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Questions tagged [number-theory]

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.

2
votes
2answers
31 views

Property prime powers

Let us suppose that $p$ is an odd prime number and and $n$ a natural number greater than $2$. If $R$ is an integer such that $R \not \equiv 1 \pmod{p^{n-1}}$ and $R^{p} \equiv 1\pmod{p^{n-1}}$, does ...
6
votes
0answers
67 views

Displaying prime numbers in two dimensions, $N$-parallelograms, and “missing primes”

First, I would like to introduce a peculiar way to display the prime numbers (greater than $9$) by means of the ten they belong to ($x$-axis), and their ending digit ($y$-axis). Here's an example of ...
3
votes
1answer
41 views

Sum of products of $m$-tuples chosen from the set of squared reciprocals

Let $S = \{1/n^2 : n \in \mathbb{N} \}$. We know $\sum S = \zeta(2) = \pi^2/ 6$. Let $f(S, m)$ be the sum of the products of all $m$-tuples chosen from $S$. That is $$f(S,m) = \sum_{X \in {S \...
1
vote
0answers
28 views

The perimeter of a triangle is a triangular number with the lengths of all three sides also triangular numbers.

Such a triangle does not exist for perimeters $1,6,10,28,55$. Are there any others?
0
votes
0answers
32 views

Find the cardinality of the set $A_p$ defined as the following : [duplicate]

For any prime number $p$, $A_p$=the set of integers $d\in \{1,2,3,\dots, n\}$ such that the power of $p$ in the prime factorization of $d$ is odd. Then \begin{align*} A_p= & \lfloor\dfrac{n}{p}\...
4
votes
0answers
52 views

About corollary of sylow $p$-subgroup of finite group.

I am facing a problem in group theory question or better say a doubt. If anyone can help me then please do try to solve this question. A couple of days ago, we were discussing a corollary of Sylow’s ...
0
votes
1answer
57 views

$n$ such that the digits immediately after the decimal point of $\pi^n$ give $n$ again

I was doing something with value of $\pi$ as I know that the beauty of numbers will always exist , doesn't matter either number is real or complex it must be beautiful. I observe something strange by ...
4
votes
1answer
89 views

Is Fermat's last theorem provable in Peano arithmetic?

The sentence $S$ which Gödel in his proof of the incompleteness theorem proves to be be unprovable in the system of Peano arithmetic can be proved (as a true theorem of PA) outside PA (and necessarily ...
20
votes
3answers
1k views

A conjecture involving prime numbers and parallelograms

As already introduced in this post, given the series of prime numbers greater than $7$, let organize them in four rows, according to their last digit ($1,3,7$ or $9$). The column in which they are ...
9
votes
0answers
108 views

Binomial Coefficients involving Prime Powers Minus 1

I would like to show the following is true; Let $p\in\mathbb{P}, \alpha,n\in\mathbb{N}$. For $$p^\alpha\mid\sum_{k=1}^{n-1}\binom{(p^\alpha-1)n}{(p^\alpha -1)k}$$ I've never worked with prime ...
1
vote
1answer
94 views

The equation $x^4 − y^4 = z^p$

In Darmon's paper The equation $x^4 − y^4 = z^p$, he says: "Factorizing the left hand side of $a^4 − b^4 = c^p$, the assumption $\gcd(a, b) = 1$ forces the three factors $a + b, a − b, a^2 + b^2$ to ...
8
votes
0answers
79 views

Are there infinite many solutions of $\ \ |\varphi(n+1)-\varphi(n)|=2\ \ $?

