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.

learn more… | top users | synonyms (1)

0
votes
1answer
34 views

Are there more non-perfect square numbers than perfect squares?

Can anything be said on this issue? I was wondering if one can find a mapping such that the cardinality of two sets of perfect and non-perfect squares can be compared. Not sure if it's a good question ...
2
votes
1answer
31 views

Explaining an integral involving the divisor function

In a 1973 paper by Martinet, Deshouilliers and Cohen, $A(x)$ is defined as $$A(x)=\lim_{N\to\infty}\frac{\#\{n\leq N\mid \frac{\sigma(n)}{n}≥x \}}{N}$$ where $\sigma(n)$ is the "sum-of-divisors" ...
0
votes
0answers
25 views

For every prime $p$ and a fixed integer $k$, are there infinitely many values of $n$ such that $p$, $p^n+k$, $kp^n+1$ are all primes?

This is similar to my last question, but may or may not be the case: For every prime $p$ and a fixed integer $k$, are there infinitely many values of $n$ such that $p$, $p^n+k$, and $kp^n+1$ are all ...
2
votes
1answer
61 views

The first step in the proof of the Pólya-Vinogradov Inequality.

The well-known Pólya-Vinogradov Inequality states: $$\forall m, n \in \mathbb{N}: \displaystyle \left|{\sum_{k \mathop = m}^{m+n} \left({\frac k p}\right)}\right| < \sqrt p \ \ln p,$$ where $\...
4
votes
1answer
71 views

Find all odd positive integers $n$, which there exists odd positive integers $x_1,x_2,..,x_n$, such that $x_1^2+x_2^2+\cdots+x_n^2=n^4$

Find all odd positive integers $n$, which there exists odd positive integers $x_1,x_2,..,x_n$, such that $$x_1^2+x_2^2+\cdots+x_n^2=n^4$$ My work so far 1) $n=3$ $$x_1^2+x_2^2+x_3^2=81$$ no ...
1
vote
2answers
62 views

Solution of Diophantine equation

Find all integral solutions of $x^2+1= y^2+z^2$. Actually I have to find all integral solution of $a(a+1)=b(b+1)+c(c+1)$. I reduced this in the above form I.e., $ (2a+1)^2+1= (2b+1)^2+(2c+1)^2$ .
0
votes
2answers
32 views

primitive polynomials and their factorisation

A polynomial with integer coefficients is called primitive if its coefficients are relatively prime. For example, $$3{x^2} + 7x + 9$$ is primitive while $$10{x^2} + 5x + 15$$ is not. (a) Prove that ...
3
votes
1answer
39 views

Is there no proof of Dirichlet's results on quadratic residues without analysis?

Wikipedia states that all known proofs of Dirichlet's results $$ L(1) = -\frac{\pi}{\sqrt q}\sum_{n=1}^{q-1} \frac{n}{q} \left(\frac{n}{q}\right) \gt 0 $$ and $$ L(1) = \frac{\pi}{\left(2-\left(\...
0
votes
2answers
40 views

proving theorem about perfect powers

Im currently studying the journal entitled Perfect Powers with All Equal Digits but One theorem: For a fixed integer $l \geq 3$, there are only finitely many perfect $l$-th powers all whose digits ...
-1
votes
1answer
34 views

writing numbers as sum of at least two consecutive odd positive integers [on hold]

Since 24 = 3 + 5 + 7 +9, the number 24 can be written as the sum of at least two consecutive odd positive integers. (a) Can 2005 be written as the sum of at least two consecutive odd positive ...
2
votes
1answer
25 views

Primitive quintuples of distinct positive integers

Are there infinitely primitive quintuples of distinct positive integers such that each integer divides the sum of the four others?
0
votes
1answer
49 views

Prove that any integer greater than or equal to $7$ can be written as a sum of two relatively prime integers, both greater than $1$.

