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.

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Solutions of equations of form $\tau(n+a_i)=\tau(x+a_i)$ where $\tau$ is the number of divisors function and $a_i$ is a diverging series

Consider the function $$\tau(n)=\sum_{d|n}1$$ which gives the number of divisors of a number. The question is: How much information does $\tau(n)$ contain about $n$? The answer is obviously: not very ...
xyz1234's user avatar
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3 votes
0 answers
64 views

How many $k$-full numbers are there?

Recall that $n\in\mathbb{N}$ is $k$-full if, for a prime number $p$ : $p\mid n$ implies $p^{k}\mid n$. In the paper I am currently reading it is said that there are $O(B^{1/k})$ $k$-full positive ...
matheg's user avatar
  • 496
0 votes
1 answer
27 views

A test based on Perrin Sequence Pseudoprimes

Let X=x^n (mod n, x^3-x-1). Then a Perrin pseudoprime (PPP) n is a composite number for which trace(X)=0. A symmetric PPP is one for which X=x. Can you construct a non-symmetric PPP for which X^3-X-1=...
Paul Underwood's user avatar
1 vote
2 answers
305 views

Concept confusion over negative and minus sign

$5-3=5-(+3)=2$ $5-3=5+(-3)=2$ The above 2 operations are equal to each other. In the first example, the $''-''$ sign belongs to the subtraction operation. We subtract two positive numbers from each ...
Bilgehan Yılmaz's user avatar
1 vote
0 answers
29 views

Prove that $Enc_{e}(\overline{a}) = \overline{a^e}$ is a one-to-one mapping

Task: Let $n \in \mathbb{N}, n \geq 2, e \in \mathbb{Z}, \left( e, \phi(n) \right) = 1$. Prove that the mapping $Enc_{e}(\overline{a}) = \overline{a^e}$ mutually unambiguously maps (one-to-one mapping)...
Jacobs Monarch's user avatar
8 votes
1 answer
199 views

Is there a sequence, defined in terms of integers and the basic arithmetic operations such that $\lim\limits_{n\rightarrow\infty} s_n = \pi$?

It's a well-known formula that $$ e = \lim\limits_{n\rightarrow\infty} (1+1/n)^n $$ and I'm wondering if there exists a similar formula for $\pi$. All the formulas I know for $\pi$ involve series or ...
Dark Malthorp's user avatar
0 votes
0 answers
60 views

Repitition in the pattern of $\tau(n)$ [closed]

I am interested in the function $\tau(n)=\sum_{d\mid n}1$ which gives the number of positive divisors of $n$. I have the following two questions related to this: Does there exist a $k$ such that $$\...
xyz1234's user avatar
  • 75
0 votes
0 answers
30 views

Investigating Primes Below a Bound That Do Not Divide Sequences Generated by Functions

I am interested in questions about the interaction between prime numbers and sequences generated by specific functions $f: \mathbb{N} \rightarrow \mathbb{N}$. I'm particularly interested in sequences ...
TheTheoremForgettor's user avatar
2 votes
1 answer
157 views

Find all $(a,b)\in\Bbb{N},$ such that $5^a +2^b +8$ is a perfect square.

Find all $(a,b)\in\Bbb{N},$ such that $5^a +2^b +8$ is a perfect square. My approach: Let $5^a +2^b +8=k^2\implies \bigg(k+5^{\frac{a}{2}}\bigg)\bigg(k-5^{\frac{a}{2}}\bigg)=8\bigg(1+2^{b-3}\bigg).$ ...
Vulch's user avatar
  • 81
2 votes
1 answer
40 views

Can we bound the lower and upper asymptotic density of these subsets of $\mathbb N$?

Suppose we partition the powers of $2$ into two sets in the following manner: $$A = \{1\}, \quad B = \{2,4,8,16,\dots\}$$ Now suppose that, for each odd positive integer $t$, we select either $tA$ or $...
tuna's user avatar
  • 547
3 votes
0 answers
58 views

Is galois cohomology invariant under inner forms and not just pure inner forms?