The solutions of the equation $$|\varphi(n+1)-\varphi(n)|=2$$ upto $\ \ n=10^8-1\ \ $ are (the first entries of the arrays) : ...
-4
votes
2answers
43 views

Number system(perfect squares) [on hold]

Can the difference of any two perfect square numbers be equal to a perfect square? How many integral pair of $(x, y)$ satisfy the following condition.. $$9x^2 - 4y^2 = 1600$$
0
votes
1answer
25 views

A number theory problem concerning effects of shifting digits [on hold]

Find a positive integer whose first digit is 1 and which has the property that, if this digit is transferred to the end of the number, the number is tripled. A full proof will be much appreciated!
5
votes
1answer
57 views

For any $n$ there's a power of $2$ which contains $n$

So I saw this problem in an Olympiad book, "Prove that for any natural number $n$, there exists a power of $2$ which contains $n$ in it. " For example, $n=19$ is in $2^{13}=8192$, $n=24$ is in $2^{...
0
votes
0answers
13 views

Square integrable Maass forms are cusp forms, local version near a cusp

Let $Y$ be a finite volume quotient of the upper half plane. I can show that a weight $0$ nonharmonic Maass form is cusp iff it is square integrable. One direction uses only local information near a ...
1
vote
0answers
70 views

Factorial of a trillion [on hold]

How do I find the last $5$ nonzero digits of the factorial of $10^{12}$? The computer can quickly find the factorial of up to $10^6$. However, above that order of magnitude it takes days. I used $$...
1
vote
2answers
46 views

Is $\dfrac{a^2+b^2}{6}\leq \dfrac{a^2+b^2}{3}+\dfrac{ab}{3}\leq \dfrac{a^2+b^2}{2}$ for any $a,b\in\mathbb R$?

Is $\dfrac{a^2+b^2}{6}\leq \dfrac{a^2+b^2}{3}+\dfrac{ab}{3}\leq \dfrac{a^2+b^2}{2}$ for any $a,b\in\mathbb R$ ? For $a$ and $b$ are both positive or both negative,I proved this. But I am not able to ...
5
votes
2answers
130 views

Does there exist positive rational $s$ for which $\zeta(s)$ is a positive integer?

Does there exist positive rational $s$ for which $\zeta(s) \in N$ or equivalently, does there exist finite positive integers $\ell,m$ and $n$ such that $$\zeta\left(1+\dfrac{\ell}{m}\right) = n$$ I ...
0
votes
0answers
13 views

How to prove the mersenne number is not pseudoprimes to the base 3?

i start think to take a small factor $p$ of $M_n$ after supposing $M_n$ is composite, then i reached that $ord_{p}(3)\mid M_{n}-1$ Now how i can continue to reached that $p$ is the same as $M_n$ ??
2
votes
2answers
62 views

If the equation $2x+2y+z=n$ has $28$ solutions, find the possible values of $n$

Let $n$ be a positive integer. If the equation $2x+2y+z=n$ has $28$ solutions, find the possible values of $n$. I tried taking $2x$ as $a$, and then $2y$ as $b$, and then finding the possibilities. ...
1
vote
2answers
60 views

Function that tells if number is prime, perfect square or other composite.

There was a math function (studied by Euler, but I might be wrong) that takes a number and maps it on one of three classes: "prime", "perfect square", "composite, but not square" represented as $-1$...
0
votes
0answers
16 views

Proof that Zsigmondy Numbers are the Maximal Overpseudoprimes

It is stated in https://oeis.org/A064078 that Zsigmondy numbers, Z, are the maximal overpseudoprimes base 2 that divide $2^n-1$, where n is the multiplicative order of 2 modulo Z ($2^n-1\equiv 0\mod Z$...
1
vote
1answer
31 views

For $a, b>1$, $(a,b)=1$ iff $a$ does not divide $b$ and $b$ does not divide $a$?

I am guessing the following statement is true: For two integers $a$ and $b$ greater than $1$, $(a,b)=1$ if and only if $a$ does not divide $b$ and $b$ does not divide $a$. Could anyone please ...
0
votes
0answers
14 views

How many different ways three two-digit numbers can be chosen so that their product ends in four or more zeros.

How many different ways three two-digit numbers can be chosen so that their product ends in four or more zeros. I could'n do more: three two-digit numbers - $XX$, $YY$, $ZZ$ $XX$, $YY$, $ZZ$ are ...
0
votes
0answers
24 views

Is the condition defining a Ramanujan graph equivalent to the Ramanujan conjecture ?

