Questions on more advanced topics of number theory. Consider first if (elementary-number-theory) might be a more appropriate tag before adding this tag.
0
votes
0answers
2 views
Proving that none of these elements 11, 111, 1111, 11111…can be a perfect square
How can i prove that no number in set S
S = {11, 111, 1111, 11111...}
Is a perfect square.
I have absolutly no idea how to tackle this problem i tried rewriting it in powers of 10 but that didn't ...
0
votes
0answers
24 views
A challenging problem on prime uncertainty interval
I have a very challenging problem to solve, seeking for good advice; I have to make an intro in the first part and then comming to the problem.
Theorem:
In an interval between a prime $p$ and its ...
7
votes
4answers
108 views
every integers from 1 to 121 can be written as 5 powers of 3
We have a two-pan balance and have 5 integer weights with which it is possible to weight
exactly all the weights integers from 1 to 121 Kg.The weights can be placed all on a plate but you can also put ...
1
vote
0answers
76 views
Period of decimal for $1/n$, odd part of $n+1$, and primes.
Let $n$ be an odd number is relatively prime to 10,such the period of the decimal expansion of $1/n$ is $n-1$ or a divisor of $n-1$, and let $c$ be the "cycle length of $n$" (defined below). If ...
1
vote
1answer
44 views
Subsets - is it possible or not?
Given $n$ numbers, you can perform the following operation any number of times : Choose any subset of the numbers (possibly empty), none of which are $0$. Decrement the numbers in the subset by $1$, ...
0
votes
3answers
55 views
How to find remainder?
$$a=r\mod (r+1) \ \ \forall r\in\{2,3,4,\dots,9\}$$
Then how do we find $'x'$ if $$a=x\mod 11$$
I get $$2a=9\mod11$$ but that does not help.
Please keep solution simple , i don't now number ...
1
vote
1answer
49 views
Why is the set $S = \{ (x,y,z) \in \mathbb{N}^3 : x^2 + 4yz = p, p \text{ prime} \}$, finite
I am looking at this proof: http://people.mpim-bonn.mpg.de/zagier/files/doi/10.2307/2323918/fulltext.pdf
We are given that this set is finite. But it is not immediately obvious to me why. The rest ...
6
votes
0answers
58 views
Find integer in the form: $\frac{a}{b+c} + \frac{b}{c+a} + \frac{c}{a+b}$
Let $a,b,c \in \mathbb N$ find integer in the form: $$I=\frac{a}{b+c}
+ \frac{b}{c+a} + \frac{c} {a+b}$$
Using Nesbit inequality: $I \ge \frac 32$
I am trying to prove $I \le 2$ to implies ...
12
votes
1answer
204 views
Is $1847^{2013}+2$ really a prime?
The number $1847^{2013}+2$ is a probable prime. Is it really prime?
I started primo, but it seems to slow for this task. I noticed that there is a faster program used to find the primes ...
0
votes
3answers
66 views
Finding the number of integer solutions, why is this wrong?
The question is to find the number of solutions such that $(x, y)$ are integers: $(x-8)(x-10)=2^y$. Here's what I did: $u(u-2)=2^y$. From the quadratic formula, $u=1+\sqrt{1+2^y}$. This is where I ...
9
votes
1answer
419 views
Can an odd perfect number be divisible by $825$?
I know that an odd perfect number cannot be divisible by $105$. I wonder if that's also the case for $825$.
5
votes
0answers
151 views
+50
Can an odd perfect number be divisible by $165$?
I know that an odd perfect number cannot be divisible by $105$ or $825$. I wonder if that's also the case for $165$ (this is actually a stronger statement).
1
vote
2answers
99 views
Möbius function help
Given some large random integer k, how much longer would it take to determine the primality of k, then to calculate mobius(k), and how much longer would it take to factor k, then to calculate ...
3
votes
1answer
186 views
Odd perfect number divisors
I have a tough one today.
Show that if $n$ is an odd perfect number, then not all of $3$, $5$, and $7$ are divisors of $n$.
