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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Proof that coprime cubes are divisible by $n$

I would like to ask if my proof is correct. Task is to prove that $$ n \big| \left( \sum_{0 < a < n \atop (n, a) = 1} a^3 \right), $$ where $n>2$. My proof: Take a number $k$ coprime to ...
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49 views

A conjecture on integers coprime to $2^n-1$ and having a prescribed Hamming weight in their binary representation

I wonder if anyone has seen this before or would have some ideas on how to go about proving it. I have done several experiments with the computer and seems to hold. For an integer $k$, denote by ...
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53 views

The number of totatives to the nth primorial, in an interval shorter than the nth primorial.

Can, and if so when can, we determine the amount of natural numbers which are relatively prime to the nth primorial in an interval of length smaller than the nth primorial? Introduction Let ...
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69 views

A different viewpoint of Hilbert's Theorem 90

Let $L/K$ be a galois extension with galois group $G$($|G| = n$) cyclic and generated by $\sigma$. Let $\beta \in L$ have $N(\beta) = 1$. $N(.)$ is the norm function from $L$ to $K$. Hilbert's ...
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47 views

Proving that for $F_k = F_{k-1} + F_{k-2}$, $F_k$ is even iff $3|k$

Consider the recursion $F_k = F_{k-1} + F_{k-2},$ $ k\geq 2,$ $F_0 = 0$ and $F_1 = 1$, then show that $F_k$ is even iff $3|k$ I tried to do a proof by induction: The statement is true for base ...
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30 views

element of an $\ell$-adic Galois representation with certain eigenvalues.

Let $\mathscr{G}=Gal(\bar{\mathbb{Q}} / \mathbb{Q})$, $E$ an elliptic curve over $\mathbb{Q}$, and consider the $\ell$-adic representation $$ \varphi_{\ell}: \mathscr{G} \longrightarrow ...
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69 views

Every positive real number has a base two expansion?

"Every positive real number has a base two expansion." This is a statement from Folland Page $8$ and I wonder why it is true. Here is my procedure to express a given positive real number $a$: Step 1: ...
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37 views

Rate of large composite numbers, which are strong probable prime to the bases $2,3$ and $5$

Here http://primes.utm.edu/glossary/xpage/StrongPRP.html is the definition and some useful informations about strong probable primes. For higher numbers, lets say near $10^{50}$, strong probable ...
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34 views

Limit of an euler product

Before I can ask my question, I have to state a couple of definitions. Let $f$ be a multiplicative function and let $$ D_f(s) = \sum_1^{\infty} \frac{f(n)}{n^s}, $$ and define $\Lambda_f(n)$ as ...
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34 views

Why does the uniqueness theorem for Dirichlet series hold for the infinite sums, while obviously not for partial sums?

I asked in a previous question whether a function, $a_n$, is unique to $F(s)$ for any Dirichlet function defined by the following $$F(s)=\sum_{n=1}^\infty{\frac{a_n}{n^s}}.$$ Its uniqueness property ...
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50 views

Proof by contradiction using asymptotic properties

The following is from this paper that discusses polynomials and classic number theory functions. Theorem: There do not exist polynomials $P,Q \in \mathbb{R}[X]$ such that ...
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Proof of the binomial theorem through Dirichlet convolution?

Here I gave a proof for $\sum_{k=0}^n\binom nk(-1)^k=0$ based on the fact that $\mu*1=\varepsilon$ (the Dirichlet identity). I am wondering if using a similar technique we can prove that ...
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61 views

“Race” of the primes modulo $1,3,7,9\ \pmod {10}$

The "race" starts with the prime $11$. The number of primes $1, 3, 7, 9 \pmod {10}$ is denoted after every occurring prime. Does the lead change infinitely often? And does every "runner" have ...
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Are there other known continued fractions that show the digits of the golden ratio?

I found a few. {16; 5, 1, 1, 5, 22} {161; 1, 4, 11, 1, 1, 3, 6, 1, 13, 8, 1, 6, 1, 4, 1, 1, 2, 1, 1, 1, 1, 13, 2, 1, 3, 8, 1, 2, 19, 1, 54, 1, 19, 2, 1, 8, 3, 1, 2, 13, 1, 1, 1, 1, 2, 1, 1, 4, 1, 6, ...
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Ways Of Finding Primes and If they are efficient

I am currently in middle school and love number theory. I try and do a proof every day and today I was working on a relatively simple one involving primes. I proved that every prime above 5 can be ...
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34 views

Is there a tighter approximation for the least prime gap of a given length?

