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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For what positive integers k does the following property apply?

Suppose you start at the origin $(0,0)$ and can move in the following ways: You can move horizontally by $k-1$: That is, $(x,y) \to (x\pm (k-1),y)$. You can move vertically by $k$: That is, $(x,y) \...
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-1 votes
1 answer
20 views

Step-by-step method for $p$-adic expansion of $\sqrt{a}$

Let $a>0$. I need to check following: Is $\sqrt{a}$ an element of $\mathbb{Q}_p$? How I can find the $p$-adic expansion for $\pm\sqrt{a}$? How I can distinguish $\sqrt{a}$ and $-\sqrt{a}$? ...
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-2 votes
1 answer
32 views

Solving $(p-n)\bmod\:\left(\sqrt{n}\right)\:=\:0$ for $n$ [closed]

I have the equation $$(p-n)\bmod\:\left(\sqrt{n}\right)\:=\:0$$ where $p$ is known. n is a perfect square. Is there a fast algorithm or method to find solutions to this equation?
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0 votes
0 answers
31 views

A square of rep units represent with summation notation.

Can anyone help me to understand how the LHS is equal to RHS. I dont understand RHS with summation notation. Why the ending limit is 4022. Thank you. $(1+10+10^2+\dots+10^{2011})^2=\displaystyle \...
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  • 9
1 vote
1 answer
107 views

Primes of the form $2n+1$

Let $\sigma(n)$ denotes the divisor function which sums the divisors of $n$, an integer $ \geq 1$. We introduce the function $f$ such that: $$f(n) = 1+(n!)^2-\sigma(n!)(n!)^2+2\sum_{k=1}^{-1+\sigma(n!)...
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-1 votes
0 answers
62 views

Need help, proving by summation by induction hypothesis. How can I prove this formula?

enter image description here After this, How can I prove this by mathematical induction? *thank you for your correction everyone. Is this right? And after that I am looking for mathematical induction?
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-3 votes
0 answers
50 views

Find all integers $a, b, c$ such that $abc$ divides $a^n+b^n+c^n$, with $n$ odd prime [closed]

Find all integers $a, b, c$ such that $abc$ divides $a^n+b^n+c^n$, with $n$ odd prime. It can be assumed that $a,b,c$ are mutually coprime. Note: I'm not interested in the case n = 1.
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  • 1,617
0 votes
0 answers
35 views

Property of the greatest common divisor

Suppose $1\le a\le n$ and $1\le b\le m$ for which $\gcd(a,n)=\gcd(b,m)=1$. Can we produce an upper bound for $\gcd(a,b)$? I was thinking that there must be some way of solving this given the ...
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0 votes
1 answer
62 views

Can anyone provide me List of prime numbers which are sum of the cubes of three natural numbers?

Pierre de Fermat discovered that if $p$ is a prime and congruent to 1 mod 4, then it can be written as the sum of the square of two natural numbers. Similarly, I was trying to find the list of those ...
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-4 votes
0 answers
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Someone to help me prove this number theory question about limits and LCM. Thanks [closed]

Suppose that p and q are integers with 0 < p, 0 < p + q, and the greatest common divisor of p and q is 1. For a positive integer k, let P(k) be the lowest common multiple of p + q, p + 2q, p + ...
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0 answers
27 views

Sum of two squares problem (from a result of theta function)

Let $r_2(n)$ count the number of integer ordered pair $(x,y)\in \mathbb{Z}^2$ such that $n=x^2+y^2$. Then we can show that $r_2(n)=4(d_1(n)-d_3(n))$, where $d_1(n)$ counts the factors of $n$ which are ...
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  • 353
0 votes
1 answer
17 views

How to calculate the number of occurences of $\sigma_1(n)$ (divisor function)?

$\sigma_1(n)$ denotes the divisor function. More precisely is it possible to calculate the number of cases where the sum of the divisors is identical? For example it exists 3 cases where the sum of ...
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-4 votes
0 answers
41 views

Does there need to be a complex part of the value of s for the Riemann Zeta Function for the Riemann Hypothesis?

