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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2
votes
0answers
42 views

Does knowing $g, g^r, g^{r^2}, g^{r^3}, \dotsc$ sometimes offer a significant advantage in finding $r$?

Is there a cyclic group $\mathcal G$ with generator $g$ for which the discrete log problem is assumed to be hard, but knowing $g, g^r, g^{r^2}, g^{r^3}, \dotsc$ for random $r$ makes finding $r$ easy? ...
0
votes
0answers
27 views

Solve $x^n\equiv1 \pmod p$ for $x$ where $n$ is odd, $p$ prime [duplicate]

The solution to $x^3\equiv1 \pmod p$ has been discussed in Solve $x^3 \equiv 1 \pmod p$ for $x$ and explained elegantly by Arturo Magidin. The discussion established the form of $p$ and $x$. What ...
2
votes
1answer
45 views

Expected value when die is rolled $N$ times

Suppose we have a die with $K$ faces with numbers from 1 to $K$ written on it, and integers $L$ and $F$ ($0 < L \leq K$). We roll it $N$ times. Let $a_i$ be the number of times (out of the $N$ ...
0
votes
1answer
22 views

Find the smallest prime number that doesn't divide any 5-digit number whose digits are in strictly increasing order.

Find the smallest prime number that doesn't divide any 5-digit number whose digits are in strictly increasing order. I have posted an answer of my own below; any alternative solutions will also be ...
2
votes
1answer
28 views

Does the $5x + 1$ sequence for 7 reach a power of 2 or does it get stuck in a period?

This is much like $3x + 1$, except that if $x$ is odd, you do $5x + 1$. If $x = 7$, then we have 7, 36, 18, 9, 46, 23, 116, 58, 29, 146, 73, 366, 183, 916, 458, 229, ... I've iterated this twenty ...
2
votes
0answers
34 views

How could one invert this sum of Stirling numbers?

In this paper, under Stirling Numbers and their Asymptotics, the author takes equation (3.1): ...
-3
votes
2answers
54 views

Find all integer numbers $n$ such that $\frac{11n-5}{n+4}$ is a perfect square.

Find all integer numbers $n$, such that, $$\sqrt{\frac{11n-5}{n+4}}\in \mathbb{N}$$ I really tried but I couldn't guys, help please.
1
vote
2answers
34 views

The number of numbers lying between 1 and 200 which are divisible by either of 2 , 3 or 5?

The number of numbers lying between 1 and 200 which are divisible by either of two , three or five?
1
vote
3answers
155 views

Find the value of $x$ such that $\sqrt{4+\sqrt{4-\sqrt{4+\sqrt{4-x}}}}=x$

Find the value of $x$, $$\sqrt{4+\sqrt{4-\sqrt{4+\sqrt{4-x}}}}=x$$ Help guys please, I have tried and I got, $x=-2, x=1$, and I think it's wrong
2
votes
0answers
78 views

Does the following conjecture regarding the Riemann Zeta function hold?

Background We define the following numbers: $\phi_i$ is the i'th number such that it cannot be expressed as $\beta^n $ where $\beta $ and $n$ is an integer greater than 1. We define the following ...
0
votes
0answers
17 views

Number Theoretic Sum of three variables. Having trouble isolating two of them.

I encountered the following sum: $$\frac{c_{mn-p}}{c_0} = \sum_{d|p}\ \frac{a_{n-d}b_{m-\frac{p}{d}}}{a_0b_0} $$ Where $$a_i=0 \text{ when } i<0 \text{ and } b_j=0\text{ when } j<0\text{ and ...
5
votes
0answers
58 views

What primes were “pending” at the time of Wiles's proof of FLT?

I would like to know what instances of Fermat's Last Theorem were pending at the time of Wiles's proof. More specifically: what families of irregular primes had been discarded as possible ...
0
votes
0answers
21 views

Rabin's cryptography - when the message $M$ isn't coprime to $n = pq$

Say the message $M$ is a product of one of the primes $p$ or $q$, won't the $gcd$ of $M$ and $n$ (the public encryption key) give me $p$ or $q$? say $p = 11$ $q=19$ $n=11*19=209$ and $M=33$. ...
-2
votes
1answer
25 views

Concept of combinations and number theory.

The number of six digit numbers of the form ababab(in base ten)each of which is a product of exactly six distinct primes?
3
votes
0answers
44 views

Approximate a large number with perfect powers

I'm dealing with number theory now and I have an interesting question. Every number can be approximated with two perfect powers, where perfect power is a number in form $$a^b$$$$a,b \geq 2, a,b \in ...
-1
votes
0answers
23 views

Diophantine Equation with gcd. [duplicate]

Find all positive integers $a,b$ such that $\gcd(3^a+1,3^b+1)$ is a multiple of $ab$. I've given this problem many attempts but I can't seem to make any progress, there doesn't appear to be any way ...
-3
votes
1answer
46 views

solve $(x-3)^2 + (x+1)^2 + (4x-5)^2=0$

solve $(x-3)^2 + (x+1)^2 + (4x-5)^2=0$ this is what I have tried $$(x-3)^2=(x+1)^2=(4x-5)^2=0$$ $$x=3, x=-1, x=\frac{5}{4}$$
-2
votes
0answers
50 views

How to compute $\mathbb Z_n \times \mathbb Z^*_m$? [on hold]

How to compute $\mathbb Z_n \times\mathbb Z_m^*$? (Here $\mathbb Z^*_m$ is the unit group mod $m$ and $(m,n)=1$.) In the paper Multiplicative properties of sets of residues it is said that by ...
1
vote
1answer
58 views

What the difference between the smallest two numbers from these numbers?

There are infinitely many integers $n$ bigger than $1$, such that if we divide $n$ by any integer $k$ where $2\leq k\leq 11$, the remainder is equal to $1$. What the difference between the smallest ...
2
votes
1answer
42 views

Determining parity of the multiplicative inverse?

Let $\mathbb{F}_p$ be a finite field of characteristic $p > 2$, for a fixed $p$. I will consider only prime fields, not $GF(p^n)$. Represent the $p$ elements of the field as integers $\{0,1,\ldots ...
6
votes
1answer
39 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, ...
6
votes
0answers
48 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}) = ...
1
vote
1answer
14 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 ...
0
votes
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 ...
-1
votes
0answers
30 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 ...
3
votes
1answer
25 views
3
votes
2answers
111 views

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

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?
2
votes
3answers
82 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 ...
5
votes
2answers
72 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 ...
5
votes
1answer
69 views
+100

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 ...
1
vote
1answer
62 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
votes
1answer
44 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
votes
1answer
47 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 ...
4
votes
4answers
77 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
votes
1answer
62 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: $$a^2 +b^2 + c^2 \equiv 1\pmod 3$$ ...
1
vote
1answer
46 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; ...
1
vote
1answer
29 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 ...
0
votes
0answers
18 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 ...
1
vote
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.
7
votes
2answers
75 views
+50

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 ...
1
vote
0answers
41 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
vote
1answer
61 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 ...
0
votes
2answers
40 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
votes
1answer
87 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 ...
0
votes
0answers
41 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} ...
0
votes
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, ...
1
vote
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, ...
2
votes
1answer
145 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
votes
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} ...
2
votes
3answers
77 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 ...