Tagged Questions

Questions on more advanced topics of number theory. Consider first if (elementary-number-theory) might be a more appropriate tag before adding this tag.

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0
votes
1answer
25 views

struck on generating operating functions.

Find the ordinary generating function associated with t 1. he problem of finding the number of solutions in nonnegative integers of the equation? 2a + 3b + 2c + d = r. where (a) r=10 (b) r=15 any ...
6
votes
2answers
471 views

Sum of digits raised to a power

Let $S$ equal the sum of the digits of $2014^{2014}$. Let $T$ equal the sum of the digits of $S$. Let $U$ equal the sum of the digits of $T$. What is $U$?
4
votes
3answers
73 views

$(n!)!>n^{n!} \forall n \in \Bbb N^{\ge 4}$

I'm not sure if this has been asked before, but I came across it in the reasoning of solving a problem for my Real Analysis homework. It was not directly part of the problem so it may be a unique ...
0
votes
1answer
29 views

How to find the sum of digits of $p^q$ when $p$ and $q$ are large integers?

Is there any formula to get direct value for this function. $F(p,q)$ = sum of digits in $p^q$ I know that i can compute $p^q$ and sum up the digits. But I want to find it when $p$ and $q$ are big ...
0
votes
0answers
8 views

Does Schinzel's hypthesis hold when you allow exponentials?

I'm curious to what extent Schinzel's hypothesis is expected to hold. Heuristically it seems that a typical doubly exponential function say $3^{3^n}+4$ probably is prime only at finitely many $n$ by ...
1
vote
0answers
27 views

Number of solutions of $xy^2-y^2-x+y=k$ [closed]

Let $k$ be a positive integer. How many solutions does the equation $xy^2-y^2-x+y=k$ have in integers? The equation can be written as $x(y^2-1)-(y^2-y)=k$, or $(y-1)(x(y+1)-y)=k$.
-1
votes
2answers
57 views

Units in the ring $\mathbb{Z}(\omega)$

If $\omega \not= 1$ is a cube root of unity in $\mathbb{C}$, show that the units in the ring $\mathbb{Z}[\omega]$ are the elements of modulus 1. Hence, or otherwise, show that $U(\mathbb{Z}[\omega]$ ...
1
vote
1answer
24 views

Chinese remainder theorem corollary.

I was puzzled regarding corollary 31.29 of the Chinese remainder theorem as presented in the chapter on number theoretic algorithms by Cormen et al. I found another person who asked the same question ...
1
vote
1answer
25 views

Solution for task using number theory and group theory.

$$p \mid b^n +1 \text{ and } p \text{ is prime } \Rightarrow$$ (1) $ p\mid b^d +1$ where $d$ is some proper divisor of $n$ such that $\frac{n}{d}$ is odd or (2) $p=1 \pmod {2n}$ I'm tried to ...
-3
votes
0answers
26 views

Find sum of digits in $n^m$ [closed]

Given n and m, two numbers, n<10, m<10000 , find sum of digits in $n^m$ For example: when n=5 m=4 , value=625, sum=6+2+5=13
1
vote
0answers
52 views

Extending the domain of the “greatest common divisor” function.

In the seventeenth century the famous mathematician Leonhard Euler extends the concept of factorial $n!=n\cdot(n-1)\cdot(n-2)\cdot\ldots\cdot 2\cdot 1$ of $\mathbb{N}$ to non-integer values. Euler ...
7
votes
2answers
53 views

Find the $k$ of the formula $k\pi ^m=\sum_{n=0}^{\infty }\frac{1}{(2n+1+\frac{a}{3})^m}+\frac{1}{(2n+1-\frac{a}{3})^m}$

$$k\pi ^m=\sum_{n=0}^{\infty }\frac{1}{(2n+1+\frac{a}{3})^m}+\frac{1}{(2n+1-\frac{a}{3})^m}$$ when $a$ an even integer number if $$a\geq 4 $$ $$gcd(a,3)=1$$ I want to find $k$ with $m$ if $m$ is an ...
10
votes
2answers
527 views

Locating a paper in Euler's complete works

I'm currently reading Disquisitiones Arithmeticae and I keep seeing references to Euler such as "Comm. acad. Petrop., 8 [1736], 1741, 141". My question: How can I go about locating this paper in ...
0
votes
0answers
20 views