Let $G, G'$ be smooth algebraic groups over $k$ (absolute Galois group $\Gamma$) which are etale inner forms of each other, that is, there exists an isomorphism $G_{k_s} \cong G'_{k_s}$ and the ...
C.D.'s user avatar
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1 vote
1 answer
62 views

Topology in the context of Pontryagin dual

Let $A$ be an abelian group. The definition of the Pontryagin dual of $A$ is $\text{Hom}_{\text{conti}}(A, \mathbb{Q}/\mathbb{Z})$. In this context, what are the topologies on $A$ and $\mathbb{Q}/\...
Pont's user avatar
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1 vote
0 answers
71 views

Cubic non-residue calculation

I am currently studying Cubic residue characters from Kenneth Ireland and Michael Rosen's "A Classical Introduction to Modern Number Theory", and this is the definition given in the book: If ...
Disha's user avatar
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0 answers
11 views

Continuous galois cohomology with twisted profinite integer coefficients

Let $k$ be a field that is finitely-generated over its prime field and $\bar k$ a separable closure. I would like to compute $$ H^1_{cts}(\mathrm{Gal}_k, \hat{\mathbb{Z}}(1)) $$ Since $\mathbb{Z}(1) = ...
Erich's user avatar
  • 225
5 votes
1 answer
105 views

Number of Salem–Spencer subsets of $\{1,2,3,\dots ,n\}$

I was wondering about sets that do not contain any $3$-term AP, and came to know that the official name of such a set is Salem–Spencer set. I was considering the question of counting the number of ...
Sayan Dutta's user avatar
  • 8,641
1 vote
1 answer
62 views

Generalized Pell equation $ x^2 - (k^2-1)y^2 = p $ and solutions recurrence relations

Solving the Pell equation $ x^2 - (k^2-1)y^2 = 1 $, the general solutions for $y$ are generated by the recurrence relation $y_{n+2} = 2k\cdot y_{n+1} - y_n, y_0 = 0, y_1 = 1$ which is the same of the ...
user967210's user avatar
-2 votes
0 answers
92 views

Prove that there is infinite number of primes which are sums of two squares.

Prove that there is infinite number of primes hich are $1$ modulo $4$ that can be represented as sum of to squares of natural numbers. It is well known that every such prime can be writen as sum of ...
nonuser's user avatar
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0 votes
1 answer
37 views

Ramification of ray class field/ ring class field

Suppose $K$ is a number field. Let $q$ be a prime of $K$. Let $ K [q] $ be the ray class field of $K$ of modulus $q$. Someone claims that $K[q]$ is totally ramified at $q$ over the hilbert class field ...
Peter's user avatar
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0 answers
15 views

Galois group of the maximal extention unramified outside a finite set of primes over a number field

In Karl Rubin's book Euler System, he gives a definition of selmer groups as below: Then he states as in lemma 5.3 on page 12 that the selmer group with upper tag is equal to a first cohomology group ...
Peter's user avatar
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3 votes
0 answers
79 views

On thickness of binary polynomials

OEIS A169945 introduces the concept of thickness of a polynomial as the magnitude of the largest coefficient in the expansion of the square of the polynomial. Considering the $2^{n+1}$ polynomials $p(...
Sayan Dutta's user avatar
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0 votes
0 answers
71 views

Eisenstein integers with norm value of 5

I am trying to find out the Eisenstein integers that have norm values of 3,5,7.... I want to see if there is some pattern. For the norm value of 3, I was able to find six Eisenstein integers. For the ...
Disha's user avatar
  • 39
3 votes
1 answer
196 views

On why solutions to $a^4+b^4+c^4+d^4 = (a+b+c+d)^4$ also come in pairs

Jacobi and Madden found infinitely many primitive solutions to, $$a^4+b^4+c^4+d^4 = (a+b+c+d)^4$$ using an elliptic curve. We will use a different approach that, like the method for, $$a^4+b^4+c^4 = d^...
Tito Piezas III's user avatar
-4 votes
0 answers
43 views

Can I read Chapter 7 of Serre's 'A course in Arithmetic' without reading the previous Chapters? [closed]

Assuming known a typical first (graduate) course in Complex analysis and Algebraic number theory, could one read chapter 7 of Serre's book without reading the ones before?
Learnmath's user avatar
0 votes
0 answers
72 views

Difficulty with proving or refuting a claim related to a recursive sequence

I am trying to prove a certain claim about a recursive sequence. The sequence is defined as follows: $$ S_{n}^{k}(a) = \begin{cases} \sqrt{a} & \text{if } n = 1, \quad a \in \mathbb{Z}^{+}, \, a ...
Apollisto's user avatar
0 votes
1 answer
34 views

The number of positive divisors of a number that are not present in another number.