For example, if I pick s = 1/2 + 0i = 1/2 (This is valid since 0 is a real number). If I plug this into the Riemann Zeta function I get a value of -1.46035450880958681289, which is not a 0 although ...
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0 votes
0 answers
72 views

When is $4a^2+4b^2+4a+4b+1$ a square?

Is it possible to determine when $$4a^2+4b^2+4a+4b+1 = ((2a+2b)+1)^2-8ab = (2a+1)^2+4(b^2+b)$$ is a perfect square, assuming $a,b \in \mathbb{Z}$ and $b>0$? I've tried writing it in a few different ...
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0 answers
57 views

Finding the positive integer of $\frac{a^2+b^2}{ab+1}$ without Vieta-Jumping

Find all positive integer values of $\dfrac{a^2+b^2}{ab+1}$. Vieta-Jumping Solution: \begin{align} &\text{let } \dfrac {a^2+b^2}{ab+1}=k. \\ \ \\ &a^2+b^2=kab+k. \\ &a^2-(kb)a+(b^2-k)=0. \...
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  • 1,202
-1 votes
0 answers
22 views

Hilbert symbol calculation

Is the next statement true? If $p$ is an odd prime, then $$(-1,-1)_p=1,$$ where $(\cdot ,\cdot)_p$ is the Hilbert Symbol. I can't related it with the definition of the Hilbert Symbol in which $$(a,b)...
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0 votes
0 answers
21 views

Does this the variant of asymptotic density satisfy Intermediate value property?

Let $A\subset \mathbb{N}$. Its upper and lower asymptotic densities are defined by $$\overline{d}(A)=\limsup\limits_{n\to \infty}\frac{\#(A\cap \{1,2,\ldots,n\})}{n}$$ and $$\underline{d}(A)=\liminf\...
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  • 361
-1 votes
0 answers
17 views

Bipartite complete graph and degree of an L-function

I just learnt that the number of vertices of the bipartite complete graph $K_{m,n}$ is $m+n$ and its number of edges $mn$, partitioning $K_{m,n}$ into two sets of dots $U$ and $V$ of respective ...
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1 vote
0 answers
26 views

Why are the fowlloing sequences important ? (Bailey)

Here's an excerpt of what Wikipedia says https://en.wikipedia.org/wiki/Bailey_pair A pair of sequences $(α_n,β_n)$ is called a Bailey pair if they are related by $$\beta_n=\sum_{r=0}^n\frac{\alpha_r}{(...
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3 votes
1 answer
67 views

Number of integer solutions of a multivariate polynomial

Given a single-variable polynomial, we all know that the number of its roots is bounded in terms of its degree. A polynomial here is a polynomial with integer coeffecients, and a root of a polynomial ...
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0 votes
0 answers
21 views

Littlewood's theorem on sign changes of $\text{li}-\pi(x)$

Is there any link to the paper " J. E. Littlewood, Sur la distribution des nombres premiers"? I cannot find anywhere on googling. What about the analogous to the theorem for small interval? ...
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2 votes
1 answer
78 views

"3n + 1 problem" variant, using (n/2 + 3)

Background Variant "3n + 1 problem" Consider a mapping similar to the original "3n + 1 problem" (Collatz conjecture): $n → n/2 + 3$ for even n (variant rule due to +3) $n → 3n + 1$...
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0 votes
1 answer
29 views

Can I define topology on Schwartz-Bruhat function space without using direct limit?

Given $F$ a Local field which is also a locally compact abelian group, we donote the Schwartz-Bruhat functions on it by $\mathcal{S}(F)$. Without using direct limit language and after simple ...
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4 votes
1 answer
57 views

Show that $\sum_{n\le x}\max(n)=O(x)$

[An Introduction to Sieve Methods and Their Applications- M Ram Murty, pg.14, Q35,36] Let $\max(n)$ denote the largest exponent appearing in the unique factorization of $n$ into distinct prime powers....
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  • 5,679
1 vote
0 answers
32 views

Symmetric $k-$ tuples of primes in arithmetic progressions

I've been working on this for a long time but Im stuck, I hope someone here can guide me. First I define a symmetric k-tuple as $\mathcal{H}=\{h_1,h_2,...,h_{k-1},h_k\}$ such that $h_1+h_k=h_2+h_{k-1}=...
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6 votes
2 answers
56 views

What is the relationship between these two versions of BSD?

