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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55 views

why $\frac{a}{b}\pmod p=\frac{a\pmod p}{b\pmod p}$

It is said this following is theorem? what's this name? and How to prove it? Thanks show that $$\dfrac{a}{b}\pmod p=\dfrac{a\pmod p}{b\pmod p},a,b\in N^{+},(a,p)=1,(b,p)=1$$
3
votes
3answers
194 views

What is the Max value of n when 185! is divided by (189^n) will give an Integer Value?

What is the Max value of n when $185!$ is divided by $(189^n)$ will give an Integer Value? Options are a) $91$ b) $30$ c) $36$ d) $24$ MyApproach: $189$=$3^3$ . $7$ When $185$/$3$=$61$ ...
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0answers
26 views

Is there a tighter approximation for the least prime gap of a given length?

This link https://primes.utm.edu/notes/gaps.html gives a definition of the maximal gaps. For a number $g$ , $p(g)$ is the smallest prime $p$ followed by at least $g$ composites. The estimate is ...
1
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2answers
42 views

$z=100^2-x^2$. Then, how many values of $x,z$ are divisible by $6$?

$z=100^2-x^2$. Then, how many values of $x,z$ are divisible by $6$? My approach: For $x=1$, $z$ is not divisible by $6$. For $x=2$, $z$ is divisible by $6$. For $x=3$, $z$ is not divisible by ...
0
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0answers
57 views

If a, b are positive integers and $(ab - 1) \mid (a² + b²)$ then prove that $q = \frac{a² + b²}{ ab - 1} = 5$. [on hold]

If a, b are positive integers and $(ab - 1) \mid (a² + b²)$ then prove that $q = \frac{a² + b²}{ ab - 1} = 5$. I know it has something to do with number theory
2
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3answers
46 views

Fastest way to perform this multiplication expansion?

Consider a product chain: $$(a_1 + x)(a_2 + x)(a_3 + x)\cdots(a_n + x)$$ Where $x$ is an unknown variable and all $a_i$ terms are known positive integers. Is there an efficient way to expand this?
4
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4answers
71 views

Books on Prime numbers

I am a graduate student and have just finished Burton's book on number theory. Now I want to read further on prime numbers. Does anyone have any suggestion?
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1answer
38 views

LCM of randomly selected integers

What is the expected LCM of 21 randomly selected positive integers under 10000000? How would someone even approach this problem? EDIT: The positive integers are chosen with replacement.
2
votes
2answers
37 views

Can this congruence be simplified?

$$p(p+1) \equiv -q(q+1) \bmod pq$$ Can this be reduced to an easier format?
3
votes
1answer
36 views

Large regions of the plane $(x,y) \in \mathbb{Z}^2$ with no relatively prime points: $\mathrm{gcd}(x,y) > 1$

For any $\ell > 0$ can you find $M, N$ such that $ \boxed{\mathrm{gcd}(x,y) > 1}$ for all $x \in [M, M+\ell]$ and all $y \in [N, N+\ell]$ ? This is related to the statement that the set of ...
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2answers
80 views
+50

Pentagonal Numbers

I recently was passing some time on Project Euler, when I came across this question. It deals with finding Pentagonal Numbers $P_j$ and $P_k$ such that $P_j+P_k$ and $P_j-P_k$ are also pentagonal ...
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votes
1answer
28 views

Is there a number which describes the approaching ratio of twin primes to other primes? Or a formula for the change in density of twin primes?

Could someone shed some light on what we know about the density of twin primes? I find that it seems to be empirically true that the density of prime gaps increases as $\log(x)$ does for any gap. ...
1
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0answers
50 views

Two irrational numbers are congruent iff the tails of their infinite continued fractions eventually coincide

We say that a real number $\alpha$ is $congruent$ to real number $\beta$ if there exist integers a, b, c and d with ad-bc=+1 or -1 and such that $$\alpha=\frac{a\beta +b}{c\beta+d}$$ I need to prove ...
5
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0answers
46 views

Integers which are the sum of non-zero squares

Lagrange's four-square theorem states that every natural number can be written as the sum of four squares, allowing for zeros in the sum (e.g. $6=2^2+1^2+1^2+0^2$). Is there a similar result in which ...
5
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0answers
82 views

Diophantine equation: $13^x+3=y^2$

$$13^x+3=y^2\iff \left(4+\sqrt{3}\right)^x\left(4-\sqrt{3}\right)^x=\left(y+\sqrt3\right)\left(y-\sqrt3\right)$$ $\gcd\left(y+\sqrt3, y-\sqrt3\right)=1$, therefore ...
0
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1answer
56 views

How to compute an integral?

I am reading the lecture notes. I am trying to understand the prove of Lemma 0.0.1.1 on page 4. From line 3 to line 4 in the proof of Lemma 0.0.1.1., how to prove that $$ \int_{F^{n-1}} ...
4
votes
1answer
60 views

Each number in $S\subseteq \{1,\ldots,2n\}$ does not divide another one, with $|S|= n$. In how many ways?

