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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94 views

If Robin's inequality ever fails, are there only finitely many colossally abundant numbers that satisfy it?

Let$\ \sigma(n)$ be the sum-of divisors function, with the divisors raised to$\ 1$. If the Riemann Hypothesis is false, Robin proved there are infinitely many counterexamples to the inequality$$\ ...
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1answer
160 views

Something related to Frobenius coin Problem/Chicken McNugget Theorem

Let positive integers $a,b,c$ be pairwise coprime. Denote by $g(a, b, c)$ the maximum integer not representable in the form $xa+yb+zc$ with positive integral $x,y,z$. Prove that $$ g(a, b, c)\ge ...
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0answers
49 views

How find this sum $S=\sum_{i=1}^{m}(-1)^{a_{i}}\cdot 2^{m-i}$ and $2^i\equiv a_{i}\pmod n$

Question: let $n$ is give odd positive integer numbers,and $a_{i}\neq 1,0\le a_{i}\le n-1$, and $$2^i\equiv a_{i}\pmod n,i=1,2,\cdots,m-1$$ where $m(m\le n)$ such $2^m\equiv 1\pmod n$ ...
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3answers
98 views

integer solutions to $x^2+y^2+z^2+t^2 = w^2$

Is there a way to find all integer primitive solutions to the equation $x^2+y^2+z^2+t^2 = w^2$? i.e., is there a parametrization which covers all the possible solutions?
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1answer
87 views

How prove this sequence $S_{n}=[2^n\cdot \sqrt{2}],n\in N$ contains infinitely many composite numbers

define sequence $$S_{n}=[2^n\cdot \sqrt{2}],n\in N$$ show that $\{S_{n}\}$contains infinitely many composite numbers where $[x]$ is the largest integer not greater than $x$ my try: since ...
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0answers
54 views

Zero to power Zero (Zero ^ Zero) indeterminable or not? [duplicate]

I want to know Zero power to Zero equal to 1 or Indeterminable. I think it cannot be exist. Please explain with proper mathematical definitions.
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1answer
73 views

Farey Sequence Vector Orthogonality Relation Question

Take the Farey sequence $\mathcal{F}_n$ for $n=39$ with values $a_m\in \mathcal{F}_n$ and put them into a vector $$ \vec v_k=\biggr(\exp(2\pi i k a_m)\biggr)_m $$ Since Merten's function for $n=39$ ...
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1answer
275 views

Number of triplets for which $\gcd(a,b,c)=1$ and $c=n$

As stated in A123323: Number of triples a,b,c with a<=b<=c<a+b, gcd(a,b,c)=1 and c=n. ...... A123323(n)=sumdiv(n, d, floor((d+1)^2/4)*moebius(n/d)) How ...
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1answer
112 views

Fermat's little theorem proof by Euler

I am reading a book, it explains the Euler's proof of Fermat's little theorem (FLT). There are 3 theorems are presented to prove FLT, I understood the first two (I will skip the proof of each ...
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2answers
201 views

Summation of factorials modulo ten

I have read that$$\sum\limits_{i=1}^n i!\equiv3\;(\text{mod }10),\quad n> 3.$$ Why is the sum constant, and why is it $3$?
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3answers
202 views

An identity involving the Pochhammer symbol

I need help proving the following identity: $$\frac{(6n)!}{(3n)!} = 1728^n \left(\frac{1}{6}\right)_n \left(\frac{1}{2}\right)_n \left(\frac{5}{6}\right)_n.$$ Here, $$(a)_n = a(a + 1)(a + 2) \cdots (a ...
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2answers
82 views

Function approximating this product

Is there any function approximating, for large values of $p$, the quotient between the product of all primes and the product of all primes $-1$? Basically: $2/1 \cdot 3/2 \cdot 5/4 \cdot 7/6 \cdot ...
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2answers
173 views

GCD and LCM of three numbers

Given two positive integers G and L, could you tell me how many solutions of (x, y, z) there are, satisfying that gcd(x, y, z) = G and lcm(x, y, z) = L? gcd(x, y, z) means the greatest common divisor ...
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1answer
118 views

Relation between the prime density and Riemann's zeros?