The BSD conjecture is usually formulated like this. If $E/\mathbf{Q}$ is an elliptic curve, then $$ \text{rank }E/\mathbf{Q} = \text{ord}_{s=1} L(E,s), $$ where $L(E,s)$ is the Hasse-Weil $L$-...
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1 vote
1 answer
61 views

Is this generalization of prime gaps also bounded?

I know that it has been proven that prime gaps are bounded. Meaning, no matter how far you go along the number line, you will keep finding consecutive primes less than a fixed distance, which I ...
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-3 votes
0 answers
45 views

Functional Equation Concerning Natural Numbers [closed]

How to I find all increasing functions f:N->N such that f(1) = 1, f(2) = 2, f(4) = 4 and f(3n) = f(2n) + f(n) for all n that are natural numbers?
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0 votes
2 answers
68 views

solve in positive integers $\frac{1}a + \frac{1}b + \frac{1}c = \frac{4}5$ [duplicate]

Solve in positive integers $\frac{1}a + \frac{1}b + \frac{1}c = \frac{4}5$ (i.e. find all triples $(a,b,c)$ of positive integers satisfying the equation). The expression is equivalent to $5(bc + ac + ...
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  • 359
0 votes
0 answers
53 views

The divisor function and prime numbers

We introduce the divisor function $\sigma_1(n)$ which sums the divisors of an integer $n$ such that $n > 1$. The question is : what are the conditions required for $n$ satisfying the following ...
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0 votes
1 answer
61 views

Prove (or disprove): $\sum_i^m2^{a_{i}}\neq \sum_j^n2^{b_{j}}$

Prove, $$\sum_i^m2^{a_{i}}\neq \sum_j^n2^{b_{j}},$$ where each $m, {a_{i}}, n, {b_{j}} \leq k$ is non-zero, unless $\text{for each } a_i, \text{there exists a } b_j \text{ such that, }a_i=b_j (\text{ ...
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0 votes
2 answers
81 views

Show that $[(n+1)(2n+1)]^n > 6^n n!$ without using induction

Show that for all n>1 we have that $[(n+1)(2n+1)]^n > 6^n n!$ . n belongs to integers . What i considered was as $(n+1)(2n+1)$ function is increasing . So we have $(n+1)(2n+1)>6 \forall n&...
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6 votes
0 answers
58 views

Sum of all integers up to $x$ with digit sum $t$

If $S(x,t)$ is the sum of all integers up to $x$ whose sum of digits is $t$, is there a way to calculate it? I mean for high arbitrary $x$. For example, $S(120,11) = 29+38+47+56+65+74+83+92+119 = 603$ ...
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-1 votes
0 answers
37 views

Dirichlet generating function of convolution that takes every k-th term

I need to find Dirichlet generating function $Z_k(s)$ defined below by Dirichlet convolution $f*g$, where arithmetic functions $f(n)$ and $g(n)$ are defined by Möbius function $\mu(n)$ and Von ...
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-1 votes
0 answers
45 views

A diophantine equation inspired in a conjecture due to Gica and Luca

In this post I consider the equation $$k\cdot x=y^2+z^2(x^2-2)-2\tag{1}$$ over odd integers $y\geq 1$ and $z\geq 1$, and over integers $k\geq 1$ and Mersenne exponents $x\geq 13$ such that $x^2-2$ is ...
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0 votes
1 answer
70 views

Number Theory Undergraduate Thesis Topics Suggestions [closed]

I am currently looking for Number Theory undergraduate thesis topics. Do you have any suggestions or any websites/resources I could check? I am not a graduate student so I am looking for a relatively ...
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1 vote
0 answers
41 views

Weighted count of Egyptian fraction representations

This question emerged during an activity I ran for some middle school students this week; basically, it's about a way to "count" - with an appropriate kind of weight - the Egyptian fraction ...
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0 votes
3 answers
85 views

$p\equiv 2\pmod 3$ is an odd prime. Prove that there are no integers $x$, $y$ satisfying $p=x^2-xy+y^2$. [duplicate]

$p\equiv 2\pmod 3$ is an odd prime. Prove that there are no integers $x$, $y$ satisfying $p=x^2-xy+y^2$. The textbook says because when $p\equiv 2\pmod 3$ is an odd prime, $\left( \frac{-3}{p} \right) ...
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0 votes
2 answers
47 views

Is there a better way to finish this number theory/algebra problem other than trial and error?