A subgroup of multiplicative group of a field

I can't seem to find a solution to this: Let $ K $ be a field of characteristic not equal to $ 2 $, $ K^* $ be its multiplicative group and $ K^{*2} = \{k^2, ~ k\in K^*\}$ Prove that if $ a \notin ...
0
votes
1answer
20 views

quadratic equation modulo some number

I read a post that $$ax^2+bx+c \equiv 1 \pmod p$$ can be solved in a similar way we solve a simple quadratic equation, just by replacing division by $2a$ by modulo inverse of $2a$ and square root of ...
2
votes
1answer
30 views

Remainder of a combination

Problem from a contest: What is the remainder when $\binom{169}{13}$ is divided by $13^5$? I thought that Wolstenholme's/Babbage's would help, but not entirely sure how.
0
votes
1answer
23 views

Representation of an inverse

I'm facing the following problem: Let $ K $ be a field of characteristic not equal to $ 2 $. Prove that if $ \alpha \in K $ is representable as $ x^2 - ay^2 $, then so is $ \alpha^{-1} $ Well, I ...
0
votes
1answer
15 views

Number theory,GCD, coprime integers

I am sorry for the bad title but I really can't think of a better one. So I was learning about the euclidean algorithm and I see a statement that is hard for me to understand. In the book that I was ...
0
votes
1answer
29 views

Is $f(n)= \sum_{1\leq i \leq n}\log(i) - \sum_{\text{p is prime},\ p\leq n} \log(p)^2$ a function of $\operatorname{O}(n^{\frac{1}{2}+\epsilon})$?

Is $$f(n)= \sum_{1\leq i \leq n}\log(i) - \sum_{\text{p is prime},\ p\leq n} \log(p)^2$$ a function of $\operatorname{O}(n^{\frac{1}{2}+\epsilon})$? if no, what do we know about its asymptotic ...
1
vote
1answer
34 views

How to find the number of divisors that are perfect squares and divisible by a number

Suppose $ n = 2^{14} 3^{9} 5^{8} 7^{10} 11^{3} 13^{5} 37^{10} $ , find the number of positive divisors that are both perfect squares and divisible by $ 2^{2}3^{4}5^{2}11^{2}$. It is quite simple to ...
3
votes
1answer
61 views

$p^3 + 2$ is prime if $p$ and $p^2 + 2$ are prime?

I'm self-learning number theory. I want to prove the following statement: $$p \text{ is prime } \land \text{ }p^2 + 2 \text{ is prime } \implies p^3 + 2 \text{ is prime }$$ I failed to do so, and I ...
0
votes
1answer
20 views

how find by simple iteration 5 decimal this function

how find by simple iteration 5 decimal this function I assumed that y1= sin(x) , and y2=5x-2 to draw them and get the intersection to get the initial x thus, I got x0= 0.5 what next how to get ...
0
votes
2answers
27 views

For which integer sequences $a_1,a_2,\dots,a_t$ is it true that a certain relationship with lcm and gcd holds?

The relationship is $\operatorname{gcd}(a_1,a_2,\dots,a_t)\cdot\operatorname{lcm}(a_1,a_2,\dots,a_t)=a_1\cdot a_2\cdot\dots\cdot a_t$. I think this holds for every sequence of length less than $3$, ...
3
votes
1answer
40 views

Need Proof of LCM inequality

Let $x_0<x_1<\dots<x_n$ be positive integers. Prove that $$ \sum_{i=0}^{n-1} \frac{1}{\text{lcm}(x_i,x_{i+1})}<1, $$ where $\text{lcm}(x,y)$ is the least common multiple of $x$ and $y$.
8
votes
3answers
103 views

How can I prove $\pi ^2=\sum_{n=0}^{\infty }\frac{1}{(2n+1+\frac{a}{3})^2}+\frac{1}{(2n+1-\frac{a}{3})^2}$

Proving this formula $$ \pi^{2} =\sum_{n\ =\ 0}^{\infty}\left[\,{1 \over \left(\,2n + 1 + a/3\,\right)^{2}} +{1 \over \left(\, 2n + 1 - a/3\,\right)^{2}}\,\right] $$ if $a$ an even integer number so ...
0
votes
1answer
25 views

shift power mod 1 of the cantor set by an irrational number and their intersections