Let $f(n)$ be the number of subsets $S\subseteq \{1,2,\ldots,2n\}$ such that $|S|=n$ and $a$ does not divide $b$ whenever $a,b \in S$ are distinct. Can we evaluate $f(n)$, at least asimptotically? ...
0
votes
1answer
58 views

Roots of $\Phi_{31}(x)$ as roots of unity.

Let $\Phi_{31}(x)$ be the $31$-cyclotomic polynomial. I want to show that $\Phi_{31}(x)$ is the product of six irreducible quintic factors in $\mathbb{F}_2$. I am running into difficulties ...
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0answers
17 views

Composition of polynomial and multiplicative is multiplicative .

I made the following problem a while ago but I can't solve it (also I don't think it's extremely hard ) : Let $f$ be a non-constant completely multiplicative function over $\mathbb{Z}$ . Assume ...
12
votes
1answer
79 views

parallel resistors

Consider the set $E_b = \left\{1, 1.2, 1.5, 1.8, 2.2, 2.7, 3.3, 3.9, 4.7, 5.6, 6.8, 8.2\right\}$. This is our base set. Let's define the set $E$ as follows: $$ E = \left\{ 10^k e \mid k=0,1,2,\ldots, ...
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0answers
14 views

Symmetry of Hecke L-function zeroes

For the Riemann zeta function, it is known by the functional equation and $\zeta(s)=\overline{\zeta(\bar s)}$ that the zeroes of $\zeta(s)$ are symmetric about the critical line $1/2$ and the real ...
2
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1answer
20 views

Is $a+r \cdot b$ an uniformly random value when $a,b$ are fixed and $r$ is random value?

Imagine we have two fixed values $a,b \in \mathbb{Z}_p$ and a uniformly random value $r\leftarrow \mathbb{Z}_p$, for large prime number $p$. Question: Is $v=a+b\cdot r$ an uniformly random value in ...
3
votes
0answers
31 views

Unit group of an imaginary quadratic ring

Let $R$ be an imaginary quadratic ring. Then, the unit group $R^{\times}$ is finite. To prove this, I worked with normal forms, algebraic integers and the fact that $R \not \subset \mathbb{R}$. But I ...
1
vote
4answers
68 views

Solving $a!+b!+c!=3^d$

The question is to find all tuples $(a,b,c,d)$ of natural numbers $c\geq b$ and $ b \geq a $ and $a!+b!+c!=3^d$. I am finding difficulty in establishing relation between $a$, $b$, $c$, and $d$. I see ...
2
votes
2answers
40 views

How many pairs of natural numbers $(x,y)$, satisfy the equation $\space xy=x+y+\gcd(x,y)$.

How many pairs of natural numbers $(x,y)$, satisfy the equation $\space xy=x+y+\gcd(x,y)$. You may assume that $x≤y$.
4
votes
1answer
39 views

Example of an nonidentity element in the kernel of the map.

This question is related to my previous question here. Let $n, m > 1$. The map $\mathbb{Z} \twoheadrightarrow \mathbb{Z}/m\mathbb{Z}$, of reduction mod $m$, induces a group homomorphism $F: ...
3
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0answers
24 views

Kronecker symbol vs. Koblitz symbol

In Koblitz, Introduction to Elliptic Curves and Modular Forms on page 188 it is defined $$\left( \frac{-1}{j}\right)=0$$ in case $j$ is even. Apart from that definition $\left( \frac{c}{d}\right)$ is ...
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0answers
43 views

Does anyone recognize this graph?

It's a plot of the following: Let $$f_{(n)} = \frac{np_n}{(p_1 + \ldots + p_n)}$$ so that $$g_{(n)} = \left|\space f_{(n)} - f_{(n-k)}\right| $$ where $n > k$ and $k = 5$ in this example. For ...
0
votes
1answer
45 views

Show that there exists $s, t \in S$ such that $\gcd(s, t)$ is a prime

Let $S$ be a set containing finitely many positive integers greater than 1 with property: for all $n \in \mathbb{Z_+}$, there exist $s \in S$ such that $\gcd(s, n) = 1$ or $\gcd(s,n) = s$. Show that ...
9
votes
1answer
59 views

Induced group homomorphism $\text{SL}_n(\mathbb{Z}) \twoheadrightarrow \text{SL}_n(\mathbb{Z}/m\mathbb{Z})$ surjective?

Let $n, m > 1$. The map $\mathbb{Z} \twoheadrightarrow \mathbb{Z}/m\mathbb{Z}$, of reduction mod $m$, induces a group homomorphism $F: \text{SL}_n(\mathbb{Z}) \to ...
2
votes
3answers
107 views

Probability that the eventually a six on a dice will appear.

Dave rolls a fair six-sided die until a six appears for the first time. Independently, Linda rolls a fair six-sided die until a six appears for the first time. Let $ m$ and $ n$ be relatively prime ...
3
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0answers
61 views

Connections between Fibonacci and natural numbers

Here are some known facts about Fibonacci numbers and then some questions regarding them . 1.Carmichael's theorem . For every $n>12$ $F_n$ has a prime divisor which doesn't divide any of the ...
2
votes
1answer
100 views

How can I solve $x^2+2=y^3$ in $\mathbb{Z}$?