Problem : In a street, all the houses are numbered continuously from $1$ to $1000$. Alice lives at number $6$. The sum of the numbers from $1$ to $5$ is equal to that of the numbers from $7$ to $8$, i....
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0 votes
0 answers
43 views

How to solve $n\alpha\equiv\beta\mod 1$ (to certain precision)?

Suppose that we're given two real numbers $\alpha, \beta \in \mathbb{R}$ to infinite precision. How can we find an/the smallest integer $n \in \mathbb{N}$ such that $$ n\alpha \equiv \beta \mod 1 $$ , ...
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3 votes
0 answers
34 views

Formal group of Abelian Varieties

I am reading Barry Mazur and Tom Weston's note Euler Systems and Arithmetic Geometry (here is the link). I have a question about the following fact in section 2.2 page 87: Let $K$ be a local field ...
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  • 105
-2 votes
1 answer
62 views

an elementary number theory question [closed]

Let $2a \in \mathbb Z ,~ a^2-b^2d\in \mathbb Z$ and $d$ be a square-free integer. (of course , a, b are rationals) How to prove that if $d\equiv 2,3 \pmod 4$, then $a,b \in \mathbb Z$, and if $d\...
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  • 87
0 votes
0 answers
14 views

find all rational solutions to $m^n = n^m$ with $|m| > |n|$ [duplicate]

Find all rational solutions to $m^n = n^m$, where $|m| > |n|$. I know how to solve the problem when $m$ and $n$ are integers, and the easiest method in my opinion is to rewrite the equation as $\...
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  • 359
0 votes
1 answer
56 views

There are infinitely many positive integers $a$ such that none of the integers $a, a+1,a+2$ are oddly powerful.

A positive integer $n$ is called oddly powerful if for every prime $p$ dividing $n$ the max power that divides $n$ must be odd. $i.e.$ if $p^k \mid n$ and $p^{k+1} \not\mid n$, then $k$ is odd. Prove ...
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1 vote
0 answers
42 views

Example of an elliptic curves of rank $r\geq 2$ with finite Tate-Shafarevich group

Is there any known example today of elliptic curves of rank $r\geq 2$ with finite Tate-Shafarevich group ? As far as I have searched the internet, I did not find such a statement. Is someone aware of ...
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3 votes
0 answers
92 views

$f(n) + f(f(n)) = 2n + 3$, find all $f(n)$. [duplicate]

Find all functions $f$ that maps from $\mathbb{N}$ to $\mathbb{N}$, and $$f(n) + f(f(n)) = 2n + 3$$ for all $n \in \mathbb{N}$ Attempt: I find a function that is $f_{1}(n) = n+1$, we can easily check ...
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  • 4,330
1 vote
1 answer
130 views

Euler's totient function and primes

I have found a formula for Euler's totient function but I have no proof. I need help to demonstrate the formula if possible. $\phi$ denotes the Euler's totient function, $a$ denotes a natural number &...
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-2 votes
0 answers
62 views

Is there a closed form for $\;\prod \limits _{d\mid n}d^{\varphi (d)}\;$?

Is there a closed form for this product? $$\prod \limits _{d\mid n}d^{\varphi (d)}$$ I tried to look something online but couldn't find anything.
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3 votes
0 answers
50 views

How do Square Free Integers link to the PNT?

It is a fairly well known fact that $Q(x) = \frac{6}{\pi^2}x + O(\sqrt{x})$ where the function $Q(x)$, denotes the number of square free integers $\le$ x. There is even a wikipedia page that gives a ...
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3 votes
1 answer
44 views

Statement in an Erdos paper about primitive sets

I was reading a paper from Erdos where he proved the following: let $(a_n)$ be a sequence such that if $a_n$ divides $a_m$ then $m=n$, and let $p_n$ be the greatest prime factor of $a_n$ then $\sum_{k=...
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