Questions on more advanced topics of number theory. Consider first if (elementary-number-theory) might be a more appropriate tag before adding this tag.

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2answers
19 views

“Almost mean” of a set of integer

I have three integers $(a, b, c)$ (but this can be generalized to any size). I want to redistribute the sum $a+b+c$ as equally as possible between three variables. In our case, we have three cases: ...
23
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0answers
158 views

How many non-rational complex numbers $x$ have the property that $x^n$ and $(x+1)^n$ are rational?

In this answer, I proved the following statement: For each positive integer $n$, there are finitely many non-rational complex numbers $x$ such that $x^n$ and $(x+1)^n$ are rational. These complex ...
-1
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0answers
36 views

Sequence with Conditions and Possible Answers

Given that $a_1$, $a_2$, $a_3$, . . . $a_n$ is a sequence of positive real numbers such that: For all positive integers $m$ and $n$, $a_{mn}$ = $a_m$$a_n$, AND there exists a positive real number $B$ ...
2
votes
1answer
51 views

Some questions about prime divisors and number of primes

For an integer $n \ge 2$, let $\omega (n)$ denote the number of distinct prime divisors of $n$ and $\pi (n) $ be number of primes not exceeding $n$. Let $a_1, \ldots, a_k$ be integers greater than ...
1
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1answer
27 views

Question around the following relation: $T(n,1) = n$, for a positive integer $n$, and for all $k\geq 1,\ T(n,k+1)=n^{T(n,k)}$.

I'm beginning the studies on number theory and then i'm facing the following problem that i couldn't solve yet: given a positive integer $n$ and being $T(n,1)=n$ and, for all $k\ge1$, ...
0
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1answer
72 views

On riemann zeta function [on hold]

What is the importance of the zeta functions at even and odd integers to prime numbers? Do these encode properties of primes like the zeros on $-1/2+ai$ line encodes or are they studied for their own ...
2
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2answers
40 views

the solution to $x^2=49+k\cdot12288$

I have computed $x^2=49+k\cdot12288$ for $k=0$ to an arbitrarily large integer and found $x$ has an integer solution only for $k=0$. Can someone proove that for $x^2=49+k\cdot12288$, $x$ has only ...
2
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3answers
74 views

If $x\equiv y \pmod{\gcd(a,b)}$, show that there is a unique $z\pmod{\text{lcm}(a,b)}$ with $z\equiv x\pmod a$ and $z\equiv y\pmod b$

If $x\equiv y \pmod{\gcd(a,b)}$, show that there is a unique $z\pmod{\text{lcm}(a,b)}$ with $z\equiv x\pmod{a}$ and $z\equiv y\pmod{b}$ What I have so far: Let $z \equiv ...
0
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0answers
78 views

Why do we care about representaion of primes?

Im currently trying to figure out the genesis of quadratic reprocity by using Cox and Lemmermeyers books. I also got a copy of some works of Fermat but it is in German. It seems like there is some ...
2
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0answers
19 views

How to prove there exists a positive integer $k,d$ such $\sum_{i=1}^{2k}a_{id}=k-2014$

Let $\{a_{n}\}_{n=1}^{\infty}$ be a non-negative integer such that: for any postive integer $m,n$, we have: $$\sum_{i=1}^{2m}a_{in}\le m.$$ Show that there exist positive integers $k,d$ such that: ...
3
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3answers
197 views

$(1+1/x)(1+1/y)(1+1/z) = 3$ Find all possible integer values of $x$, $y$, $z$ given all of them are positive integers.