Soft question: Where is the connection between the zeros of Riemann's $\zeta$-funciton and the density of prime numbers? Is there a short answer to this question, to get the overview? I once had a ...
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2answers
453 views

Show that the odd prime divisors of the integer $n^2+n+1$ which are different from $3$ are of the form $6k+1$.

1)The odd prime divisors of the integer $n^4+1$ are of the form $8k+1$. My attempt: Let $p$ be odd divisor of $n^4+1$.Then $n^4+1 \equiv 0$ (mod p) $\Rightarrow n^4 \equiv -1$ (mod p) $\Rightarrow ...
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1answer
233 views

Pythagorean triples with additional parameters

I want to find solution in $\mathbb{Z}$ to the following quadratic Diophantene equation: $$na^2 + kb^2 = c^2$$ where $n,k,a,b,c \in \mathbb{Z}$, $n,k > 0$ and $(n,k) = 1$ I know that for some ...
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2answers
142 views

Limit of $\lim_{n\to\infty}\sum_{k=1}^{n}\frac{\sigma{(k)}}{k!}=$?

find the limt value $$\lim_{n\to\infty}\sum_{k=1}^{n}\dfrac{\sigma{(k)}}{k!}= ?$$ where $\sigma(n)$ is defined as $$\sigma{(N)}=\sum_{d|N}d$$ this problem is my frend ask me, I don't know this have ...
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0answers
120 views

Why these two problems lead to same answers?

Suppose these two problems: Problem 1: $$\min_{X,P} \quad\max_{1\leq l\leq L-1} \quad {|\sum_{1\leq i\leq N_p}^{N_p}x_ie^{\frac{2\pi l}{N}p_i}| \over {\sum_{i=1}^{N_p} x_i^2}} \quad \equiv \quad ...
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1answer
130 views

A Diophantine equation and decimal digits

Solutions of the Diophantine equation $a10^n+(a+1) = (2^{m+1}-1)*2^{m+1}$ are 12=3*4, 56=7*8, 67100672=8191*8192. Are there more solutions/examples like that or a generalization of the ...
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2answers
73 views

Sequences and Languages

Let $U$ be the following language. A string $s$ is in $U$ if it can be written as: $s = 1^{a_1}01^{a_2}0 ... 1^{a_n}01^b$, where $a_1,..., a_n$ are positive integers such that there is a 0-1 ...
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5answers
175 views

Does an integer $9<n<100$ exist such that the last 2 digits of $n^2$ is $n$?

Does an integer $9<n<100$ exist such that the last 2 digits of $n^2$ is $n$? If yes, how to find them? If no, prove it. This problem puzzled me for a day, but I'm not making much progress. ...
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1answer
63 views

Help with proof : $Cl_F$ with only one equivalance class implies $D_F$ is a PID

I am self-studying a set of legitimately downloaded notes on algebraic number theory. They are somewhat akin to "Ireland and Rosen," Ch. 12. I would appreciate help in understanding a proof (in the ...
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2answers
147 views

$N^2=2M^4-2p^2e^4$ has no integer solution

If $\gcd(M,e)=\gcd(N,e)=1$ and $p$ is prime and $p‎\equiv 5 \mod(16)$ then how I can show that $N^2=2M^4-2p^2e^4$ has no integer solution.
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1answer
274 views

Class number/ quadratic field/divisibility-$2$

As I am poor in construction of mathematical problem, I am not getting good answers from MSE members. However, this time I constructed the following problem in best possible way. So, I hope I will get ...
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1answer
151 views