Any and all help is appreciated. Thanks very much.
4
votes
3answers
140 views
Sums of powers being powers of the sum
I'm looking for literature on solving problems of the form
$$
n_1^\alpha+\cdots+n_k^\alpha=(n_1+\cdots+n_k)^\beta
$$
for positive integers $n_1,\ldots,n_k$ and fixed parameters $k$ and ...
4
votes
1answer
64 views
Distribution of Digit Products
A digit product $P(n)$ of a natural number $n$ is given by the product of its decimal digits. For example:
$$P(1234) = 24,\;\;\; P(24) = 8,\;\;\; P(8) = 8$$
$$1\times2\times3\times4 = 24, \;\;\; ...
1
vote
1answer
30 views
Checking if a solution exists across two inequalities
If I have
$ay < x < by$
$cx < y < dx$
With $a,b,c,d$ as known (they are real-valued, can be positive or negative or 0) and $x,y$ unknown, is there a methodical way to see if a solution ...
4
votes
2answers
51 views
What does an discriminant of an algebraic number field mean intuitively?
If $E/F$ is a finite extension of fields and $\alpha_1,\ldots, \alpha_n$ is a basis
of $E/F$, the discriminant of $\{\alpha_1,\ldots, \alpha_n\}$ is $$\det(\operatorname{Tr}_{E/F}(\alpha_i\alpha_j))$$
...
1
vote
3answers
35 views
Show that $\left( \frac{q}{p} \right) \equiv q^{(p-1) / 2} \mod p$, where $\left( \frac{q}{p} \right)$ is the Legendre Symbol
Show that if $p$ is any odd prime then
$$\left( \frac{q}{p} \right) \equiv q^{\frac{p-1}{2}} \mod p.$$
stating any theory that you use. In particular, you may assume the existence of a ...
2
votes
1answer
37 views
Identify all $a \mod 35$ such that $35$ is a pseudoprime to base $a$
Identify all $a \mod 35$ such that $35$ is a pseudoprime to base $a$.
By defintion, $\gcd(a,n) = 1$ for $a,n \in \mathbb{Z}$, then $a^{n-1} \equiv 1 \mod n$ means that $n$ is a pseudoprime to ...
0
votes
1answer
21 views
Shortcut in calculating examples of elements of a given order?
My question is:
Find all possible orders of elements of the group of units $G_{31}$. Give an example of an elememt of each possible order.
I did the question, but I felt I did it a long way. As ...
4
votes
1answer
61 views
Family of elliptic curves with trivial torsion
I'm wondering, if it is true that the torsion subgroup of $y^2=x^3+p$ (for $p$ some prime, greater than 2), is always trivial?.
I was trying to prove this using Lutz-Nagell, but I can't quite get it.
...
2
votes
0answers
61 views
+50
The generating function for Bernoulli polynomials
The generating function for Bernoulli polynomials is given by:
$$\frac{ue^{ux}}{e^u-1}=\sum_{n\geq 0}B_n(x)\frac{u^n}{n!}$$
Now, I have the following expression:
...
3
votes
2answers
49 views
Classification of nonzero prime ideals of $\mathbb{Z}[i]$
I know the classification of Gaussian primes: let $u$ be a unit of $\mathbb{Z}[i]$. Then the following are all Gaussian primes:
1) $u(1+i)$
2) $u(a+ib)$ where $a^2+b^2=p$ for some prime number p ...
1
vote
2answers
66 views
If $x$ and $y$ are in this given sequence, can $2^x+2^y+1$ be prime?
The sequence:
$3, 11, 13, 17, 19, 29, 37, 41, 53, 59, 61, 67, 83, 97, 101, 107, 113, 131, 137, 139, 149, 163, 173, 179, 181, 193, 197, 211, 227, 257, 269, 281, 293, 313, 317, 347, 349, 353, 373, 379, ...
