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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Determining parity of the multiplicative inverse?

Let $\mathbb{F}_p$ be a finite field of characteristic $p > 2$. Now, lets divide the elements of the field in two groups by their parity: the negative elements are the odd numbers in $\mathbb{F}_p$ ...
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17 views

Does this zeta like series exist?

Background After some inspection of the Riemann Zeta function I tried to make my own series. We define the following series: $$ \alpha(s) = (\ln(2))^s + (\ln(3))^s + (\ln(4))^s + (\ln(5))^s + ... ...
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0answers
15 views

$F[[T]] \times F[[1/T]]$, fundamental domain.

Let $p$ be a prime number. Here is a link which shows how to see that $$(\mathbb{F}_p((T)) \times \mathbb{F}_p((1/T)))/\mathbb{F}_p[T, 1/T]$$is compact using an adelic result. (Here $\mathbb{F}_p[T, ...
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0answers
22 views

What interesting results can be drawn from this expression?

What interesting results can we draw from this expression? $$\zeta(s)=\left(1+\sum_{r>0}\sum_{\{r\}}\prod_{j=1}^t \frac {(-1)^{i_j}}{{i_j}!}\left(\frac {P({r_j}\cdot ...
5
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0answers
33 views

$y^2 = x^3 - 26$, exist ideal satisfying conditions?

For the solution $(x, y) = (3, 1)$ of $y^2 = x^3 - 26$, does there necessarily exist an ideal $I$ of the integer ring $\mathbb{Z}[\sqrt{-26}]$ of $\mathbb{Q}(\sqrt{-26})$ such that $(y + \sqrt{-26}) = ...
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1answer
13 views

Imprimitive Dirichlet Characters

I've started to read the fifth chapter of "Multiplicative Number Theory" by Harold Davenport and I got stuck at some point. Let me elaborate the part that i didn't quite understand. Let $\chi$ be ...
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0answers
29 views

For what $n$ can this sum be an integer? [duplicate]

Consider the well known $\sum_{k=1}^{n} \frac{1}{k}$ sum. My question is simple: How can we choose $n$ in order to make the sum integer? My approach: The first obvious solution is $n=1$. I tried ...
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0answers
24 views

How does Graham knows his number is really the upper bound to the dimension problem?

I know initially he stated that the answer is somewhere between 6 and Graham's number. How does he know that for Graham's number dimensions it is really impossible to color the lines that way? I know ...
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1answer
22 views
3
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2answers
106 views

Showing there is no triplet of positive integers $(a,b,c)$ satisfying $a^7+b^7=7^c$

Show that $$a^7+b^7=7^c$$ has no positive integer solutions $(a,b,c)$. I've posted a general and way too long approach as an answer. How may one prove the claim more briefly and specifically?
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3answers
74 views

Is there always a square between two consecutive cubes?

Is there always a square between two consecutive cubes? I just thought of this question. It seems really simple and the answer is probably yes. Edit: I should have given this more than 2 seconds of ...
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3answers
89 views

Find the smallest natural number $n$

Find the smallest natural number $n$ such that rightmost digit is $6$ and when we deleted that digit $6$ and add it to the left of the number we get $4n$. Example of the operation: $123456$ becomes ...
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2answers
68 views

$L$-function, easiest way to see the following sum?

What is the easiest way to see that$$\sum_{(m, n) \in \mathbb{Z}^2 \setminus \{0, 0\}} (m^2 + n^2)^{-s} = 4\zeta(s)L(s, \chi)?$$Here $\chi$ is the homomorphism $(\mathbb{Z}/4\mathbb{Z})^\times \to ...
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0answers
14 views

Extending 2-adic valuation to real numbers

When proving Monsky's theorem, one of the steps, which, from what I have so far seen, no proof can avoid, is extending the 2-adic valuation to all real numbers, so that it still satisfies ...
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1answer
60 views

Expansion of factorial of a natural number as a summation

Factorial of any natural number $n$ (i.e. $n\in N$) can be expanded as a summation $$n!=1+\sum_{i=1}^{n-1}(i\times i!)$$$$=1+1\times 1!+2\times 2!+3\times 3!+4\times 4!+..............+(n-1)\times ...
2
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1answer
41 views

Fermat's Theorem and primitive $n$th roots of unity

Fermat's little theorem states that if $p$ is a prime number, then for any integer $a$ such that $p \not \mid a$ we have that $a^{p − 1} \equiv 1 \operatorname{mod} p$. Suppose $p =17$, then we know ...
2
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1answer
42 views

How do I calculate these sum-of-sum expressions in terms of the generalized harmonic number?