Find all possible integer values of $x$, $y$, $z$ given all of them are positive integers and $$(1+1/x)(1+1/y)(1+1/z) = 3.$$ I know $(x+1)(y+1)(z+1) = 3xyz$ which is no big deal. I can't move ...
0
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2answers
17 views

Congruence with binomial

I tried to prove this by induction on $k$. But I did not manage Let $p$ be a prime. For every $k\in\{0,\cdots,p-1\}$, one has $\binom{p-1}k\equiv(-1)^k\pmod p$. By Wilson theorem, it suffices to prove ...
0
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0answers
10 views

Ring structure of tuples mod k

Consider a vector of n integers $$A= a_1, a_2, ... a_n$$ Such that for another vector $$B= b_1,b_2... b_n$$ $$AB^T \equiv 0 \mod k$$ For an integer k. I was playing around with these structures ...
10
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0answers
89 views

How find all postive integer number such $(n+k)\nmid \binom{2n}{n}$

Question: Find the all integer $k$,such there are exist infinitely many $n$ such $$(n+k)\nmid \binom{2n}{n}$$ This is china 2014 (CMO problem 4),it's have been end exam three hours ago. I ...
24
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0answers
173 views
+50

Is There An Injective Cubic Polynomial $\mathbb Z^2 \rightarrow \mathbb Z$?

Earlier, I was curious about whether a polynomial mapping $\mathbb Z^2\rightarrow\mathbb Z$ could be injective, and if so, what the minimum degree of such a polynomial could be. I've managed to ...
1
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1answer
57 views

The ring is a principal ideal domain, especially an integral domain.

The following holds for the ring $ \mathbb{Z}_p, p \in \mathbb{P}$: The ring $ \mathbb{Z}_p $ is a principal ideal domain, especially an integral domain. I try to understand the following proof: ...
1
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1answer
85 views

Set of integer p-adics-Proposition

Proposition: "$\mathbb{Z}_p$ contains only the ideals $0$ and $p^n \mathbb{Z}_p$ for $n \in \mathbb{N}_0$. It holds $\bigcap_{n \in \mathbb{N}_0} p^n \mathbb{Z}_p=0$ and $\mathbb{Z}_p \ ...
19
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1answer
480 views

Find $a,n\in \mathbb N^{+}:a!+\dfrac{n!}{a!}=x^2,x\in \mathbb N$

Find $a,n\in \mathbb N^{+}:a!+\dfrac{n!}{a!}=x^2,x\in \mathbb N.$ I find $\{n,a\}=\{4,1\}\{4,4\}\{5,1\}\{5,5\}\{7,1\}\{7,7\}\{20,11\}.$ (These are all if $n<300$.) ...
0
votes
1answer
54 views

Why is this function an embedding?

We have the canonical function $\epsilon_p: \mathbb{Z} \to \mathbb{Z}_p, x \mapsto (\overline{x})_{k \in \mathbb{N}_0}=(\overline{x}, \overline{x}, \overline{x}, \dots )$. The function $\epsilon_p: ...
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0answers
33 views

Evaluate the following sum using the hyperbola method

This is an exercise from Iwaniec and Kowalski's book Analytic Number Theory: Prove that $$\displaystyle \sum_{n \leq x} \tau(n^2 + 1) = \frac{3}{\pi}x \log x + O(x).$$ The constant $3/\pi$ is quite ...
4
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5answers
65 views

An integer when divided by $5$ and $13$ leaves residues $4$ and $7$ respectively

Find an integer when divided by $5$ and $13$ leaves residues $4$ and $7$ respectively. (Without Modular Arithmetic). I don't know if it is right, but i got this $$n=5x+4=13y+7$$ ...
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votes
1answer
119 views

Outline approach to Collatz 3n+1 conjecture / Criticism needed

// Instead of trying to show there are no loops and no sequences that increase without bounds, consider how any "deviant set of sequences" must partition the natural numbers into two infinitely large ...
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2answers
134 views

Higher dimensional analogues of the argument principle?

I know there are higher dimensional analogues of the argument principle. (See http://en.wikipedia.org/wiki/Variation_of_argument) But I do not have books about it and I cannot find anything of value ...
4
votes
2answers
98 views

Goldbach's conjecture and number of ways in which an even number can be expressed as a sum of two primes

Is there a functon that counts the number of ways in which an even number can be expressed as a sum of two primes?
1
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1answer
52 views

primes of the form $4k+3$ and sums of squares [duplicate]

It is well-known that if $p$ is a prime of the form $4k+3$ and $p|x^2+y^2$ then $p|x$ and $p|y$. I forget what is the name of this result, and where can I find a proof (please provide a link).
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2answers
37 views

Find $x$ as the given $n$th term in the Fibonacci sequence?