Let $C$ be the Cantor ternary set and consider the shift $T_a$ mod 1 of the interval $[0,1]$ for an irrational number $a\in[0,1]$. I'm wondering whether $T_a^k(C)\cap T_a^l(C)=\emptyset$, $k,l\in ...
0
votes
0answers
12 views

About the complexity of Mersenne numbers

In this page: http://www.mersennewiki.org/index.php/Lucas-Lehmer_Test#Proof_of_the_Lucas-Lehmer_test In the end of this page I read this paragraph: The Lucas-Lehmer test, when used with the Fast ...
-1
votes
0answers
13 views

Assuming a natural number $m>50$, how many values of $m$ exist so that it divides $n^{n+1} + 1$ [closed]

Assuming a natural number $m>50$, how many values of $m$ exist so that it divides $n^{n+1} + 1$ where $n\ge0$ ($n$ is also natural)
4
votes
3answers
52 views

if $k$ is a positive integer and $G$ a finite group such that $G=\{x^k:x\in G\}$ , then is it true that g.c.d.$(|G|,k)=1$ ?

If $G$ is a finite group of order $n$ and $k$ is a positive integer such that g.c.d.$(n,k)=1$ , then I know that $G=\{x^k:x\in G\}$ ; is the converse true ? that is if $k$ is a positive integer and ...
7
votes
1answer
71 views

$x_1^2 + x_2^2+x_3^2 = p$

Prove that there are infinitely many prime numbers $p$ such that $x_1^2+x_2^2+x_3^2 = p$ has no solutions. So my attempt is the following. Let's look at residues modulo $8$. $x^2$ is either $4$ or ...
2
votes
2answers
46 views

Why $2^{3^n}=-1 \mod 3^{n+1}?$

Why $2^{3^n}=-1 \mod 3^{n+1}?$ By using the Euler totient function $\varphi$ I have got that $$ 2^{\varphi(3^{n+1})}=2^{3^{n+1}-3^n}=2^{2 \cdot 3^n}=1 \mod 3^{n+1}. $$ How to prove now that the ...
5
votes
1answer
151 views

Does this sequence of sets eventually contain all primes?

I was on Reddit earlier and answered a question about the usual proof that there are infinitely many primes: multiply any finite set of them, add 1, factor, and you get factors that are not in the ...
0
votes
1answer
19 views

Composition of binary quadratic forms as matrix operations

It is easy to see that any binary quadratic form $a^2 + 2bxy + cy^2$ is the same as $XAX^T$ where $X = [x, y]$ and $A = \begin{bmatrix}a & b\\b & c\end{bmatrix}.$ The composition of two ...
0
votes
0answers
31 views

Collatz Conjecture Format [closed]

Perhaps anyone has noticed larger forms of the Collatz Conjecture; $m$ and $q$ in the form: $$ f(n) = \begin{cases} n/m & \text{if } n \equiv 0 \mod m \\ (m+1)n + m-1 & \text{if ...
11
votes
1answer
165 views

How to evaluate $\sum_{\gcd (p,q)=1} \frac{1}{p^2q^2}$?

How do find the following sum $$ \sum_{\gcd (p,q)=1} \frac{1}{p^2q^2} $$
6
votes
2answers
85 views

Determine $\phi(2^{399}+1)$

This is a question I had on an exam so there is no access to calculators or software. I checked the answer using Maple after and as expected, the answer is very large and some of the prime factors are ...
2
votes
1answer
24 views

Is $f(n)= n - \sum_{p\leq n} \log(p)$ a function of $\operatorname{O}(n^{\frac{1}{2}})$?

Is $$f(n)= n - \sum_{\text{p is prime},\ p\leq n} \log(p)$$ a function of $\operatorname{O}(n^{\frac{1}{2}})$? I have written a Matlab program and it gave me the bellow plot that the $x$-axis is $n$ ...
4
votes
1answer
24 views

Is there an $n$ such that $p|n^2+1$ with $2n<p<2n+\sqrt n$?