Prove that any integer greater than or equal to $7$ can be written as a sum of two relatively prime integers, both greater than $1$.For example, $22$ and $15$ are relatively prime, and thus $37 = 22+...
1
vote
1answer
23 views

Prove that among any 12 consecutive positive integers there is at least one which is smaller than the sum of its proper divisors

Prove that among any 12 consecutive positive integers there is at least one which is smaller than the sum of its proper divisors. (The proper divisors of a positive integer n are all positive integers ...
3
votes
0answers
86 views

When is $\frac{b^b+b}{ab^2+9}$ an integer? [duplicate]

Find all possible Integer values of $a,b$ (where both are integers, be it positive or negative) such that $$\frac{b^b+b}{ab^2+9}$$ is also an integer. (Not necessarily positive or negative). I could ...
0
votes
1answer
50 views

For every prime $p$, are there infinitely many integers $k$, such that $p$, $p+k$, and $kp+1$ are all primes?

Please help me proved or disprove the conjecture below. Thanks. For every (fixed) prime $p$, there are infinitely many integers $k$ such that $p$, $p+k$, and $kp+1$ are all prime? I wasn't exactly ...
0
votes
1answer
41 views

Sum of $n$ products

Evaluate $$S = \sum_{k=1}^n k(k-1)\cdots (k-p),$$ where $n$ and $p$ are positive integers. I was wondering about this question because doing the positive version of the question, which is ...
5
votes
3answers
133 views

What are some fields that intersect topology and number theory? [on hold]

I see that number theory is studied from the algebraic and analytics aspects, but I have not seen any approach from topology or axiomatic set theory (using them to investigate the properties or ...
0
votes
1answer
66 views

Condition that for a given set of numbers and given divisor all finite sums from this set contain all possible remainders

Given $q \in \mathbb{N}$ and ${a_1, a_2, ...}$ where each $a_j \in \mathbb{N} \cup{\{0\}}$ define $A_p=$ {set of all finite sums of $\{a_1 ... a_p\}$ such that each $a_j$ will appear either $1$ or $0$ ...
3
votes
2answers
64 views

Prove That $n(n+1)$ Can Never Be a Square Number by Showing the Atleast One of Exponents in the Prime Power Decomposition Isn't Even

Can someone show me how to prove that when $n>0$, $n(n+1)$ can never be a square number by demonstrating at least one of the exponents in the prime power decomposition is not even? Here's what I ...
2
votes
1answer
91 views

What level of mathematics do I need to study the Collatz Conjecture?

I recently came across the Collatz Conjecture and I'm really intrigued by its tautological simplicity and complexity. I'm under no illusions that I can make any progress with a proof for it but I ...
4
votes
0answers
73 views

$\{\{(2^n+3^m)\alpha\}:n,m\in\mathbb{N}\}$ is dense in [0,1]??

Let $\alpha$ be an irrational real number. I wonder whether $\{\{(2^n+3^m)\alpha\}:n,m\in\mathbb{N}\}$ is dense in [0,1] in which $\{x\}$ means the fractional part of x. This is equivalent to the ...
1
vote
0answers
20 views

Dirichlet character to prime power modulus

Let $p$ be an odd prime number and let $\alpha \geq 1$ be an integer. Let $\chi$ be a real, primitive Dirichlet-character mod $p^{\alpha}$. How does one show that $\alpha = 1$? If we choose an ...
1
vote
1answer
43 views

What is the least possible value of $n$ such that $n^2+n+17$ is composite? [duplicate]

The question is- Find the least $n$ for which $n^2+n+17$ is composite. I tried to factorize it and show that it has a factor greater than $1$.But I could not factorize it and I also found that ...
2
votes
1answer
32 views

Existence of primitive roots

Fix $p$ prime . Exist $q$ prime that p is primitive root $Z/qZ$ ? I think that this is true (numerical evidence).
3
votes
1answer
91 views

Find last 5 significant digits of 2017!