Prove that $\left \{ (x,y)\in\mathbb{Z}^2:x^2+2=y^3 \right \}\subseteq \left \{ (-5,3),(5,3) \right \}$.
2
votes
2answers
47 views

Proof that: $a=bq+r ,-\frac {|b|}{2}<r≤\frac {|b|}{2}$ [closed]

Proof that: Let $a,b$ any integers, with $b≠0$, Then there exist unique integers $q$ and $r$ surch that $$a=bq+r ,$$ where $$-\frac {|b|}{2}<r≤\frac {|b|}{2}$$ Note corolario: Let ...
2
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0answers
29 views

Besides the $3x + 1$ problem, for which similar problems are still unresolved regarding trayectory?

Generalize the $3x + 1$ problem as $cx \pm 1$, where $c$ is a positive odd integer and $x$ is a positive integer iterated through the function as far as possible to discover a cycle. If $x$ is even, ...
0
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2answers
61 views

Proof that every positive integer has at most one prime factor greater than it's square root?

I read the statement in the title somewhere but I can't find any proof. For a positive integer $n$, why can't there be 4 numbers $a, b, c, d$ ($b$ and $d$ are prime) for which $a < \sqrt{n} < ...
2
votes
0answers
35 views

Legendre symbol identity

I try to solve the following problems ($p$ is an odd prime) Find the sum $$\sum_{a=1}^{p-1}a \cdot \left (\frac{a}{p} \right)$$ Find the sum $$\sum_{a=1}^{p-1} 2^a \cdot \left (\frac{a}{p} \right)$$ ...
3
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1answer
54 views

Power and Divisibility

Find all positive integers $n$ such that $7^n-1$ is divisible by $6^n-1$. Trying small cases, it seems that there's no such positive integer.
5
votes
2answers
74 views

Proving the irrationality of the concatenation of the $n$th powers of primes

Note: The apostrophes are meant to separate different groups of digits. Like, $0.{1^2}'{2^2}'{3^2}'{4^2}'\cdots=0.14916\cdots$. I wasn't able to come up with something better. It is easy to show ...
0
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0answers
18 views

Is this approach to Lehmer's totient problem going anywhere?

So Lehmer's totient problem states that $k\phi(n) = n-1$ and $k$ is integer and $n$ is composite. This is my first, obviously futile attempt. But what do you think of the approach to the problem? Is ...
4
votes
1answer
68 views
+50

Variation of the Josephus problem

Suppose we have a circle of $2n$ people, where the first $n$ people are good guys and the people $n+1$ to $2n$ are bad guys. Can we always choose an integer $q$, such that if we execute successively ...
3
votes
2answers
85 views

Integer solutions to $x^2 + dy^2 = c$

I'm trying to find all integer solutions of an equation $x^2 + dy^2 = c$ with $d,c \in \mathbb{Z}_{>0}$. I'm well aware of the methods that exists when $d \in \mathbb{Z}_{<0}$ or when $c$ is a ...
1
vote
1answer
25 views

Sum of digits of polynomial smaller than of factorial

I'm trying to prove this : Let $f \in Z[X]$ then for sufficiently large $n$ we have $$s(f(n))<s(n!)$$ where $s$ is the sum of digits function. What I have so far : I thought this must be true ...
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0answers
46 views

Elliptic curve is self dual.

How to prove $E[p^\infty] \cong Hom ( T_E, \mathbb{Q}_p/\mathbb{Z}_p(1)) $ where $T_E$ denotes the Tate module of $E$ ?
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1answer
30 views

Integer solutions of a quadratic equation with combined variables

I'm having problems with finding all possible integers solutions of particular equations, like this one for example: $x^2 -xy + 2y^2 = 29$. What sets me off, is the term $xy$, I don't know how to deal ...
0
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1answer
16 views

Lattice points in simplices - reference request

I found this paper http://homepages.math.uic.edu/~yau/35%20publications/An%20upper.pdf which, in formulas (1.2) and (1.3), relates the number of non-negative and positive integer values that are ...
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0answers
22 views

Optimal bounds of given curve [closed]

How to find optimal bounds for the heights of integral solutions on a given any Weierstrass equation? Please explain with an example. regards, S. Nair.
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2answers
31 views

Prove that $\gcd(\frac{m}{a},\frac{m}{b}) = 1$ if $m$ is the $\mathrm{lcm}$.

My ideas were rewrite the terms in parenthesis in a such way that any other common divisor of $m/a$ or $m/b$ divides $1$.
7
votes
3answers
60 views

Is there any element of order $51$ in the group $U(103)$

Does there exist an element of order $51$ in the multiplicative group $U(103)$ ? Now if the element exist say $x$ then it satisfies the equation $$x^{51}\equiv 1\pmod {103}$$ . ...
2
votes
0answers
59 views

Arithmetic progression of squarefree integers?

Let $x$ be a given positive integer. I'm intrested in the longest arithmetic progression of squarefree integers within the interval $(x,x^2)$. Both constructive and nonconstructive results. For ...