Questions on more advanced topics of number theory, such as quadratic residues, primitive roots, prime numbers, non-linear Diophantine equations, etc. Consider first if (elementary-number-theory) might be a more appropriate tag before adding this tag.

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Is this a recurrence for the Mertens function plus 2?

If we define a symmetric array: $$T(1,1)=3,\; T(1,2)=2,\; T(2,1)=2$$ $$T(1,k)=\frac{-T(n,k-1)-\sum\limits_{i=2}^{k-1}T(i,k)}{k+1}+T(n,k-1)$$ $$ ...
4
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136 views

Hecke operators as endomorphism of Jacobians of modular curves

Let $p$ be a primes that does not divide $N$, then $T_p$ defined an endomorphism $J_0(N)\to J_0(N)$. what is $T_p^\vee$? In other words, we naturally have $J_0(N)^\vee \xrightarrow{T_p^\vee} ...
4
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284 views

Prime zeta definition, multiplication by zero

Wikipedia has a page about the prime zeta function which is defined as follows: $$P(s)=\sum_{p\;\text{prime}} \frac1{p^s}$$ I entered this additional definition: Define a sequence: ...
4
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417 views

p-adic numbers and binomial coefficients

Let $\alpha\in \mathbb{Z}_p$ be an $p$-adic integer and define for $n\in \mathbb{Z}_{\geq 0}$ $${\alpha\choose n} := \frac{\alpha(\alpha-1)\cdot\ldots\cdot(\alpha-n+1)}{n!}.$$ This is again a ...
4
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92 views

find valuations

consider $f(x)=x^4+5x^3+1$. Let $\alpha$ be a root of this polynomial and consider $K=\mathbb{Q}(\alpha)$. Which are the absolute values on $K$ extending the usual absolute values on $\mathbb{Q}$? Are ...
3
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45 views

Approximate a large number with perfect powers

I'm dealing with number theory now and I have an interesting question. Every number can be approximated with two perfect powers, where perfect power is a number in form $$a^b$$$$a,b \geq 2, a,b \in ...
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34 views

Discriminant of $\mathbb{Z}[a,b]$

Let $K$ be an algebraic field extension of $\mathbb{Q}$. If $a\in \mathcal{O}_K$ is integral, then $Disc(\mathbb{Z}[a])=\prod_{i<j}( a_i - a_j)^2$ where the $a_i$ are the conjugates of $a$. Is ...
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61 views

'Randomness' of inverses of $(\mathbb{Z}/p \mathbb{Z})^\times$

Suppose you are given the group $(\mathbb{Z} / p \mathbb{Z})^{\times}$, where $p$ is prime. Let $A_p$ denote the sequence whose $j$th element is the inverse of $[j]$. For instance, if $p = 7$, the ...
3
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18 views

NP-hardness of solving congruence equations in several variables

You are given the following equation modulo $N$ (where the $\beta_i$'s are given integers modulo $N$, and the $x_i$'s are unknown integers modulo $N$): $$\beta_1x_1 = \beta_2 x_2 = \ldots = \beta_l ...
3
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39 views

Do the second-last-digits of the primes $\ge 11$ form a transcendental number?

Suppose, the number $x$ is constructed from the second-last-digits from the primes $\ge 11$ The first $1996$ digits of $x\ =\ 0.11112...$ after the decimal point are : ...
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46 views

Is there an advantage in using continued fractions for Catalan or Fibonacci-Lucas primality tests?

I am studying the basic theory about continued fractions and as usual after reading basic concepts I reviewed here at MSE former questions to learn more. While reviewing the questions and answers, I ...
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37 views

Partitions of integers, a series for infinite product $(1+q)(1+q^3)(1+q^5)\cdots$

Show that $$ (1+q)(1+q^3)(1+q^5) \cdots = 1+ \sum_{k=1}^\infty \frac{q^{k^2}}{(1-q^2)(1-q^4)(1-q^6) \cdots (1-q^{2k})}.$$ How would one proceed combinatorially. What I know is that the left-hand ...
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48 views

Problem for number theory

Here it is: $c , n \in \mathbb{N}$ and $x_1,x_2,\ldots,x_n \in \mathbb{N}\cup \{0\}$ $c= 1 x_1 + 2 x_2 + \ldots + n x_n$ How many solutions $\{x_1,x_2,\ldots,x_n\}$ are there? I do not know number ...
3
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69 views

Does $\pi$ contain infinitely many “zeros” in its decimal expansion?

Some number doesn't contain $"7"$ in its decimal expansion. For example Liouville's constant $$L=\sum_{n=1}^\infty\frac{1}{10^{n!}}=0.11000100....$$ contains only $0$ and $1$. It is well-known ...
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93 views

Very tentative proof of Beal's Conjecture?