Is there an integer $n$ such that $n^2+1$ is divisible by a prime $p$ with $2n<p<2n+\sqrt n$? It's complicated to describe my interest, but these are near-missed for arc-cotangent reducible ...
0
votes
0answers
21 views

making /adjusting a perodic signal given an equation

I know if I have the sin wave equation Asin(2pif*t+phase) I can increase/decrease/adjust the periodic frequency of the signal by changing f. But if I have the following equation below how can I also ...
5
votes
2answers
103 views

Proving this inequality $\sqrt[n]{x^n+\sqrt[n]{(2x)^n+\sqrt[n]{(3x)^n+\cdots}}}< (x+\frac{1}{n-1})$

How can I prove this inequality $$\sqrt[n]{x^n+\sqrt[n]{(2x)^n+\sqrt[n]{(3x)^n+\cdots}}}< \left(x+\frac{1}{n-1}\right)$$ if $n$ and $x$ are positive integer number $$x>=1$$ $$n>1$$
6
votes
0answers
58 views

Does each irreducible polynomial over the integers represent at least one prime? [duplicate]

There's not much more to the question: If $f(x) \in \mathbb{Z}[x]$ is an irreducible polynomial, is there a simple proof that there must be some $x$ for which $f(x)$ is a prime number (positive or ...
4
votes
1answer
104 views

$0=\frac{13+13^2+13^3+\cdots}{1+2+3+\cdots}$ using infinite sums?

This is not homework, just curiosity. My question arose from the apparent absurdity that $\zeta(-1)=-\frac{1}{12}$, even though $\sum_{n=1}^\infty \frac{1}{n^z}$ only makes sense when $Re(z)>1$. ...
1
vote
0answers
43 views

Golbach's partitions: is there always one common prime in G(n) and G(n+6) , n greater or equal to 8 (or a counterexample)?

I am trying to find a counterexample for the following expression when d=6. (G(n) = Goldbach partition of the even number n) ${\forall}$ n=2*k / k${\in}$N, n${\geq}$8 ${\exists}$(${p_i}$,${p_j}$) / ...
2
votes
1answer
93 views

Contradicting statements about the Riemann zeta function at positive odd integers

I have found two contradicting statements about the value of $\zeta(k)$ when $k=2n+1$ and $n\in\mathbb{Z_0^+}$. Which one is correct? "The Riemann zeta function for odd integers has no known ...
1
vote
2answers
24 views

How find this sum $\sum_{i=1}^{n}(r(i)+\sigma{(i)}$

Give the postive integer number $n$,for any $i=1,2,\cdots,n$ (1):Let $r({i})$ be the remainder of the division of $n$ by $i$, (2): $\sigma{(n)}$ is denote the sum of the (positive) ...
1
vote
1answer
66 views

Number Sense Math

I'm third grade. Shay. I need help explaining how to get this answer. Question: Kyle is thinking of a four digit number in which all the digits are different and ¾ of the digits are odd. What could ...
1
vote
1answer
17 views

Quadratic Residue modulo $nm$

Let $m$ and $n$ be relatively prime and $b \in (\mathbb Z/ mn \mathbb Z)^\times$. Then $b$ is a quadratic residue modulo $mn$ if and only if $b$ is a quadratic residue modulo $m$ and modulo $n$. I ...
1
vote
0answers
27 views

Algorithm to compute prime factors of n from whether x is a square?

Suppose there exists an algorithm that takes input $x \in \mathbb{Z}_{n}^{*}$ and returns the square root of $x$ if $x$ is a perfect square and nothing otherwise. Use this algorithm to compute the ...
2
votes
1answer
40 views

Possible values of $\gcd(a+b, a\times b)$

Main Question: Let $N \in \mathbb{N}$. What are the possible values of $\gcd(a+b, a\times b)$ given that $\gcd(a,b) = N$? Fact 0. If $\gcd(a,b) = N$, then $N \leq \gcd(a+b, a\times b) \leq ...
-4
votes
1answer
40 views

Is there anyone who could help me with this problems? [closed]

Let $R=\mathbb{Z}[\sqrt7]=\{ a+b\sqrt7 ~| a,b\in \mathbb{Z}\} $ define addition and multiplication by $(a_1+b_1\sqrt7)+(a_2+b_2\sqrt7)=(a_1+b_2)+(b_1+b_2)\sqrt7$ ...