Rational Terms
Does anyone have a counterexample to the following conjecture?
Conjecture:
Suppose that the terms $a_{n}$ is a sequence of rational numbers and further that for all real numbers $r$, there are ...
- 19
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1
answer
53
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Is the set of all rational additive partitions of a rational number countable?
We usually call additive partitions the set, we call it $P$, of all the ways to write a positive integer $n$ as a sum of positive integers. Formally:
\begin{equation}
P_n = \left\{ (a_1 ,...,a_n)\in\...
- 620
4
votes
3
answers
92
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Sum of signed permutations of digits equals zero
After playing around with signed permutations lately, (as part of studying properties of antisymmetric tensors and wedge products which are not really related to what I want to ask about), I noticed ...
- 341
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2
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70
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Show if $f:\mathbb{Z}_p\to\mathbb{Q}_p$ is continuous such that $f(n)=(-1)^n$, then $p=2$.
Let $\mathbb{Q}_p$ denote the p-adic rationals and same for the integers. Suppose $f:\mathbb{Z}_p\to\mathbb{Q}_p$ is continuous, with the additional property that at $n\in\mathbb{Z}^{\geq 0}$, $f(n)=(-...
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1
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29
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How we can show $\Lambda (s,\chi) < \exp(C|s| \log |s|)$ as $|s| \rightarrow \infty$?
We know that the functional equation for Riemann-Zeta is $\psi(s)=\frac{1}{2}s(s-1)\pi^{-1/2s}\Gamma(1/2s) \zeta(s)$ and $\psi(s) < \exp(C|s| \log |s|)$ as $|s| \rightarrow \infty$. Does it make ...
- 167
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votes
0
answers
61
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Eigenvalues of matrices corresponding to imaginary part of non-trivial Riemann zeroes.
I have an amateur interest in mathematics and in 2020 after being inspired by a Video on $\zeta$ function by Grant Sanderson, and doing further reading into topics like Hilbert-polya conjecture I ...
- 51
1
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1
answer
27
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Numerator of zeta $Z(C/\Bbb{F}_q, T)$ and Cayley-Hamilton theorem and Frobenius
Let $C$ be a genus $g$ curve. I know numerator of congruent zeta function $Z(C/\Bbb{F}_q, T)=\sum \frac{\sharp C(\Bbb{F}_{q^n})x^n}{n}$ is product of characteristic polynomial of Frobenius, $det(1-...
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6
votes
1
answer
109
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Formula for the ratio of Numbers not divisible by the first n primes
I was playing around with prime numbers, and specifically how many prime numbers we can exclude from an interval from being primes. It is easy to see, that after 2, the ratio of numbers is $1-\frac{1}{...
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1
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24
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when the fractional ideal $S^{-1}I$ of the localization $S^{-1}A$ of a dedekind domain $A$ is principal [closed]
if a fractional ideal $S^{-1}I$ of the localization $S^{-1}A$ of a dedekind domain $A$ is principal, is that implies conclusion about the fractional ideal $I$ in $A$
- 27
3
votes
1
answer
117
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Find all positive number $n$ such that $n-1 \mid 1 + 10^n + 10^{2n}$
Find all number $n\in\mathbb{N}$ such that
$$n-1 \mid 1 + 10^n + 10^{2n}$$
If $n-1$is prime, it's clear that $n-1\mid 10^{n-2} - 1$ if $n\ne 3, 6$. So, $$0\equiv 1 + 10^n + 10^{2n} \equiv 1 + 10^2 + ...
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0
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23
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Formula for difference between prime counting functions [closed]
How can I prove that, for $x>y>0$,
$$\pi(x)-\pi(y)=\int_y^x \frac{1}{\ln t}dt+O\left(\frac{xe^{-\sqrt{\ln x}}}{(\ln x)^{3/4}} \right) $$
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