With a given $n$ and I am trying to find the value of $x$, as in: $$Fib(x)=n$$ Using the formula for Fibonacci sequence, where $\varphi$ is the Golden Ration ($\approx1.61803399\ldots$) $$Fib(z) = ...
2
votes
3answers
108 views

Proving no rational satisfy $p^2 = 2$

In Rudin's analysis example 1.1, he tried to show the following Let $A$ be the set of all positive rationals $p$ such that $p^2<2$ and let $B$ consist of all positive rationals $p$ such that $p^2 ...
2
votes
3answers
72 views

Find all intergers such that $2n^2+1$ divides $n^3+9n-17$

Find all intergers such that $2n^2+1$ divides $n^3+9n-17$. Answer : $n=(2 \ and \ 5)$ I did it. As $2n^2+1$ divides $n^3+9n-17$, then $2n^2+1 \leq n^3+9n-17 \implies n \geq 2$ So $n =2$ is ...
17
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1answer
303 views

Is there any integer solutions of $3x^3+3x+7=y^3$?

$3x^3+3x+7=y^3$ $x, y \in \mathbb{N}$ Having thought about it two hours, and I'm still not sure how to show there actually aren't any integer solutions. EDIT Another formulation of this problem: ...
4
votes
2answers
234 views

Did I do this proof right?

I am not sure if I did the proof right, so I wanted to see how most of you did this. I am trying to solve this problem: Let $x, y \in \mathbb N$ be relatively prime. If $xy$ is a perfect square, ...
6
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1answer
72 views

Prove that $\{\frac{\phi (n)}{n}\}_{n \in \Bbb N}$ is dense in $[0,1]$

suppose that $\phi(n)$ is Euler function. prove that, $\{\frac{\phi (n)}{n}\}_{n \in \Bbb N}$ is dense in $[0,1]$ (if $A_n=\{1 \leq m \leq n | m \in \Bbb N ; gcd(n,m)=1\}$ then $\phi(n)=|A_n|$) I ...
1
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0answers
36 views

How to prove taht a product of two complete residue system is not a complete residue system?

Claim. Let $n$ be a natural number and $A=\{0,1,2,3,\cdots,n-1\}$ be a complete set of residues modulo $n$. Let $\sigma$ be a permutation of $A$. Show that the set $C=\{\sigma(i)i:i\in A\}$ is not a ...
1
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4answers
61 views

Is there numbers that don't fit in our sets of numbers?

It is said that the first numbers we used were natural numbers like $0$, $1$ ,$2$... in $\mathbb{N}$. Then we discovered negative numbers $-1$,$-2$... , and classified them all in $\mathbb{Z}$. Then ...
1
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2answers
43 views

Chinese remainder theorem other way around

I need to solve the following equation: $x^2\equiv 1 (\textrm{mod }1000)$ According to the chinese remainder theorem I can rewrite this as: $x^2 \equiv 1 (\textrm{mod }8)$ and $x^2 \equiv 1 ...
16
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0answers
334 views
+50

On the problem of polynomial bijection from $\mathbb Q\times\mathbb Q$ to $\mathbb Q$

The question titled "Polynomial bijection from $\mathbb Q\times\mathbb Q$ to $\mathbb Q$" which was posed on MathOverflow attracted quite a lot of attention (and may be the question with most wrong ...
2
votes
1answer
44 views

Find the sum of digits of m?

Let $m$ be the number of numbers from set $\{1,2,3,\dots,2014\}$ which can be expressed as difference of the squares of two non negative integers. The sum of digits of $m$ is... My attempt: I ...
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3answers
69 views

Can $\mathbb{Z}/n\mathbb{Z}$ (not $(\mathbb{Z}/n\mathbb{Z})^{\times}$) be a group under multiplication?

I was wondering why we usually say $\mathbb{Z}/n\mathbb{Z}$ is a group under addition and invent notation like $(\mathbb{Z}/n\mathbb{Z})^\times$ specifically for the multiplicative group modulo $n$. ...
2
votes
1answer
26 views

When is it possible to find a relatively prime pair among $n$ numbers?