This link https://primes.utm.edu/notes/gaps.html gives a definition of the maximal gaps. For a number $g$ , $p(g)$ is the smallest prime $p$ followed by at least $g$ composites. The estimate is ...
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69 views

Two irrational numbers are congruent iff the tails of their infinite continued fractions eventually coincide

We say that a real number $\alpha$ is $congruent$ to real number $\beta$ if there exist integers a, b, c and d with ad-bc=+1 or -1 and such that $$\alpha=\frac{a\beta +b}{c\beta+d}$$ I need to prove ...
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20 views

Symmetry of Hecke L-function zeroes

For the Riemann zeta function, it is known by the functional equation and $\zeta(s)=\overline{\zeta(\bar s)}$ that the zeroes of $\zeta(s)$ are symmetric about the critical line $1/2$ and the real ...
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31 views

Kronecker symbol vs. Koblitz symbol

In Koblitz, Introduction to Elliptic Curves and Modular Forms on page 188 it is defined $$\left( \frac{-1}{j}\right)=0$$ in case $j$ is even. Apart from that definition $\left( \frac{c}{d}\right)$ is ...
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65 views

Does anyone recognize this graph?

It's a plot of the following: Let $$f_{(n)} = \frac{np_n}{(p_1 + \ldots + p_n)}$$ so that $$g_{(n)} = \left|\space f_{(n)} - f_{(n-k)}\right| $$ where $n > k$ and $k = 5$ in this example. For ...
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38 views

Besides the $3x + 1$ problem, for which similar problems are still unresolved regarding trayectory?

Generalize the $3x + 1$ problem as $cx \pm 1$, where $c$ is a positive odd integer and $x$ is a positive integer iterated through the function as far as possible to discover a cycle. If $x$ is even, ...
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116 views

Twin prime conjecture (Goldbach-Collatz remix)

Assuming Goldbach's conjecture, let's denote $r_{0}(n)$ for any integer $n$ greater than $1$ the smallest non negative integer $r$ such that both $n+r$ and $n-r$ are primes. Let $f$ be the map ...
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17 views

Prime Factorisation Probability Decrease: upper bound

Suppose we have a probability distribution $p$ over $\{0, 1, 2\}$, with probabilities $p_0$, $p_1$ and $p_2$, and $p_0 + p_1 + p_2 = 1$. Now suppose we repeatedly choose an element randomly from this ...
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33 views

Rational analogue of expansion to base b

As is well known, we can expand every positive integer $n$ to a base $b \in \Bbb N$ in the form $$n = \sum_i a_ib^i ,\ \ \ 0\leq a_i \leq b_i-1$$ uniquely. Less well known is that we can do this for ...
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23 views

Non-CM totally imaginary number fields

Is there a name for the totally imaginary number fields that are not CM-fields? Any important subclass of number fields with that property, or perhaps a reference where those field are studied in ...
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40 views

Can this combinatoric sum be simplified?

Base cases: $F(n,k,d) = 0$ if $d=0$ and $n>0$ $F(n,k,d) = 1$ if $n=0$ Expression: $$F(n,k,d) = \sum_{s=0}^{\min(k,n)}\binom{n}{s}F(n-s,k,d-1)$$ I am trying to compute the value of $F(n,k,10)$ ...
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29 views

Periodic tribonacci-like sequence

How to prove that if $a_n =[(t_{n-3} + 2t_{n-2} + t_{n-1}) a_{1} + (t_{n-3} + t_{n-2} + 2t_{n-1})] \quad (\text{mod}10)$ and $a_{1}, a_{2}, a_{3}$ are consecutive numbers and $t_{1}=0, t_{2}=1$ and ...
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63 views

Integer $2n^2+2$ as the sum of 2,3,4, and 5 squares

If $n-1$ and $n+1$ are both primes, establish that the integer $2n^2+2$ can be represented as the sum of 2, 3, 4, and 5 squares. I managed to solve 2 and 4 squares, since: $$2n^2+2 = ...
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Stirling numbers of the second kind, general formula

I found out about the Stirling numbers (first and second) when I studied a way to smooth the factorial function. This is the way I want to define them. Approach the factorial $(1+n)!$ as a product and ...
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72 views

If $(a_n)$ is increasing and $a_n^{1/c^n}\to\infty$, then $\sum\frac1{a_n}$ is irrational?

I $\DeclareMathOperator\lcm{lcm}$am trying to generalise the result from this question: If $(a_n)$ is increasing and $\lim_{n \to \infty} a_n^{1/2^n} = \infty$, show that $\sum_{k=1}^{\infty} 1/{a_n}$ ...
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Modified arithmetical functions from a modified Möbius function

In this post when $n>1$ we assume that $n=\prod_{k=1}^{\omega(n)}p_{n}^{e_{k}}$ its factorization in prime powers, where $\omega (n)$ is the number of distinct primes. It is well known the ...
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26 views

Calculating $n$-th $q:P(q)=p \in \Bbb P$

Let $P(x)$ denote the number of ways of writing an integer $x$ as a sum of positive integers (where permutation of the array of integers in the sum doesn't count). Ex: $P(1)=1, P(2)=2,P(4)=5$. Let ...
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If $p\mid 2^n\pm1$ with $p$ and $n$ relatively prime, then $p$ is a Wieferich prime iff $p^2$ also divides $2^n\pm 1$