Since there are less powers of $5$ than of $2$ and since $10 = 2 \cdot 5$, I counted the number of zeros in $2017!$: $\left \lfloor{ \frac{2017}{5^1}}\right \rfloor + \left \lfloor{ \frac{2017}{5^2}}\...
0
votes
0answers
18 views

maximize a sum of unit fractions (without containing a subset of sum 1)

Let $ u \ge 2 $ be fixed. Then consider: $ S(u)=\max\left\lbrace \sum_{i=1}^{u+1} \frac{c_i}{t_i} \, \middle| \, 2 \le t_1 \le t_2-1 \le \ldots \le t_{u+1}-1, \, t_i \in \mathbb{N}, \, c_i \in \...
0
votes
1answer
33 views

How to find a modulus equation? [on hold]

Let $x$ and $q$ be an integer.Also, $a$ and $b$ are integers. We know the two modulus equations. i) $x \equiv y$ mod $p^a -1$ ii) $x \equiv z$ mod $p^b -1$ Then how to find $x$? Can we find $x$ ...
0
votes
4answers
58 views

Reduce Number of digit [on hold]

I have 24 digit number like 281564785148270616103860.I want to reduce this number with any optimization technique to 12 digit number . So i can use this to decode with the same technique.Any answer ...
1
vote
0answers
39 views

Explanation about proof of euler's function

Explanation about reduced residue system theorem Theorem 3-8 $$\varphi(m)=m\prod_{p|m}(1-1/p)$$ Proof: By reduced residue system theorem 3-7, if $$m=\prod_{i=1}^{r}p_i^{a_i}$$ then $$\varphi(m)...
0
votes
1answer
33 views

Fractional part of a square root

Let $a$ be a positive integer and let $\{x\}$ denote the fractional part of $x$. Prove that $\left \{\sqrt{a^2+1}\right \}$ is smaller than any element from the set $S = \{\{\sqrt{x}\}\mid x < a^2 \...
8
votes
1answer
30 views

Selecting disjoint subsets with the same sum from a set of ten distinct two digit numbers

My question is the following: Is it possible to select two disjoint subsets whose members have the same sum from a set of ten distinct two-digit numbers (in the decimal system)? I guess the answer ...
5
votes
2answers
57 views

Explanation about reduced residue system theorem

I need an explanation of the following theorem Theorem 3-7 $If (m,n)=1$ then $\varphi(mn)=\varphi(m)\varphi(n)$ Proof: Take integers m,n with $(m,n)=1$, and consider the numbers of the form $mx+ny$. ...
4
votes
1answer
37 views

Relating the class number of a field, and of its normal closure

Suppose I take a number field $ K $, not necessarily Galois, with class number $ h_k $ (over $ \mathbb{Q} $). Write $ \overline{K} $ for the normal closure of $ K $. What, if anything, can be said ...
-2
votes
2answers
33 views

Distinct Factors

Consider $(6-a)(6-b)(6-c)(6-d)(6-e)$ are five distinct factors of $45$. What is $a+b+c+d$ The problem I am facing is that I am supposing $b = 1$, $c = 5$, $d = 3$ The problem is coming in supposing ...
3
votes
2answers
155 views

Number of pairs of co-prime positive integers $(a, b)$ such that $\frac{a}{b} + \frac{14b}{a}$ is an integer?

How many pairs of positive integers $(a, b)$ are there such that $a$ and $b$ have no common factor greater than $1$ and $\frac{a}{b} + \frac{14b}{a}$ is an integer? The problem I'm facing in this ...
-1
votes
0answers
45 views

General formula for composite numbers

I have derived General formula of composite numbers in the form: formula of composite numbers, except divisible by 2 and 3: Positive integers contained in two 2-dimensional arrays $P1(i,j)=6i^2-1+(...
1
vote
0answers
62 views

How does $3n+1$ change the proximity of $n$ to a power of two?