I know that $$\sum_{m=2}^k\sum_{n=1}^{m-1}(nm)^{-s}=\frac 12((H_k^s)^2-H_k^{(2s)})$$ and $H_k^s=\sum_{n=1}^kn^{-s}$ But, how would I go about finding identities in terms of the harmonic number like ...
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4answers
74 views

what is the greatest integer that divides $p^4-1$ for every prime number p greater than 5

what is the greatest integer that divides $p^4-1$ for every prime number p greater than 5(this is a gre subject math problem) I think that $p^4-1=(p^2+1)(p-1)(p+1)$,so 8 must divide all the $p^4-1$ ...
5
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1answer
59 views

Find all primes $a,b,c$ and integer $k$ satisfying the equation $a^2 + b^2 + 16 c^2 = 9k^2 +1$

This was a problem in this year's Junior Balkan Olympiad. So here's what I did first: If $a,b,c,k$ satisfy the conditions, then they satisfy the congruence: $$a^2 +b^2 + c^2 \equiv 1\pmod 3$$ ...
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1answer
42 views

If Wieferich primes are finite…Then what?

I am wondering if $1093$ and $3511$ are the only Wieferich Primes, then what would it imply? (A wieferich prime is a prime satisfying the congruence $2^{p-1}\equiv 1\ mod \ p^2 $). I know of 3 cases; ...
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1answer
26 views

Identity for $L(s,\chi)L(s,\bar\chi)$

I was told recently that there is an identity roughly of the form $$L(s,\chi)L(s,\bar\chi)=\zeta(s)^2$$ If true, it seems like there should be a short proof of this. Could someone supply a ...
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0answers
16 views

Rational Right Triangle Problem & Elliptic Curves

If we know the sides of a right triangle with rational sides and area=7, from this triangle, how can we get the right triangle with rational sides and area=14? Or the question can be the other way ...
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1answer
21 views

Remainder regarding some identity regarding primes.

How do I show the following identity: $$\sum_{p\ is \ prime} \log (1-1/p^s)=\sum_{n=2}^\infty (\pi(n)-\pi(n-1))\log(1-1/n^s)$$ A hint is best. Thanks.
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1answer
42 views

Quadratic field, $O_K/\mathfrak{p} = \mathbb{F}_p$, $O_K/pO_K$ is a finite field of order $p^2$.

Let $K$ be a quadratic field $\mathbb{Q}(\sqrt{m})$ where $m$ is a square free integer, and let $p$ be a prime number which does not divide $2m$. Where can I find a reference to a proof of the ...
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0answers
39 views

Dirichlet characters with values in a finite field

Although the classical Dirichlet characters are complex valued, it seems to me rather useful that the characters attain values in a finite field; thus homomorphisms from $\mathbb{Z}_N^*$ to ...
1
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1answer
59 views

Infinitude of the primes $p\equiv1 \operatorname{mod} n$

$\textbf{Theorem:}$ Fix $1 < n \in \mathbb{Z}$. There are infinitely many primes $p\equiv1 \operatorname{mod} n$. $\textbf{Proof}$ Recall that the $n$-th cyclotomic polynomial $\Phi_n(x)$ is ...
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2answers
37 views

Can you express the fraction 1/0 using imaginary units in any way possible? [on hold]

Although in basic textbooks, 1/0 is undefined or something along those lines, can it be expressed using complex numbers (i)? One way I propose is to use a function such as 1/x and use that function in ...
7
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1answer
86 views

What is the Galois group of $\mathbb{Q}_p[\zeta] / \mathbb{Q}_p$, where $\zeta$ is a $p^r$th root of unity?

Let $\zeta$ be a $p^r$-th root of unity and $\mathbb{Q}_p$ the p-adic numbers. I would like to know the Galois automorphisms of $\mathbb{Q}_p[\zeta] / \mathbb{Q}_p$. I already know, the degree of ...
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0answers
40 views

$\eta(1) = \ln(2)$ proof using Abel's Theorem

Hi I was just wondering how does one justify $\eta(1) = \ln(2)$. Looking at the power series for $\ln(1+x)$ we have \begin{equation} \ln(1+x)= \sum_{n=1}^{\infty} \frac{(-1)^{n+1}x^{n}}{n} ...
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1answer
27 views

Elementary number theory proofs using functions

The functions $f$ and $g$ are defined by $f(x) =$ remainder when $x^2$ is divided by $7$. $g(x) =$ remainder when $x^2$ is divided by $5$. (a) Show that $f(5)=g(3)$ (b) If $n$ is an integer, ...
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0answers
40 views

Does the Collatz conjecture imply the Well-Ordering Principle?

It seems to me that it does, even trivially so. My reasoning is as follows: Suppose that H is a nonempty set of positive integers. Case 1. 1 is a member of H. Then H obviously has a smallest member, ...
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1answer
136 views

Solve $x^n+z^n=(x+1)^n$ for $n\ge 3$ without FLT

Is there a way to prove that for $x,z,n \in \mathbb{Z}$, $x > 0$, $z > 0$, $n > 2$, the equation $$ x ^ n + z ^ n = (x + 1) ^ n $$ has no solution, without using Fermat's Last Theorem? ...
6
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1answer
67 views

Is $\pi(n)$ a Rational Function?