The Wolfram Mathworld article on Wieferich primes states: $2^{p-1}-1\equiv 0 \mod p.$ If the first case of Fermat's last theorem is false for exponent $p$, then $p$ must be a Wieferich prime ...
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35 views

Properties of a Semi-modulo! operation

Let $A$ be an integer with its representation in base $p$ ($p$ may be prime number but not necessarily) described as: $$A=(a_ma_{m-1}\ldots a_1a_0)_{p}$$ We know $A\equiv (a_n\ldots a_1a_0)_{p}\pmod ...
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Need my CRT work spot-checked

So I have a bunch of equations that look like this: $$k + tx \equiv a \bmod m$$ Where $t$ is the common variable I am solving for among the equations (each equation may have different values for ...
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86 views

Are logarithms of prime numbers algebraically independent?

From Baker's theorem it follows that a linear combination of natural logarithms of prime numbers with non-zero algebraic coefficients can never be zero. Has it been proved that the set of all natural ...
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Does $F\otimes G\in\mathcal{M}$?

Let $\mathcal{M}$ be the class of automorphic L-functions which belong to the Selberg class. Let $F$ and $G$ be elements of this class, and define $F\otimes G$ by $a_{p}(F\otimes G)=a_{p}(F).a_{p}(G)$ ...
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How many composite pairs $(6n-1, 6n+1)$ in the range $[5, 6(1+35t)+1]$ for large $t$

I would like to find out that how many composite pairs $(6n-1,\, 6n+1)$ are their in the range $[5, 6(1+35t)+1]$ for large $t$. Total composite pairs should be a function of t. For example, ...
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35 views

Number of solutions to quadratic congruence

For every positive integer $b$, show that there exists a positive integer $n$ such that the polynomial ${x^2} - 1 \in (\mathbb{Z}/n\mathbb{Z})[x]$ has at least $b$ roots. My efforts Let $n = ...
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Hasse-Weil zeta function of projective hypersurfaces

Assume $f$ is a homogeneous integer polynomial in $n\geq 3$ variables such that the hypersurface $f=0$ is irreducible over $\mathbb{Q}$ (but not necessarily over $\overline{\mathbb{Q}}$ so for example ...
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35 views

Complex integration over a simple pole

A paper I am reading addresses the following integral: $$\int^\infty_{-\infty}\frac{F'}{F}(1+it)h(t)dt$$ where $F$ is a function of $s\in\mathbb C$ with a simple pole at $s=1$, and $h$ is a smooth, ...
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What if the p and q used in keys generation of Pailler cryptosystem are composite?

I've seen a few implementations of Paillier cryptosystem that uses probable primes to choose $p$ and $q$. Assuming that a keypair is generated with $p$ and $q$ that are coprime and that $pq$ is ...
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35 views

How to prove that sums of even powers is divisible by p

For $n\leq (p-2)$ I want to prove that $\sum_{k=0}^{p-1} (r+k)^{n} \equiv 0 \pmod{p}$ It is easy to see that it is true for odd n, since $(-a)^k \equiv -a^k$, and you can just pair up terms since ...
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What is a good site or book for understanding base/radix?

I have a hard time, understanding what base is, how to convert from one to another, and why is the conversion so? And I can't find a site or book that explains all these in detail.
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Possible maps $(\mathbb Z[i]/\mathfrak p)^\times\to\mu_4$

Let $\mathfrak p$ be a maximal ideal of $\mathbb Z[i]$ not dividing 2. Is it true that the only maps from the cyclic group $(\mathbb Z[i]/\mathfrak p)^\times$ the the fourth roots of unity are powers ...
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Why does the number of ways that $n$ can be summed with at least one $1$ equal the partition function for $n-1$?

For some reason I was counting the number of partitions of $n$ that have at least one $1$ as an addend. The beginning sequence for these numbers, starting with $n=1$, is $\{1, 1, 2, 3, 5, 7, 11, 15, ...
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39 views

Modular fractions: $5 \big| 3- \frac 12$

I've read a lot here about how modular fractions are valid as long as the denominator is invertible, but they always cause me trouble understing this part: From the definition of congruence: $$ a ...
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46 views

Erdős' papers on Analytic Number Theőry

My adviser has often mentioned that Paul Erdős' works on Analytic Number Theory contain a myriad of techniques that any number theorist must know. What are some of his papers in Analytic Number Theory ...
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51 views

Is this Riemann Zeta function integral formula known about?

I discovered that $$\zeta(s)=\int_0^1\frac{(-\log(1-x))^{s-1}}{x(s-1)!}dx.$$ Is this an obvious result that is not worth much interest or is this new and unique?
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Fourier coefficients of eigenforms

How does one prove that the fourier coefficients of a normalized eigenform for Hecke operators $T_p$ on $S_k(N)$ all lie in a fixed number field? If the proof is lengthy, a reference to a book that ...