Inequality for binomial coefficients

Let $m \leq n, n \leq N$ and $0\leq k \leq m$. I am wondering what is the dependence of $n$ and $N$ that for all $m, k$ $$ \frac{{N-m \choose n-k}}{{N \choose n}}\leq 1. $$ Thank you for your help.
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95 views

How can we show that $\operatorname{ord}_{p}\left(\binom{2n}n\right) \le \frac{\log 2n}{\log p}$

How can we show that $$\operatorname{ord}_{p}\left(\binom{2n}n\right) \le \frac{\log 2n}{\log p}$$ where $p$ is a prime number and $n$ is a natural number My attempt: $$2^{n} \le \prod ...
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2answers
1k views

The Binary Representation in Number Theory?

I was reading "Concrete Mathematics" and confronted the Joseph Problem. I'm astonished that the recurrence equation in binary representation is so simple. i.e. for J(n) say J(345), $\text{J(345) = ...
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1answer
239 views

How to prove $\sum_{ d \mid n} \mu(d)f(d)=\prod_{i=1}^r (1-f(p_i))$?

I have to prove for $n \in \mathbb{N}>1$ with $n=\prod \limits_{i=1}^r p_i^{e_i}$. $f$ is a multiplicative function with $f(1)=1$: $$\sum_{ d \mid n} \mu(d)f(d)=\prod_{i=1}^r (1-f(p_i))$$ How I ...
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1answer
98 views

Given $a$ and $n$, find $y \equiv \frac{n!}{a^x} \bmod a$

Take $n!$ and find $x$, where $a^x$ is the greatest power of $a$ who divides $n!$ Then find $y$, where $y \equiv \frac{n!}{a^x} \bmod a$ For example, if $a=3$ and $n=6$ then ...
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1answer
520 views

If $n$ is an odd pseudoprime , then $2^n-1$ is also odd pseudoprime

I have some problems understanding the following proof: Definition: A composite number $n \in \mathbb{N}$ is called pseudo prime if $n \mid 2^{n-1}-1$ holds. Theorem: If n is a odd pseudo prime ...
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4answers
569 views

two elementary games in number theory

I solved these two problems from a programming challenge website: numgame and numgame2. These two problems are very similar. In the first one, the position is a number $n$ and each player can ...
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2answers
245 views

A problem on Number theory

You are given three non-negative integers $A$, $B$ and $C$, find a number $X$ (say) satisfy $X^A \equiv B\pmod{2C + 1}$ and $0 \le X \le 2C$. I am inquisitive about how to approach this one?
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35 views

How can we show the assertion?

Every natural number not of the form $4^n(8m+7)$ where $m$ and $n$ are natural numbers, can be represented as sum of three squares.
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31 views

Is the Euler prime of an odd perfect number a palindrome (in base $10$), or otherwise?

Let $N = {q^k}{n^2}$ be an odd perfect number given in Eulerian form (i.e., $q$ is prime with $\gcd(q,n)=1$ and $q \equiv k \equiv 1 \pmod 4$). (That is, $2N=\sigma(N)$ where $\sigma$ is the ...
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1answer
67 views

Diophantine equation in $a, b, c,$and $d$

I'm looking for positive integers $a, b, c,$ and $d$ such that $$ (ad - bc)(ac + bd) \: | \: abcd$$ One partial solution that I found is $$(a, \, b, \, c, \, d) = (2x + 1,\, 2x, \, 2x + 2, \, 2x + ...
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2answers
109 views

Solve $a^{2013}+b^{2013}=p^n$ for all postiive integers a,b,n and prime numbers p

Solve $a^{2013}+b^{2013}=p^n$ for all positive integers $a$, $b$, $n$ and prime numbers $p$. I got that you can take $\gcd(a,b)=1$ and solve for it.
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1answer
64 views

How to find solutions in set of intgers?