Are there some two-variable polynomials $P(n,\log n)$ and $Q(n,\log n)$ which we have the bellow equation for prime counting function $\pi(n)$ for $n \in \mathbb{n}$? $$\pi(n) = \Bigl{\lfloor} ...
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1answer
42 views

If $a|(p+1)$ for all but finitely many $p=3 (\text{ mod } 4)$ then $a$ divides $4$

I have the following question: Let $a$ be an integer such that $a$ divides $p+1$ for all but finitely many primes $p=3 \text{ mod } 4$ Can we conclude that $a$ must divide $4$? How we can prove ...
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3answers
75 views

Sum of super exponentiation

$f(x,n)=x^{2^{1}}+x^{2^{2}}+x^{2^{3}}+...+x^{2^{n}}$ Example: $f(2,10)$ mod $1000000007$ = $180974681$ Calculate $\sum_{x=2}^{10^{7}} f(x,10^{18})$ mod $1000000007$. We know that $a^{b^{c}}$ mod ...
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0answers
35 views

what is the sum of these 1230 sequences such as 1,2,1,2,2,1,2,2,2,1,2,2,2,2… A. 2411 B. 2412 C. 2413 D. 2414 [on hold]

what is the sum of these 1230 sequences such as 1,2,1,2,2,1,2,2,2,1,2,2,2,2..... A. 2411 B. 2412 C. 2413 D. 2414
2
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3answers
104 views

Binomial Coefficient Computation by Dividing Consecutive Terms

If I take the binomial coefficient: $$\frac{n!}{k! (n-k)!}$$ and I want to know the result of 10 choose 4 and I proceed to do the computation $$\frac{7*8*9*10}{1*2*3*4}$$ by first dividing and then ...
3
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1answer
42 views

Prime Zeta Function proof help: Why are these expressions not equal?

I was trying to create a formula for the Prime Zeta function and I partially succeeded except for one frustrating error. I was only able to formulate an approximation. Consider the following sum: ...
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1answer
46 views

How does the Riemann summation connect with prime numbers [on hold]

The Riemann sum goes as the sum of $\dfrac{1}{1^s}+\dfrac{1}{2^s}+\dfrac{1}{3^s}+\ldots$ It has been said that if $\frac{1}{2}+ix=0$ for $s$ ($x$ can be any number), then the sum will have massive ...
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0answers
27 views

Why does the Harmonic series diverge? [duplicate]

I am aware of Oresme's proof of its divergence(http://mathworld.wolfram.com/HarmonicSeries.html), but this proof could be applied to the sum of all natural numbers and it would still be valid. Yet, ...
4
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1answer
56 views

Substitutions that transform Fermat Equations to Elliptic Curves

I was reading Chapter 1 of Elliptic Curves - Number Theory and Cryptography by Lawrence C Washington. He was considering Fermat equations $$a^4+b^4=c^4\text{ and }a^3+b^3=c^3.$$ For the 1st equation, ...
2
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2answers
56 views

Number Theory- Mathematical Proof

I am trying to prove a theorem in my textbook using another theorem. What I need to show that if a,b, and c are positive integers, show that the least positive integer linear combination of a and b ...
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0answers
30 views

Notation for indexing the factorizations of a number?

Background Given any $n \in \mathbb{N}$, the ordered factorization count of $n$ can be computed and is traditionally written $H(n)$. This is, essentially, the number of unique decompositions of $n$ ...
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1answer
63 views

prove that this number contains two equal digits

We delete the first digit from the number $7^{1996}$ and then we add it to the remaining number, repeat this until we get a number consisting of $10$ digits, prove that, this number contains two equal ...
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1answer
15 views

Definition of “Contractive Invariant Plane”

Can someone please explain the definition of a contractive invarient Plane found in: the paper It is nearly at the very beginning of the Introduction. By contractive do they mean a contractive map? ...
6
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1answer
46 views

Cyclotomic polynomials, properties.

Let $F$ be a field of characteristic prime to $n$, and let $F^a$ be an algebraic closure of $F$. Let $\zeta$ be a primitive $n$th root of unity in $F^a$. I know that the monic polynomial $\Phi_n(X)$ ...
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0answers
7 views

Comparing conway chains

See https://en.wikipedia.org/wiki/Conway_chained_arrow_notation for the details how conway chained arrow notation works. I want to calculate the approximate value $n$ such that $$n\rightarrow ...
3
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1answer
94 views

To find positive integers $n$ such that $\dfrac {n(n+1)(n+2)}6$ is a perfect square [duplicate]

How many positive integers $n$ are there such that $\dfrac {n(n+1)(n+2)}6$ is a perfect square ? I know $n=1 , 2$ works ; are there any more ? Are there only finitely many such $n$ ?
6
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1answer
133 views

To solve $n(n+1)(n+2)=6m^3$ in positive integers $m,n$

How to find all positive integers $m,n$ such that $n(n+1)(n+2)=6m^3$ ? I can see that $m=n=1$ is a solution , but is it the only solution ?
2
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0answers
15 views

Show that the equation has a natural solution [duplicate]

let $n$ be a natural number and $r$ , $s$ be rational such that $n=s^2+r^2$ show that there are natural numbers a,b such that $n=a^2+b^2$