2
votes
2answers
30 views
If $17 \mid \frac{n^m - 1}{n-1}$ find the values of $n$ where $m$ is even but not divisible by $4$
Let $m, n \in \mathbb{Z}_+$ with $n > 2$, and let $\frac{n^m-1}{n-1}$ be divisible by $17$. Show that either $m$ is even:$ m \equiv 0 \mod 17$ and $n \equiv 1 \mod 17$. Find all possible values ...
1
vote
1answer
38 views
Routine question about derivatives of automorphic forms being L^2
I consider Automorphic forms on $G = SL_2 (\mathbb{R})$, which are $\Gamma$-invariant, $K$-finite, $Z(g)$ finite, and of moderate growth. If I have such an automorphic form, which happens to be in ...
2
votes
1answer
38 views
If $y<x$ is there a way to phrase it in terms of $y>$something?
For instance I know $y<x$ implies $-y > -x$ but is there a way to phrase it in terms of $y>$ something (that does not itself contain $y$)?
7
votes
4answers
88 views
Find what values of $n$ give $\varphi(n) = 10$
For what values of $n$ do we get $\varphi(n) = 10$?
Here, $\varphi$ is the Euler Totient Function. Now, just by looking at it, I can see that this happens when $n = 11$. Also, my friend told me ...
1
vote
4answers
80 views
Help me prove $\left(\int_{0}^{\infty} t^{50} e^{-t} \,\mathrm dt \right)^{1/2}$ isn't a perfect square
I need some help in proving
$$\left( \displaystyle\int_{0}^{\infty} t^{50} e^{-t} \,\mathrm dt \right)^{1/2}$$
isn't a perfect square. The only way I can think is repeated integration by parts ...
1
vote
1answer
74 views
Pythagorean triples and the unit circle
use the lines through the point $(1, 1)$ to describe all the rational points on the circle $x^2+y^2=3$. Why isn't this possible?
1
vote
1answer
42 views
Prove that if $n$ is sufficently large, there is a prime gap $G(p_k, p_{k+1})$ with $p_k \leq n$ and $p_{k+1} - p_k > \frac{1}{2}\ln(n)$
Let $p_k ( k \geq 1)$ be an enumeration of all the positive primes with $p_1 = 2$ and $p_k < p_{k+1}$ for all $k \geq 1$. Prove that if $n$ is sufficiently large, then there is a prime gap ...
-2
votes
0answers
46 views
Generating function for integers
What is the generating function for the numbers of partitions of an
integer in which each part is either even or divisible by 3?
1
vote
2answers
34 views
general solution of equation and relation
I am interested in learning the below question in some elementary way. Please discuss this problem and help me to get mind free state.
How to get solutions for $x^2 - 10y^2$ = $1$? I would like to ...
6
votes
3answers
56 views
Tensor product of a number field $K$ and the $p$-adic integers
In a paper by J. Jones and D. Roberts (http://math.la.asu.edu/~jj/localfields/database.pdf) we are introduced to an isomorphism $K \otimes \mathbb{Q}_p \cong \prod\limits_{i=1}^g K_{p,i}$, where $K$ ...
2
votes
1answer
25 views
How to calculate $\sum_{i= 0}^{k-1}\left \{ \frac{ai+b}{k} \right \}$
How to calculate: $$F=\sum_{i= 0}^{k-1}\left \{ \frac{ai+b}{k} \right \}$$
where $a,b \in \mathbb Z$ and $(a,k)=1$; $\left \{ \frac{ai+b}{k} \right \} =$ fraction part of $\frac{ai+b}k$
Such problem ...
2
votes
2answers
54 views
Counting integer solutions to $x^2 + y ^2 < n$
How can I count the number of integer solutions, $\mu(n)$, of $x^2+y^2 < n$, and then hopefully look at behavior as $n \to +\infty$?
3
votes
1answer
52 views
$\mathbb{Z}[i]$ is a Dedekind domain
I know that $\mathbb{Z}[i]$ is a PID, and that every PID is a Dedekind. But I want to show that $\mathbb{Z}[i]$ is a Dedekind, without using PID.