I just learnt that a $ d $-regular graph $ G $ is a Ramanujan graph if $\lambda(G)\leq 2\sqrt{d-1} $, and that this condition is equivalent to the Ihara zeta function fulfilling the analog of RH. ...
0
votes
0answers
41 views

To find minimum value of “$b$” in the equation $x^2+ax+b =0$ for some given conditions.

Qs. For a natural no. '$b$', let $N(b)$ denotes the no. of natural numbers '$a$' for which the equation $x^2+ax+b =0$ has integer roots. What is the smallest value of $b$ for which $N(b) =20$? Now, ...
2
votes
0answers
68 views
+50

find the maximum of the value $c$ such $ \{ a^2 \} + \{ b^2 \} \leqslant 2 - \frac{c}{(a + b)^2} $

Suppose that $a, b$ are postive numbers and $a + b \in \mathbb{Z} {}_+$. Find the maximal constant $c$, s.t. $$ \{ a^2 \} + \{ b^2 \} \leqslant 2 - \frac{c}{(a + b)^2} $$ for all $a, b$. Here $\{ x \}$...
0
votes
2answers
27 views

Identify elements involved in a log product

Since we know that $\log_2 ab = \log_2 a + \log_2 b$, is there a way to figure out numerical values of $a$ and $b$ (or even $\log_2 a$ and $\log_2 b$) if we are just given the value of $\log_2 ab$?
-1
votes
1answer
24 views

NUMBER system and counting

Let $S =\{1,2,3,...,20\}$ be the set of all positive from $1$ to $20$ suppose that $N$ is the smallest positive integer such that exactly eighteen numbers from $S$ are factors of $N$ and the only two ...
4
votes
3answers
74 views

If $S(n)$ is an odd integer, what is the sum of all possible $\frac1n?$ [on hold]

If $n$ is a positive integer, let $S(n)$ be the sum of all the positive divisors of $n$. If $S(n)$ is an odd integer, what is the sum of all possible $\frac1n?$ The function $S$ is multiplicative ...
39
votes
4answers
4k views

A conjecture involving prime numbers and circles

Given the series of prime numbers greater than $7$, we organize them in four rows, according to their last digit ($1,3,7$ or $9$). The column in which they are displayed is the ten to which they ...
0
votes
0answers
17 views

Motivation behind proof of Behrend's theorem on size of AP free subset

Behrend's theorem regarding AP states that There exists an absolute constant $c$ such that for all sufficiently large integers $N$ there exists a subset $A$ of $\{1, 2, \cdots, N \}$ with at ...
-9
votes
0answers
96 views

I want help to test my prime theorem [on hold]

Formula for primes In number theory, a formula for primes is a formula generating the prime numbers, exactly and without exception. Such formula which is efficiently computable. A number of ...
0
votes
1answer
26 views

Linear Diophantine Equation Signs

How to determine the coefficient signs for the solution of a linear diophantine equation? Take $24x + 69y = 33$ for example. I know the solution is $x = 33 − 23k , y = −11 + 8k$, and I understand ...
2
votes
1answer
39 views

Does there exist a prime p for each natural n such that all polynomials in Zp of nth degree or less with coefficients in [-n,n]∩Z split?

Does there exist a prime $p$ for each natural number $n$ such that all $n$th degree or less polynomials in $\mathbb{Z}_p$ with coefficients in $[-n,n] \cap \mathbb{Z}$ split? Motivation: If you have ...
0
votes
0answers
21 views

A = 15 (mod B); B = 19 (mod A); Find the sum of all R such that A+B = R (mod |A-B|)

The actual question is phrased as such: A divided by B has a remainder of 15, B divided by A has a remainder of 19. Find the sum of all remainders that result from the division $\frac{A+B}{\left | A-B ...
0
votes
0answers
17 views

Can we prove $B (n) = \frac{1}{4} G (n - 1) G (n)$ is an indicator function that takes on the value 1 for 'bad' Gram points?