How many positive divisors are there of 30^2024 which are not divisors of 20^2021? I have tried many ways to try to get a pattern for this problem but I can't. I know that 30 has 8 divisors and 20 has ...
Ahmed Amir's user avatar
2 votes
1 answer
208 views

Proving for every $f \in \mathbf{Z}[x]$ (with a few conditions) there will be composite $f(n)$ with only "large" divisors

Let $f(x) = a_k x^k + a_{k-1}x^{k-1} + \cdots + a_1x + a_0 \in \mathbf{Z}[x]$ be irreducible with no common factor among $f(1), f(2), f(3), \cdots$ Suppose, $k \geq 3$, $a_k \geq 1$ or, $k = 2$, $...
Antosha's user avatar
  • 177
0 votes
0 answers
16 views

Trying to find a solution to a problem related to the recursive sequence $S_{n}^{k}(a)$

I'm working on a problem related to a recursive sequence, and I'm having some trouble. The sequence $S_{n}^{k}(a)$ is defined as follows: $$S_{n}^{k}(a) = \begin{cases} \sqrt{a} & \text{if } n = ...
Apollisto's user avatar
-1 votes
1 answer
68 views

Non negative integral solutions when coefficients are involved and inequality among parameters

What is the number of non negative integral solutions of $5w+3x+y+z=100$, such that $w≤x≤y≤z$. I have the answer key to this but I do not understand how to even start. The second inequality condition ...
A shubh's user avatar
  • 83
0 votes
0 answers
42 views

A sum involving the floor function

Consider a rational number $q=\frac{m}{2k}$ where $m$ and $2q$ are coprime. Is there a close formula for the sum $\sum_{h=1}^{k-1}\lfloor \frac{hm}{2k}\rfloor$? I know that there is a formula for the ...
Flying pencil's user avatar
0 votes
1 answer
75 views

Elementary proof that If $n+1$ is prime in $\mathbb{Z}$, then $1+\sqrt{-n}$ is prime in $\mathbb{Z}[\sqrt{-n}]$?

If $n+1$ is prime in $\mathbb{Z}$, then $1+\sqrt{-n}$ is prime in $\mathbb{Z}[\sqrt{-n}]$. I believe this statement is true based on the answer here: https://math.stackexchange.com/a/3610560/ However, ...
RitterSport's user avatar
  • 1,513
0 votes
1 answer
118 views

Find all positive integers satisfying $n \mid 5^n+1 $

Find all positive integer satisfying $5^n+1 \vdots n$ Here is my attempt: Introducing $p$ as the smallest prime divisor of $n$. Let $ord_p(5)=h \in \mathbb{N^*}$ Therefore $h \mid 2n; h \mid p-1$ on ...
khanh20's user avatar
  • 27
0 votes
0 answers
57 views

How to make arithmetic function continuous?

Suppose that we have an arithmetic function $f(x)$ defined as follows: What are the methods in the literature that will make this function continuous and differentiable? However, it should be noted ...
Severus' Constant's user avatar
1 vote
0 answers
29 views

Prove the p-adic valuation of n! equals a floor function sum

I've seen related questions, but none quite like this one. For a rational number $r$ let $[r]$ be the largest integer less than or equal to $r$, e.g., $[\frac{1}{2}] = 0, [2] = 2$, and $[3\frac{1}{3}] ...
maxphi's user avatar
  • 11
0 votes
0 answers
26 views

Seeming contradiction between Genus theory on quadratic fields, and computations of sizes of class groups for those quadratic fields

The Fundamental Theorem on Genera of Quadratic Fields (Satz 100 in Hilbert's Zahlbericht) says as follows: "an arbitrary set of $r$ units $\pm 1$ is the character set of a genus of the field $\...
pedroelpanda's user avatar
0 votes
1 answer
57 views

Profinite completion of local Mordell-Weil group

Let $K$ be a finite extension of $\Bbb{Q}_p$. Let $E/K$ be an elliptic curve over $K$. Let $E(K)$ be the Mordell-Weil group of $E/K$. Let $\widehat{E(K)}$ be the profinite completion of $E(K)$, that ...
Pont's user avatar
  • 5,696
1 vote
2 answers
101 views

How to check if something is perfect square or not.

While finding eigenvalues of a particular matrix, I end up with the following: $\sqrt{p^{2\alpha}-4p^{\alpha}+8p^{\alpha-1}+4}$, where $p$ is an odd prime and $\alpha \geq 1$. The next step is to ...
Akhil P's user avatar
  • 11
-1 votes
0 answers
45 views

Show sin(1°) is an algebraic number. [closed]

I am reading Herstein's Topics in Algebra where I found this exercise. The second part of the question is to show sin(m°) is algebraic for every integer m
Khalid Muzaffar's user avatar
0 votes
0 answers
43 views

Divisibility of Graham Norton's Number (Graham's number + 1) [closed]