One strategy coul be to show that ...
1
vote
2answers
91 views
Is every prime number less than twice the previous prime number?
And if so, how do you prove it? (for example 7 is less than 2 times 5, 11 less than 2 times 7, and so on).
1
vote
3answers
38 views
Inverse | Modulo | Power
Describe the inverse of $5$ modulo $18$ as a positive power of $5\pmod{18}$.
I've got that the inverse of $5$ is $11$, but is this question asking to find a $t$ such that
$$ 11=5^t\pmod{18}?$$
2
votes
1answer
50 views
Showing an elliptic curve has infinitely many points over $\mathbb{Z}_p$
I stumbled upon this question, and I can't think of how to do it, or what kind of results to use. The question is as follows:
Let $$y^2=x^3+ax+b$$ be an elliptic curve ($a,b$ integers), and let $p ...
7
votes
3answers
633 views
Proving a statement regarding a Diophantine equation
FINAL EDIT : Prove that if $p^z|n^2-1$
$$p^{x-z}(p^{z}-1)=\dfrac{ n^2-1}{p^z}-3$$ doesn't hold for any chosen values of $p,x,n$ and $z$.
Here $p>3$ is an odd prime , $x=2y+z, \ ...
0
votes
0answers
34 views
Which theorem addresses relation between prime p and order of group {p^i mod q}
Imagine, the sequence p^i, i is natural number. This sequence will start repeating as we increase i modulo q (e.g the units digit start repeating in a power sequence base 10). Lets call the period of ...
3
votes
4answers
106 views
Proof by induction; $a^n$ divides $b^n$ implies $a$ divides $b$
I want to prove by induction that $a^n \mid b^n$ implies that $a \mid b$ holds for all integers $n\geq 1$.
clearly for $n=1$ this is true, since if $a \mid b$, then $a \mid b$.
Suppose this is true ...
5
votes
1answer
170 views
Efficiently determining if a discrete log exists
Finding a discrete log in a finite cyclic group, like $(Z_N)^x$, seems hard and in some cases a solution may not even exist. Consider $N=15$ and the generator function $2^k=m \bmod 15$. This will ...
1
vote
2answers
73 views
$2^{q-1}\equiv 1\pmod{q}.$
The question is asking to show that $q$ must be prime given
$$ 2^{q-1}\equiv 1\pmod{q}. $$
0
votes
1answer
32 views
Which numbers of [0,1) have a unique base g expansion?
Good evening,
i know that is question is rather standard, but unfornunately I have not much knowledge of number theory.
Take $2 \leq g\in \mathbb{N}$. I know that every $x \in [0,1)$ can be ...
5
votes
0answers
88 views
Sum of rational numbers given some properties
Let $R(n)$ denote the sum of all positive rational numbers whose numerators and denominators are less than or equal to $n$ and have no common factors. I have estimated this sum to be
$$
\begin{align*}
...
7
votes
0answers
123 views
Numbers of the form $(p_{1}^{\alpha_{_{_1}}})^{2}+(p_{2}^{\alpha_{_{_2}}})^{2}+\cdots+(p_{n}^{\alpha_{_{_n}}})^{2}=(p_{m}^{\alpha_{_{_m}}})^{2}$
I'm looking for numbers of the form
$$(p_{1}^{\alpha_{_{_1}}})^{2}+(p_{2}^{\alpha_{_{_2}}})^{2}+\cdots+(p_{n}^{\alpha_{_{_n}}})^{2}=(p_{m}^{\alpha_{_{_m}}})^{2}$$
where $p_{i}$ are prime numbers, ...
6
votes
1answer
71 views
Proof of $\lim_{s\to 1^+}\frac{\sum_p p^{-s}}{\ln(s-1)}=-1$
I am looking for a proof of $$\lim_{s\to 1^+}\frac{\sum_p p^{-s}}{\ln(s-1)}=-1.$$
If the proof is too long, a direct reference is fine. Here the sum $\sum_p$ denotes the sum over all prime numbers.
...