Is there a simple way to find set of solutions in integers. For example, find integer solution of (1) $x^2 + y = y^5 + x$ (2) what is Extend Ellenberg’s approach? For what type of equations can be ...
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1answer
79 views

Integer $k$ such that $k!$ has 99 zeros

For how many positive integers $k$ does $k!$ has 99 zeros. The question is not difficult,since if $k$ the first for $k!$ to have 99 zeros, then since $k+1,\cdots,k+4$ are not divided by 5, so the ...
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99 views

Number theory proofs relating to units

Moderator Note: This has been claimed to be a current contest question. It is being locked while we investigate. What is a counterexample for the proposition: If u ∈ Um has order n1 and u2 ∈ Um ...
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44 views

the non-zero integer roots of an inequality

Let $a,b,c,d$ be real numbers, and $ad-bc \ne 0$, given $$|(ax+by)(cx+dy)|\le \frac{1}{2}|ad-bc|.$$ does there exist non-zero integers $x,y$ which satisfy the above inequality ?
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1answer
72 views

Can you also conclude also that $X^{\phi{(n)}}-[1]=\prod_{[a] \epsilon (Z /_n Z)^x}(X-[a])$?

Conclude that the difference of the polynomials from $X^{\phi{(n)}}-[1]$ $\prod_{[a] \epsilon (Z /_n Z)^x}(X-[a])$ is a polynomial of degree strictly less than $\phi{(n)}$ and that it has ...
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3answers
208 views

Problem Heron of Alexandria.

Meaning of the problem is to find two right triangles equal perimeter, but with a predetermined magnification area. That is necessary to solve a simple system of equations. ...
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2answers
68 views

Division in ring $Z[\frac{-1+\sqrt{-3}}2]$

Let $w=\frac{-1+\sqrt{-3}}2$,Find $q,r \in Z[w]$ such that $3+5w=(2-w)q+r$ What's the best way to approach this kind of questions?
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3answers
470 views

Congruences of the form $x^2-a \equiv 0$ (mod pq)

Problem: Let p and q be distinct primes. What is the maximum number of possible solutions to a congruence of the form $x^2-a \equiv 0$ (mod pq), where as usual we are only interested in solutions that ...
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3answers
73 views

How to prove some statements about divisibility and the $\gcd$ function

Struggling with some number theory homework. Could use a helping hand. The two statements are as follows $\gcd(c, ab) \mid \gcd(c,a)\gcd(c,b)$ If $c \mid ab$ and $\gcd(a,b)=1$, then ...
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5answers
135 views

Relationships between the elements $(a,b,c,d)$ of a solution to $A^2+B^2+4=C^2+D^2$

I have reduced a certain equation (in positive integers) to the equation $$A^2 + B^2 + 4 = C^2 + D^2. \quad(\star)$$ Assume the positive integers $(a,b,c,d)$ are any solution to $(\star)$. Are there ...
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0answers
40 views

Primality Test with some condition

Given a prime number p, how can I quickly determine the primality of 10p+a, where a is an integer between 0 and 9? O(1) test is preferred Thanks!
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4answers
162 views

number theory $\gcd(a,bc)=\gcd(a,c)$

Suppose $\forall (a;b;c;d) \in \mathbb{Z}^4$ as $\gcd(a,b)=\gcd(c,d)=1$: How can I prove that $\gcd(a,bc)=\gcd(a,c)$? Also, how to prove that $\gcd(ac,bd) = (\gcd(a,d))*(\gcd(b,c))$?
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3answers
253 views

Basic Questions about the Properties of Mertens Function

I am astounded by how little information about Mertens function M(n) (partial sums of the Möbius function) is on the Internet. Thus, I would be thankful if someone could clear up some of my confusion. ...
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1answer
139 views

Show that $B$ is a basis for $K$ as a vector space over $\mathbb{Q}$

Let $K$ be a number field of degree $d$, and let $B = \{b_1,\ldots,b_d\}$ be a subset of $K$ of cardinality $d$ such that the matrix $(\mathrm{Tr}(b_ib_j))^d_{i,j=1}$ has non-zero determinant ...