Let \begin{equation} g (n) = 2 \pi e^{1 + W \left( \frac{8 n + 1}{8 e} \right)} \end{equation} be the approximate value of the $n$-th Gram point. Let \begin{equation} G (n) = \frac{Z (g (n))}{...
2
votes
0answers
27 views

Conjecture about congruences arising from a special semiprime

Let $k$ be a positive integer such that $p=2k+1$ and $q=4k+1$ are both prime. Consider the number $$N=pq$$ I proved that for every positive integer $a$ coprime to $N$ we have $$a^{N-1}\equiv 1\mod N$$ ...
0
votes
1answer
40 views

For any prime number $p$, let $A_p$ be the set of integers $d\in \{1,2,\dots, 999\}$ s. Then what is the cardinality of $A_p$?

For any prime number $p$, let $A_p$ be the set of integers $d\in \{1,2,\dots, 999\}$ such that the power of $p$ in the prime factorization of $d$ is odd. Then what is the cardinality of $A_p$? I ...
-1
votes
1answer
65 views

A Problem on Prime and Composite Numbers relation

I was fighting with this question about 5 days and I don't able to get an mathematical proof. Question:"Let imagine a natural number and a prime number 'q' and 'k' respectively such that (k>q and 'k' ...
-1
votes
1answer
46 views

Cool Modular Arithmetics. [on hold]

Let $a$ be the remainder when $1124^{2017}$ is divided by $2017$. Find the value of $a^{2048}$ $(\text{mod } 45)$
0
votes
0answers
29 views

given a Diophantine equation $ax+by=k$ for which $k$ values ​have a solution

given a Diophantine equation $ax+by=k$ and $a,b$ are natural numbers and there $gcd(a,b)=3$ . for which $k$ values ​​have a solution? i try do some examples like $3x+3y=2$ and we see that there is ...
1
vote
1answer
35 views

Reference for Euler totient function identity?

I'm trying to find a reference for an identity I found on the Wikipedia page on the Euler totient function: Let $n,m > 1$ be integers and let $\omega(m)$ be the prime omega function, then $$ \sum_{...
3
votes
1answer
42 views

First four numbers of a sequence are $2,0,1,8,$. Each next is the last digit of the sum of the preceding four numbers. Will $2,0,1,8$ show up again?

Let $a_n$ is a sequence of numbers. The firs four numbers are $2,0,1,8$ and each following number is the last digit of the sum of the preceding four numbers. The first ten numbers are $2,0,1,8,1,0,0,9,...
2
votes
1answer
40 views

Continuous automorphisms of $\mathbb{C}_p$

Goodmorning my queston is: Every continuous automoprhism of $\mathbb{C}_p$ is on the form $\sigma(x)=lim_{n\to \infty}\bar{\sigma}{(x_n})$ where $x =lim_{n\to \infty}x_n $ and $\bar{\sigma}\in ...
0
votes
0answers
21 views

ratio of Sum of divisors and number

If I plot the sum of the divisors of n divided by n, and plot that against n on a log scale, there are some interesting features in the figure. I can see hockey stick-like lines going through the ...
2
votes
0answers
34 views

Is there a distinguishable characteristic between the summation / continuous fraction method in algebraic and transcendental numbers?

I try to illustrate the formation and derivation of algebraic numbers and transcendental numbers. I found that both categories of numbers can be made by continuous summation or division/fraction ...
1
vote
0answers
42 views

The sieve formula choosen in Zhang's breakthrough work in Twin Prime conjecture

In the breakthrough work of the proof of weak twin prime conjecture, Goldstone, Pintz and Yildirim as well as Zhang use the following modified Selberg sieve: $v=\lambda^2$ where $\lambda(n)$takes ...
1
vote
0answers
51 views

Some confusion with Godel, Escher, Bach

In the first part of chapter 3 'figure and ground', Hofstadter is discussing a formal system to represent multiplication, then making use of this system to create another for composite numbers. RULE: ...