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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Polynomial Evaluation

My question is to some extent related to cryptography, but I'd like the mathematicians answer my question, please (as their answers are usualy more clearer than cryptographers). Consider I have a ...
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Computation of a two variable combinatorial function

For every $t\geq1$ and $1\leq k\leq t-1$, $G(t,k)$ is a combinatorial sum satisfying the following recurrence: $$G(t,k) = 2t-1 + \sum_{h=1}^{t-k-1} G(t-k,h)\, (2k+2h-1) \;.$$ Is it possible to ...
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Some number theory questions.

Prove, that the product of $3$ , and $4$ following natural numbers can never be a number with the form of $x^k$ , where $x$ and $k$ are natural numbers, and $k$>1 (for example $9$ has this form, ...
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33 views

Division algebra over 2-adic fields

Let $D$ be the quaternion division algebra and $O$ be a maximal $\mathbb{Z}$-order in $D$, say the Hurwitz quaternion integers. It can be proved that $D$ and $O$ split at odd primes, that is ...
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53 views

Pell's equation and binary hyperbolic forms.

We define the function $f:\mathbb Z_p \times \mathbb Z_p \rightarrow \mathbb Z_p$ such that $f(x,y)=x^2-cy^2$ and $0\neq c=a^2+b^2$. Is it true that $f$ is hyperbolic? In other word's is there any ...
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Link between partition function and ordered partition function

The partition function $p(n)$ measures the number of partitions of $n$, or the number of ways in which natural numbers can be summed to produce $n$, without regard to order. For example, the ...
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103 views

Confused by a proof about harmonic numbers

I've been puzzled by a step in D'Aurizio's proof concerning the finiteness of a certain subset $J_p$ of $\mathbf{N}$ (its precise definition is irrelevant to my question). For the interested, his ...
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72 views

Proof of Frobenous Coin Problem lower bounds (Chicken McNugget theorem)

For the Frobenius Coin Problem, where $n = 3$ http://en.wikipedia.org/wiki/Coin_problem Does anyone know a proof for Davidson's formula? The one that states that the lower bound for the Frobenius ...
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27 views

Prove that if k and j are relatively prime to n, then so is k * j modulo n

GCD(kj, n) = 1 as kj and n don't share common prime factors. kj=qn + r for some q let's construct linear combinations ks + nt = 1 ju + nv = 1 Multiplying left-hand and right-hand sides (kj) su + ...
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Averages of $L(1,\chi)$

Let $(\frac{m}{n})$ denote the usual quadratic Jacobi symbol and $\mu(n)$ be the Moebius function. The series $$ \sum_{\substack{m,n \in \mathbb{N} \\ m,n\equiv 1 \mod{4}}} ...
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On the irreducibility of a polynomial!

Let $ p_1,p_2,\ldots,p_n $ be distinct prime numbers. Prove that the polynomial $$ f(x)=\prod_{e_1,e_2,\ldots,e_n=\pm1}(x+e_1\sqrt{p_1}+e_2\sqrt{p_2}+\cdots+e_n\sqrt{p_n}) $$ Is irreducible in $ ...
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51 views

Rational points and resolution of singularities

Suppose $X$ is an algebraic variety over a field $F$ of characteristic 0. By resolution of singularities, there is a nonsingular variety $Y$ over $F$ with a proper birational morphism $Y \rightarrow ...
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69 views

Reasoning about $z^n = x^m + y^m$

Let $z,n,x,y,m$ be positive integers with $z \ge 5$ and $m \ge 3$ and $m$ odd. Does it follow that: $z$ cannot be prime if $p \ge 5$ and $p | z$, then either $p > m$ or $p|m$ Here is my ...
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56 views

Square Root of a second degree polynomial

I need to know for what value of $x \in \mathbb{N}$ does the polynomial $ax^2+bx+c$ with $ a,b,c\in\mathbb{N}$ equal the square of an integer. If you prefer, i need to know what value of $x$ will be ...
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54 views

Strange diophantine equation involving seventh powers

Consider the diophantine equation $$\frac{x^7 - y^7}{z^7 - t^7} = 3$$ where $x$, $y$, $z$ and $t$ are integers. Can one prove that the number of coprime 4-tuples satisfying this equation is finite?
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What is the smallest number written as a sum of cubes?

What is the smallest number of the kind $\overline{999a}$, which can be presented as a sum of two natural cubes? ($a$ is a digit). I do NOT multiply below (when I write $999a$) $$999a = x^3 + y^3$$ ...
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Maybe Fermat primes are infinite?