This is part of an attempt to prove Collatz's conjecture. I proved a modification of Collatz's conjecture, where instead of $3n+1$ if $n$ is odd, you do $n+1$. In Collatz's conjecture, if you get to a ...
2
votes
3answers
56 views

The arithmetic function $\lambda(n)=(-1)^{a_1+\cdots +a_k}$

Define $\lambda(1)=1$, and if $n=p_1^{a_1}\cdots p_k^{a_k}$, define $$\lambda(n)=(-1)^{a_1+\cdots +a_k}$$ How can I see that $$\sum_{d\mid n}\lambda(d)=\begin{cases} 1 \,\,\text{ if $n$ is a square}\\...
1
vote
2answers
23 views

Inequality on sequence of integers whose only prime factors are $2$ or $3$

Let $M=\lbrace 2^i 3^j | i,j \geq 0\rbrace$ and denote by $m_k$ the $k$-th element of $M$ ; so $m_1=1,m_2=2,m_3=3,m_4=4,m_5=6\ldots$. Is it true that $3m_k\geq 2m_{k+1}$ for every $k>1$ ? My ...
0
votes
1answer
9 views

Why is the running time of the trial division $O(f \cdot (log N)^2)$?

I saw this being cited in a few paper,but none of them seems to explain why this is the case. Maybe because it is quite trivial, but I am not sure why exactly... Here $f$ is the size of the factor. I ...
1
vote
1answer
61 views

Conjecture concerning modular arithmetic

Below $0\notin\mathbb N$. I want a proof or a counter-example of the following (corrected) conjecture: Suppose $p$ is the smallest prime dividing $n\in\mathbb N$ and suppose $kn+ap=m!$, where $...
2
votes
1answer
42 views

maybe this conjectures is also hold?

for any irrational $x\in(0,1)$,and positive integer $n$,there exsit prime numbers $p_{1},p_{2},\cdots,p_{n}$ where $$p_{1}<p_{2}<\cdots<p_{n}$$ such $$0<x-\sum_{i=1}^{n}\dfrac{1}{p_{i}}&...
1
vote
1answer
30 views

Proof about complete residue system

Theorem 3-5. If $a_1,a_2,\ldots,a_m$ is a complete residue system $\pmod m$ and $\gcd(k,m)=1$, then $ka_1,ka_2,\ldots,ka_m$ also is a complete residue system $\pmod m$ Proof: We show directly that ...
0
votes
1answer
48 views

I need an explanation of a theorem about congruence

Theorem 3-5. If $a_1,a_2,\ldots,a_m$ is a complete residue system $\pmod m$ and $\gcd(k,m)=1$, then $ka_1,ka_2,\ldots,ka_m$ also is a complete residue system $\pmod m$ Proof: We show directly that ...
6
votes
1answer
186 views

Number Theory and d-Self-Contained Numbers

Given any natural number $N = a_{n}a_{n-1}\ldots a_{1}$, let us associate to it the set $S_{N} = \bigcup_{j=1}^{n}\{(a_{j},j)\}$. We're going to define a d-self-contained number as any natural number ...
0
votes
0answers
12 views

Last digits of a Power of 3 using digits 0, 4, 8 not counting the last

I am looking for a pattern continuation in the last $x$ digits (except the final digit) of $3^n$ using only multiples of 4 (0, 4, 8). This is narrowed down to 01, 03, 09, 81, 83, 89. What are the next ...
0
votes
0answers
33 views

Find all the solutions that make the expression a perfect square

The expression: $$(2x^2+2x+2y^2+2y+1)(2x^2+2x-2y^2-2y)\tag{1}$$ where: $x>y,\qquad x,y$ are non-negative integers. Is it possible to find all $x,y$'s that make $(1)$ a perfect square number? ...
1
vote
1answer
35 views

Proof about mod properties

Let $m>1$ be fixed. Show that if the integers $a_1,a_2,...,a_k$ have any two of the following three properties, they also have the third and hence constitute a complete residue system (mod m) a) ...
2
votes
0answers
34 views

Number of solutions of some trigonometric equations

Let $N > 1$ and let $S$ be a subset of the integers in the (real) interval $[1, N]$. Can we prove that there are only finitely many solutions $x \in [1,N] \setminus \mathbb{Z}$ to the equation $$ \...
3
votes
2answers
44 views

Find roots of polynomial in a finite field

I need to build a field $L$ of 121 elements and find how many roots polynomial $g=x^9-1$ has in $L$. Then to find all these roots. So, $121=11^2$ this is power of prime. We can build finite field of ...