Suppose I have a set of $n$ numbers and their gcd is $d$. If I divide every number by $d$, is it possible to find a pair that is relatively prime? Intuitively yes, but how do I prove it? I tried ...
6
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2answers
211 views

Ratio of sum of Euler's totient to $n$: $\lim_{n \to \infty} {\log \left( \sum_{k=2}^n \varphi(k) \right) \over \log(n)}$

This is more a casual/recreational question... It seems to me, that the limit as given in the subject line $$\lim_{n \to \infty} {\log \left( \sum_{k=2}^n \varphi(k) \right) \over \log(n)} = \log_n ...
0
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0answers
27 views

Is there an online calculator in which you can type a number and have it tell you if it could be a Lychrel number or not?

Say you type 7326 into it, it runs a few calculations and tells you it reaches 99099 in three iterations. But if you type in a number like 887, it runs a reasonable number of iterations (say, twenty) ...
17
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0answers
134 views

Irreducibility of cyclotomic polynomials via schemes

A few months ago, someone told me there existed a scheme theoretic proof of the irreducibility of cyclotomic polynomials. I've tried coming up with a proof, and when that didn't really yield anything ...
7
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1answer
130 views

Proving that $T$:$(x_1,…,x_n) \rightarrow (\frac {x_1+x_2}{2},\frac {x_2+x_3}{2},…,\frac {x_n+x_1}{2})$ leads to nonintegral components

Start with $n$ paiwise different integers $x_1,x_2,...,x_n,(n>2)$ and repeat the following step: $T$:$(x_1,...,x_n) \rightarrow (\frac {x_1+x_2}{2},\frac {x_2+x_3}{2},...,\frac {x_n+x_1}{2})$ ...
4
votes
1answer
60 views

Twin Primes between $n$ and $2n$

Is it theoretically possible for there to always be a twin prime pair between $n$ and $2n$ for all sufficiently large $n$ (assuming of course that there are infinitely many twin primes) or would this ...
1
vote
1answer
29 views

Sum of Coefficients and Number of Terms in Trinomials and Quadrinomials

I already know how to find the sum of coefficients in a binomial, but how do you do it for a trinomial/quadrinomial (after like terms are added)? Example Problem: $(wa+xb+yc+zd)^n$ (all variables are ...
2
votes
1answer
63 views

A sufficient condition for $\eta$-quotients to be modular forms

Let $f$ be the form : $$f(\tau)=\prod_{M\mid N}{\eta(M\tau)^{a_M}} \quad (\tau \in \mathcal{H})$$ Generally, we said that $f$ is an $\eta$-quotient when $(a_M)$ is a sequence of integers. One can ...
21
votes
2answers
2k views

Fermat's Last Theorem (Case n = 3) Question

A very simple question. We all know that there are no solutions to $x^3 + y^3 = z^3$ for integer $x$, $y$ and $z$, $xyz\neq 0$, but are rational $x$, $y$ and $z$ possible? Thanks.
8
votes
5answers
192 views

If $(m,n)\in\mathbb Z_+^2$ satisfies $3m^2+m = 4n^2+n$ then $(m-n)$ is a perfect square.

I came across this question on another forum. The question is: $$ \text{If $m,n\in \mathbb{Z}_+$ such that $3m^2+m=4n^2+n$, then $(m-n)$ is a perfect square.}$$ I have managed to partially prove ...
2
votes
2answers
119 views

The intersection points are collinear

It is given a hexagon inscribed in a conic section. I want to prove that the pairs of opposite site intersect at three points of the projective plane that are collinear. How could we do this? ...
1
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3answers
146 views

this is a conjecture or a result? every arithmetic progression contains a sequence of $k$ “consecutive” primes for possibly all natural numbers $k$?

Writing a little better the previous question: is it true that if we let $a$ and $b$ be coprime integers, then the arithmetic progression : $a + bh: h\in {\mathbb Z}$, contains a sequence of $k$ ...
4
votes
1answer
74 views

What motiveted Gauss to formulate his theorem on quadratic reprocity?

Im trying to connect his work on quadratic reciprocity with some simple question, like solution to certain diophantine equation or representing primes. Any ideas? I find it hard to imagine that he out ...