A Fermat prime is a prime $p$ in the form $2^{2^n}+1$, for some integer $n\ge 0$. It is actually unknown if there infinitely many such primes. Despite the title, here we propose an argument in its ...
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Numbers in the form $10^n + 1$ with square divisors

Basically, describe every number in the form $10^n + 1$ with square divisors meaning at least one of it's divisors is a square. Of course, there's infinite, but give a general algorithm for finding ...
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44 views

Bunyakovsky conjecture for cyclotomic polynomials

This article on Wikipedia: http://en.wikipedia.org/wiki/Bunyakovsky_conjecture says: In fact, it can be shown that if for all natural number $ n $, there exists a natural number $ x > 1 $ such ...
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23 views

Asymptotic for the height of the derivatives of a rational function

Let $\phi=\frac{P(z)}{Q(z)}$ be a homogeneous rational function of degree $d\ge 2$ over $\overline{\mathbb{Q}}$. If $h$ is the absolute logarithmic height, it seems that for each $z\in ...
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Surjectivity of the derivation map on Washnitzer Algebra

Let $K$ be a non-archimedean field of characteristic zero and $||.||:K\to \mathbb{R}_{\geq 0}$ be its absolute value. Define the Washnitzer Algebra as: $$W_n=\{\sum_{u\in \mathbb{Z}_{\geq 0}^n} \in ...
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43 views

Zeros of the Ramanujan sum for finite fields

Let $\mathbb{F}_q$ be a finite field with $q$ elements, let $N$ be a positive divisor of $q-1$, and let $\xi_N$ be an element of $\mathbb{F}_q^*$ of order $N$. One can similarly define the Ramanujan ...
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Estimating modulus from integer approximation

I want to calculate $NM^{-1}\bmod q$ as efficiently as possible. However, I have only $N+\delta$,$M+\mu$ with $\delta,\mu>0$ arbitrarily small. How accurately can I estimate $NM^{-1}\bmod q$? ...
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Proof of Chevalley–Warning theorem

How to prove Chevalley–Warning theorem (http://en.wikipedia.org/wiki/Chevalley%E2%80%93Warning_theorem) by using Fulton's trace formula (#$|X(\mathbb{F}_p)| \equiv \sum (-1)^i Tr(Frob_p|H^i(X, ...
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54 views

Strict total ordering

I'm not able to understand how the below relation is example of "strict total order". Consider a set $X = 2^Y$ where $Y = \{1,2,3,4,5,6,7,8,9\}$. The expected order of $X$ is for all $x, y$ ...
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39 views

Hensel's lemma in $n $ variables

I'm trying to find a proof for the following formulation of Hensel's lemma: $$\text{Let } f \in \mathbb{Z}[x_1, \dots, x_n], a = (a_1, \dots, a_n) \text{ be such that (with } p \text{ prime)}$$ $$ ...
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117 views

Any formula for the exact number of primes below a given bound?

Reading The music of the primes, the author relates that Riemann had figured out a formula giving exact number of primes up to a certain bound with no errors. Does such formula really exist? If ...
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111 views

Sum of two squares - Number of steps in Fermat descent

If a prime $p$ can be written as the sum of two squares, then one can construct this representation via Fermat descent if we know an $x$ such that $x^2 \equiv -1 \mod p$. Is there a possibility to say ...
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Orbits of left-multiplication from $\mathrm{SL}_2(\mathbb Z)$ on $\mathbb Z^{2\times 2}$

I am trying to learn moduli space of elliptic curves from different resources. But only a spacial class of elliptic curves where the lattice in the plane has "integer vectors" as generators. To study ...
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Definition of Natural Numbers with one statement.

Is there a way that we can use mathematical or logical axioms to define natural numbers with one statement? In planar geometry there are several basic notions which can define everything with a finite ...
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Independence of FLT over weak systems

It is known that Fermat's last theorem can be proven in finite-order arithmetic (e.g. accoridng to this site). This is still an extremely high upper bound on proof complexity (for example, compared to ...
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An elliptic curve for the multigrade $\sum^8 a_n^k = \sum^8 b_n^k$ for $k=1,2,3,4,5,9$?

I. The first solution to, $$\sum^6_{n=1} a_n^9 =\sum^6_{n=1} b_n^9$$ $$13^9+18^9+23^9-5^9-10^9-15^9 = 9^9+21^9+22^9-1^9-13^9-14^9$$ was found in 1967 by computer search by Lander et al. It stood ...
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Repeating decimal algorithm

I was working on a problem where I needed to prove things in base 10, like: "11 divides $a$ if and only if 11 divides $a_0-a_1+a_2-\cdots$" where ...
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Does Idele group of norm 1 preserved by the norm?