I'm a high school student, so please point out my mistakes nicely and in layman's terms :) Thanks! Ok. Beal's Conjecture: If $$a^x+b^y=c^z$$ where $a$, $b$, $c$, $x$, $y$, $z$ are whole numbers; $x, ...
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20 views

Special $\omega(n)$-sequence

Let $k$ be a natural number, $\omega(n)$ the number of distinct prime factors of $n$. The object is to find a number $n$ with $\omega(n+j)=j+1$ for each $j$ with $0\le j\le k-1$. In other words, a ...
3
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43 views

Order of prime ideals over split primes in the class group.

Let $P$ be a prime ideal of $\mathcal O_K$ ($K$ a quadratic field) and let $P$ have norm $p$ where $p$ is a split prime. Is it possible for the ideal class $[P]$ to have order less than three? I feel ...
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69 views

Zariski density of points over completion

I have a simple question which I couldn't find a reference to. Let $X$ be a smooth projective irreducible variety over $\mathbb{Q}$. Suppose we base change to $\mathbb{Q}_p$ (the $p$-adics) and ...
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26 views

Lists of negative discriminants by class group?

Is there a handy listing of the discriminants of imaginary quadratic fields having a given ideal class group? It would be nice to use such a resource as a source of examples. For example, we're all ...
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50 views

Solving exercise 1.10 in Silverman's AEC

Please note that although there is a very similarly titled question Exercise 1.10 from Silverman "The Arithmetic of Elliptic Curves" this question received no answers. Let $p$ be an odd prime and ...
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20 views

Town Network - Show there is a Tetrahedron of the Same Transport

Question: There are 18 towns such that between each pair of towns there is either a train or bus service (not both). Prove that there are 4 towns such that all 6 of their pairwise connections use the ...
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54 views

Are there any known special properties of a number located between twin primes?

With the exception of $4$, every number located between twin primes is divisible by $6$. This one is obvious, but are there any other properties that can be ascribed to such numbers? A property may ...
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56 views

degrees of L-functions and dimensions of Shimura Varieties

I try to grab a (very) little understanding of what a Shimura variety is, and although I still don't understand the formal definition of that notion, a few vague ideas have come to my mind. Hence ...
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1k views

$ \lim_{n \to \infty} \sum_{i=1}^n\{ n / i \} (i/n \{ n / i \} - ln(n) -2){\mu{(i)}} +\sum_{i=1}^n 1/2( log(2 \pi [n/i])){\mu{(i)}} = \beta $

I'm just a physics undergraduate. I think (in an non-rigorous way) I have managed to prove: $$ \lim_{n \to \infty} \sum_{i=1}^n\{ n / i \} (i/n \{ n / i \} - ln(n) -2){\mu{(i)}} +\sum_{i=1}^n 1/2( ...
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32 views

On the existence of a polynomial in several variables with only one zero

Given an ordered field $\mathbb{K}$, then for all $n>1$ there exists $f\in\mathbb{K}[X_1,\ldots,X_n]\ s.t.\ \mathcal{V}^{\mathbb{K}^n}(f):=\{p\in\mathbb{K}^n:f(p)=0\}=\{(0,\ldots,0)\}$ For ...
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55 views

Notation Question in proof of Kronecker-Weber theorem

I am trying to understand this proof of the Kronecker-Weber theorem by Franz Lemmermeyer http://arxiv.org/pdf/1108.5671.pdf. The set-up is this: $K/\mathbb{Q}$ is a cyclic extension of prime degree ...
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66 views

How prove $\sigma(4^p-1)<(2^{p+1}-1)^2$

If $p$ is an odd prime numbers, show that $$\sigma(4^p-1)<(2^{p+1}-1)^2$$ where $\sigma(n)$ stands for the sum of divisors. I thought of using the formula for $\sigma(n)$: If ...
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131 views

If $\alpha = \prod_{i = 1}^{\infty} a_i \in \bar{\Bbb{Q}}$ can we write $\alpha = a_n \alpha'$ with $\text{den}(\alpha) = \text{den}(\alpha')$?

If $\alpha$ is any algebraic number, there is an integer $d > 0$ such that $d\alpha$ is an algebraic integer, and the minimum such $d$ is called the denominator of $\alpha$, written ...
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36 views

Merten's function

I am tasked with applying the Wiener-Ikehara Theorem to achieve a bound of little o(x) on Merten's function $\sum_{n=1}^x \mu (n)$. My problem is the Wiener-Ikehara Theorem applies to Dirichlet series ...
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21 views

Estimating special values of the Riemann zeta function on the critical line

If $p,q$ are primes, is it necessarily true that $$\left|\zeta\left(\frac{1}{2} + i\frac{p}{q}\right)\right| > (p+q)^{–(p+q)} ?$$ (Here $\zeta$ is the Riemann zeta function.)
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26 views

Can we treat finite and infinite primes in scheme theory?

In Number Theory, I know we have analogy to geometry by considering infinite primes not only Spec $\mathcal{O}_K$ (and I heard the point of view bring the Riemann-Roch theory). My question is can we ...
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222 views

Question about the elements of a reduced residue system relative a primorial $p_n\#$

I've been dividing up the elements of reduced residue system relative a prime $p_n$ into congruence classes modulo $p_{n+1}$ and I noticed that each congruence class is represented. If $r$ = the ...
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50 views

How many distinct factors of $n$ are less than $x$?