The British TV show 'QI' (Series G, Episode 6, 'Genius') mentioned 'Graham Norton's Number', which is Graham's number + 1. This number clearly has a final digit of 8. What factors, other than 2 would ...
Geddes's user avatar
  • 109
3 votes
1 answer
39 views

$\Gamma(N)$ -inequivalent cusps clarification

We know that $\Gamma(N)$ has at most $[\Gamma(1):\Gamma(N)]$ inequivalent cusps given possibly by $g_i \infty$ where the $g_i$ are coset representatives of the subgroup $\Gamma(N)$. Then I don't ...
Meliodas's user avatar
  • 101
2 votes
0 answers
102 views

Integral domain containing non-prime irreducibles (hence not a UFD) where all factorizations into irreducibles are unique

An integral domain is called a UFD if (1) every non-zero non-unit element factors into irreducibles, and (2) every element that factors into irreducibles does so uniquely (up to units and order). It ...
RitterSport's user avatar
  • 1,513
1 vote
1 answer
104 views

A Conjecture Relating Modulo Arithmetic and the Riemann Zeta Function.

I recently created a function that has perplexed many of my fellow amateur mathematicians. It goes something like this: $$f\left(g(x)\right)=\frac{1}{N^{2}}\sum_{n=1}^{N}\left(Ng(x)\operatorname{mod}n\...
Gabriel Turner's user avatar
0 votes
0 answers
72 views

The chunking aspect of repunit prime factors [closed]

While others have already mentioned the divisibility of decimal repunits, $$m := \frac{{10}^n - 1}{9}.$$ ...
RARE Kpop Manifesto's user avatar
1 vote
0 answers
62 views

What is the abelianization of $\operatorname{Aut}(F_2)$?

Let $F_2$ be the free group of rank 2. What is the abelianization of $\operatorname{Aut}(F_2)$? There is a surjection $\operatorname{Aut}(F_2) \rightarrow\operatorname{ GL}_2(\mathbb{Z})$, so we at ...
stupid_question_bot's user avatar
0 votes
1 answer
54 views

Coefficients of a rational function that depend meromorphically on a parameter

Let $D \equiv 1, 2 \, (\textrm{mod }4)$ be a positive, squarefree integer. Let \begin{align} r_D(n) = \{(x, y) \in \mathbf{Z}^2 \mid x^2 + Dy^2 = n\} \end{align} for any positive integer $n$. Consider ...
Joseph Harrison's user avatar
0 votes
0 answers
33 views

Tighten Corollary of Dirichlet's Simultaneous Approximation Bound (d = 3)

For ease of notation, below when I write $|x| \mod N$, I am referring to the absolute value of the integer with smallest absolute value in the same residue class as $x$. For example $|6| \mod 7$ is $1$...
mathmasterzach's user avatar
0 votes
0 answers
28 views

Calculating the number of suspicious pairs of intruders in the hotel [closed]

I have an exercise in the "Mining of Massive Datasets" book, and i need help with solving it. Let's say we believe that "intruders" are operating somewhere, and we want to expose ...
Lauren's user avatar
  • 1
2 votes
0 answers
31 views

Are the semiballs induced by even p-norms the integrands of integral representations of trascendental numbers?

$\frac{π}{2}$ is a trascendental number, given by the integral: $$\frac{π}{2}=\int_{-1}^1\sqrt{1-x^2}dx$$ The integrand is also the curve of a semiball of the form: $$||(x,y)||_2=1$$ Induced by the 2-...
Simón Flavio Ibañez's user avatar
1 vote
1 answer
52 views

Dirichlet series and Euler product

For a multiplicative function $f$, show that we have \begin{equation}\sum_{n=1}^{\infty}\frac{f(n)}{n^s}=\prod_p\left(\sum_{\nu=0}^{\infty}\frac{f(p^{\nu})}{p^{\nu s}}\right).\end{equation} My ...
turkey131's user avatar
4 votes
0 answers
139 views

Probability that one random number among many has a unique prime factor

If I sample $N+1$ integers $x, x_1, \ldots, x_N$ uniformly and independently from $\{1, \ldots, M=2^k\}$, what is the probability that $x$ contains a prime divisor that does not divide any of the $\{...
user432944's user avatar
0 votes
1 answer
54 views

Count lattice points enclosed in curve

Let $$ (\log x_1)^2+ (\log x_2)^2 + \cdot\cdot\cdot + (\log x_n)^2= r $$ For $r=0,1,2,…$ The goal is to count the number of lattice points enclosed by the curve/surface for $r$. In this way the ...
John Zimmerman's user avatar

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