I should explain my question in detail as of now I'm sure it makes no sense. Let $K$ be a global field (in particular I care about the characteristic $p$ case.) Then its Idele group $I_K$ has a ...
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FLT (Fermat): Combinatorial approaches?

Such a simple equation like $x^n+y^n=z^n$ is bound to have a nice/natural combinatorial interpretation. One very crude one is: Let the number of ways of choosing $n$ objects from $x$ objective, ...
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Finding a basis for a particular integer lattice

The following problem arose in the context of string theory. I hope someone here might provide some guidance or a solution... Our starting point is: i) an integer-lattice $L\subset\mathbb R^n$ ...
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Non-trivial odd characters mod m

I am stuck with this problem of marcus: I proved it when the charater is even. But I cannot prove the given formula when the character is odd. Please help.
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Can you subtract two integers using multiplication and division?

The multiplication operation is traditionally defined as "repeated addition", and division (with remainder) can be defined using repeated subtraction. Can we define subtraction the other way round? ...
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Using Perron's formula for asymptotic behaviors

I happen to read this post about trying to get the formula of $\sum_{n=1}^N n^m$ for Perron's formula. The general Perron's formula is $$\sum'_{n\le x} a(n)=\frac{1}{2\pi i}\int_{\text{Re ...
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51 views

Two closest sums of pairs of reciprocals

Trying to obtain a better bound for a problem from this bounty question, I obtained the following problem. Let $n\ge 3$ be a natural number. The problem is to estimate (in particular, asymptotically) ...
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Using a Gauss sum to show that $p$ is of the form $x^2 + xy +3y^2$ if and only if $p \equiv 1, 3, 4, 5, 9 \pmod{11}$

Let $p \neq 11$ be an odd prime, and $\zeta$ an $11$th root of unity. Let $g$ be the Gauss sum $$g = \sum\limits_{i=1}^5 \zeta^{i^2} = \zeta + \zeta^4 + \zeta^9 + \zeta^5 + \zeta^3$$ We may ...
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About RSA-Factoring

Respected All Today I do not ask any problem on mathematics but only willing to know about the following. According to RSA website found here it is declared that RSA factoring challenge is no longer ...
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Unramified extension of a imaginary quadratic field.

Let $K$ be an imaginary quadratic number field over $\Bbb{Q}$. Let $K(\sqrt{a})$ be an extension with $a$ an integer. Let $d$ be discriminant of field $K$. Then how to show that $K(\sqrt{a})$ is ...
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Galois group of splitting field of a polynomial

Let $f$ be a polynomial with integer coefficients and irreducible over $\Bbb{Q}$. Let $p$ be a prime. Suppose $f(mod $ $p$) can be written down as a profuct of $r$ distict irreducible polynomials ...
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54 views

On $p^{\log_q n}$, where $p$ and $q$ are distinct primes

Let $p,q$ be distinct primes, $n>1$ an integer with $\log_q n $ irrational. It was, and probably still is, a conjecture that $p^{\log_q n}$ is non-integer. What progress has been made towards it?
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Discrete logarithm - factorization of modul

I am solving discrete logarithm problem $mod N$. $N$ is composite number, i found its factors - lot of small primes and two big primes ($> 2^{50}$). Does the factorization of $N$ somehow help me? I ...
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Analytic continuation of the Dirichlet $\eta(s)$ series to $\Re(s) \gt -1$. Why does this work?

Take the known Dirichlet $\eta(s)$ series, $$\displaystyle \eta(s) = \sum _{n=1}^{\infty } \left( {\frac {1}{(2\,n-1)^{s}}} - \frac{1}{(2\,n)^s}\right), \qquad \Re(s)>0$$ and add $\displaystyle ...
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On integer $n>1$ and prime $p$ such that $p<n$ , $p$ does not divide $n$ and $n-p$ is a prime

Let $n>1$ be a given integer and $p$ be a prime less than $n$ and not dividing $n$ ; so $p$ and $n$ are co-prime ; hence $n-p$ and $n$ are also co-prime ; I would like to ask when is $n-p$ also is ...
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How to calculate the $(3)$ and $(4)$?

In Gérald Tenenbaum's book "Introduction to Analytic and Probabilistic Number Theory" Cambridge University Press 1995, On the page of 97-98, I Can calculate the $(1)$ and $(2)$, but I do not know how ...
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Extension of valuation

This is a question from Milne's 'Algebraic Number Theory'. Let $K$ be a valued field with absolute value $|\cdot|$ and $L=K(\alpha)$ a finite separable extension of $K$. Let $\hat{K}$ be the ...