For some (squarefree) integer $n$ and some integer $x$, I would like to find an expression that gives, for all $n$ and $x$, a good upper bound on the function $$f(n, x) = \sum_{d|n, d < x} 1$$ ...
3
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46 views

the different of finite extension of p-adic numbers

Let $F=\mathbb Q_p$ be the field of p-aidc numbers. Let $\xi_n$ be a $n$-th primitive root of unity. Now consider the finite extension of fields $K/F$ where $K=\mathbb Q_p(\xi_n)$. I want to find the ...
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68 views

Do there exist any cycles for these number sequences?

We define, for $k\in\mathbb{N}$, the sequence $\left(S_{k,n}\right)_{n\in\mathbb{N}}$: $$S_{k,1}=k,\;\;\; S_{k,n+1}=p_1q_1\cdots p_mq_m \text{ (written out in decimal)}$$ Where $p_1^{q_1}*\cdots ...
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62 views

Inertia field of a compositum.

My question is regarding composite field extensions. The problem is motivated but not explicitly stated in the chapter 4 exercises of Marcus' Number Fields, in particular ex. 31 c) We first provide ...
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42 views

A bound on the squares of primes

If $p_n$ is the $n$th prime, what is the best known upper bound on $m$ such that $p_n \cdot p_{m+n} < p_{n+1}^2$?
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61 views

For what values of $x$ will $ax^2+b$ be perfect squares?

I need help on the this. Suppose that $ax^2+b$ is given where $a, b\in \mathbb Z$. Can we determine all the values of $x\in \mathbb Z$ such that $ax^2+b$ will be a perfect square ? Please help me. ...
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29 views

<Reference Request> Research done on whether the Euler prime can be the largest factor of an odd perfect number

(Note: This has been cross-posted to MO.) Good day! I would like to request for references to research done as to whether the Euler prime of an odd perfect number can also be its largest factor. ...
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94 views

Question regarding the prime factors of $2^{35} - 1$

Question regarding the prime factors of $2^{35} - 1$ I just wanted to make a few things clear; 1) It is true to state that this cannot be a Mersenne prime (A number of the form $2^r - 1$ where ...
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35 views

Number of digits $d$ in $d^k$

The title says it all really. For example, how many occurences of $6$ are there in $6^k$? It starts 6, 36, 216, ... so 1, 1, 1... The question can now be generalized into any digit or group of digits. ...
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54 views

Generalizing in Mathematics

I was reading the book "Fermat's Last Theorem" by Simon Singh when it hit me that this theorem is so contrived, andyet it lead to several important breakthroughs in mathematics and especially the ...
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93 views

Questions about decimal expansion being able to represent all real numbers

I read this in several books, and there's a Wikipedia article unquestionably stating that reals must be representable by means of regular language generated from finite alphabet. My questions are: ...
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151 views

Some Diophantine problems for equal sums with high powers

Given rationals $R = a,b,c,d,e,f$. Define, $$F_n = a^n+b^n+c^n-(d^n+e^n+f^n)$$ If $F_\color{red}1=0$, is there a rational solution to $7F_3x^4+7F_5x^2+F_7 = 0$? Then for $k=1,2,8$, ...
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72 views

Algorithm for Converting Rational Into Surreal Number

I'm writing a library in Haskell that represents the class of surreal numbers, which like many things can be read about here. I've run into a problem in converting between other classes of numbers ...
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75 views

the Teichmüller character

Let $d \geq 2$ be an integer, $K$ a number field containing the $d$-th roots of unity $\mu_d(\mathbb{C})$ and $\mathfrak{p}$ a prime ideal of $K$ not dividing $d$. Let $\mathbb{F}_q$ be the residue ...
3
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80 views

Number of $n$-permutations for which ${\tau}^k = id$

I am curious about the formula(any closed form) for the number of $n$-permutations $\tau$ such that ${\tau}^{n-1} = id$. How about for the case ${\tau}^n = id$ ?
3
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60 views

Finding the sum of the coefficient squared of a polynomial

Given a polynomial $$f(x) = \sum_{k = 0}^n a_kx^n = a_0+a_1x+ \ldots + a_{n-1}x^{n-1}+a_nx^n $$ is it possible to find a general expression for $$ \sum_{k = 0}^n a_k^2 ?$$ For example, $\sum_{k = ...
3
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80 views

Group generated by two polynomials

The $Homeo(\mathbb{R})$ be the group of all the homeomorphisms on $\mathbb{R}$, with the group operation of composition. Let $f(x)=x^3+\alpha$ and $g(x)=x^3+\beta$ be two elements of ...
3
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87 views

A Tale of Two Quadratic Identities (Pell-like)

Question is at the end. Let all variables be integers. For some constants $a,b,c,d$, assume we have initial solution {$m,n$} to, $$a m^2 + b m n + c n^2 = d\tag{1}$$ Identity 1: $$a x^2 + b x y + ...