Questions on more advanced topics of number theory. Consider first if (elementary-number-theory) might be a more appropriate tag before adding this tag.
1
vote
0answers
35 views
Extending a rational entry matrix to an orthogonal matrix.
Suppose $M \leq N$ are positive integers and let $r_1,\ldots,r_M$ represent the rows of an $M\times N$ matrix $A$ in which: (i) the rows of $A$ are orthogonal and having the same norm, (ii) the ...
0
votes
2answers
29 views
Square-Trangular Numbers Checking Answer
Problem: The first 2 numbers that are both squares and triangles are 1 and 36. Find the next one and if possible, the one after that.
Answer: 1225, 41616
Problem: Can you figure out an efficient way ...
2
votes
1answer
28 views
Question about the definition of representability of a quadratic form
Say I have an integral quadratic form $q$ on $\mathbb{Z}^r$ and another integral binary quadratic form $\tilde{q}$. What does it mean for $q$ to primitively represent $\tilde{q}$? I can't seem to find ...
13
votes
3answers
351 views
Fermat Last Theorem for non Integer Exponents
We now that Fermat's last theorem is true so there are not positive integer solutions to
$$x^n+y^n=z^n$$
for $n\in\mathbb{N}$ and $n>2$.
But what about if $n\in\mathbb{R}$ or $n\in\mathbb{R}^+$?
4
votes
1answer
51 views
$p^{3}+m^{2}$ is square of a number.
Well i thought it is a nice problem so i will post it here.
1) Prove that for every natural numbers $m$, There is at most two primes $p$ where $p^{3}+m^{2}$ is the square of a number.
2) Find all ...
0
votes
0answers
123 views
How to prove these two ways give the same numbers?
How to prove these two ways give the same numbers?
Way 1:
...
4
votes
0answers
45 views
Free algebra over $\mathbb{Z}/N\mathbb{Z}$
Let $A$ be a commutative finite free $\mathbb{Z}/N\mathbb{Z}$ algebra of rank 2 with unit discriminant.
I have two questions :
1) Why is it true that $A/pA$ is isomorphic to either ...
3
votes
1answer
54 views
Minimum number of coconuts
Three friends namely $A$, $B$ and $C$ collected coconuts with the help of monkey and fell asleep. At night, $A$ woke up and decided to have his share. He divided coconuts into three shares, gave the ...
10
votes
2answers
192 views
$x^2+x+1$ is the cube of a prime.
Please help me find all natural numbers $x$ so that $x^2+x+1$ is the cube of a prime number.(Used in here)
3
votes
4answers
40 views
Proving $\frac{1}{2}(5x+4),\;2 < x,,\;\text{isPrime}(n)\Rightarrow n = 10k+7$
How is it possible to establish proof for the following statement?
$$n = \frac{1}{2}(5x+4),\;2<x,\;\text{isPrime}(n)\;\Rightarrow\;n=10k+7$$
Where $n,x,k$ are $\text{integers}$.
To be more ...
3
votes
1answer
39 views
Knowing that $n= 3598057$ is a product of two different prime numbers and that 20779 a square root of $1$ mod $n$, find prime factorization of $n$.
Knowing that $n= 3598057$ is a product of two different prime numbers and that 20779 is a square root of $1$ mod $n$, find prime factorization of $n$.
What I have done so far:
$n = p \cdot q$
...
4
votes
1answer
65 views
Property of natural numbers involving the sum of digits
How can you prove that every natural number $M$ or $M+1$ can be written as $k + \operatorname{Sum}(k)$, where $\operatorname{Sum}(k)$ represents the sum of the digits of some number k.
Example:
$$
...
0
votes
0answers
26 views
number property [duplicate]
How can you prove that every natural number M or M+1 can be written as k + Sum(k), where Sum(k) represents the digits sum of number k.
Example:
248 = 241 + Sum(241) = 241 + 2 + 4 + 1
-4
votes
0answers
79 views
Is not a perfect square
If $\frac{m+2}{2}<n<m$ where $m,n$ posivite integers , show that the $2^{2n-2}-2^m+1$ is not a perfect square.
The question changed!!!
4
votes
1answer
28 views
Trying to understand Theorem 2.27 in a recent paper on the Chebyshev function
In February 2013, Sadegh Nazardonyavi and Semyon Yakubovich posted on arxiv: Sharper estimates for Chebyshev's functions $\vartheta$ and $\psi$.
I have a question about Theorem 2.27 on page 22.
My ...
0
votes
1answer
51 views
probability of divisibility
Let S be the sum of k randomly selected integers between 1 and n.
What is the probability of S being divisible q?
Can this be expressed in a closed form?
This is the generalization of one of the ...
0
votes
4answers
29 views
Prove that $a^{(p-1)/2} $$\equiv$-1 (modp). Deduce that if $a, b$ are primitive roots modp, then $a\times b$ is NOT a primitive root mod p.
Let $a$ be a primitive root mod the odd prime p. Prove that $a^{(p-1)/2} $$\equiv$-1 (modp). Deduce that if $a, b$ are primitive roots modp, then $a\times b$ is NOT a primitive root mod p.
Here ...
3
votes
1answer
54 views
How to prove the existence of infinitely many $n$ in $\mathbb{N}$,such that $(n^2+k)|n!$
Show there exist infinitely many $n$ $\in \mathbb{N}$,such that
$(n^2+k)|n!$ and $k\in N$
I have a similar problem:
Show that there are infinitely many $n \in \mathbb{N}$,such that
...
4
votes
0answers
64 views
Sum of rational numbers given some properties
Let $R(n)$ denote the sum of all positive rational numbers whose numerators and denominators are less than or equal to $n$ and have no common factors. I have estimated this sum to be
$$
\begin{align*}
...
1
vote
2answers
46 views
How to find the limit for the quotient of the least number $K_n$ such that the partial sum of the harmonic series $\geq n$
Let $$S_n=1+1/2+\cdots+1/n.$$ Denote by $K_n$ the least subscript $k$ such that $S_k\geq n$. Find the limit $$\lim_{n\to\infty}\frac{K_{n+1}}{K_n}\quad ?$$
4
votes
3answers
72 views
Deduce that there exists a prime $p$ where $p$ divides $x^2 +2$ and $p≡3$ (mod 4)
I am revising for a number theory exam and have a question that I am struggling with, any help would be greatly appreciated.
First I am asked to show that for an odd number $x$, $x^2+2 ≡3$(mod 4).
...
-3
votes
0answers
29 views
Theory number gcd [duplicate]
Taking the example
Prove $\left(\left(a^{m}-1\right)/(a-1),a-1\right)=(a-1,m)$
Demonstração
$d = (a^{m−1} + a^{m−2} + \ldots + a + 1, a − 1)=
(a^{m−1} − 1) + (a^{m−2} − 1) + \ldots + (a − 1) + m, a ...
-1
votes
1answer
27 views
Prove $\gcd\left((a^{2m}−1)/(a+1),a+1\right)=\gcd(a+1,2m)$ [duplicate]
Show or prove that
$$
\gcd\left(\frac{a^{2m}−1}{a+1},a+1\right)=\gcd(a+1,2m),
$$
and that
$$
\gcd\left(\frac{a^{2m+1}+1}{a+1},a+1\right)=\gcd(a+1,2m+1).
$$
3
votes
3answers
147 views
Does the decimal representation of $\pi$ contain almost all numbers?
Let $\Pi \in \Sigma^\infty$ be the decimal representation of $\pi$ where $\Sigma=\{0..9\}$. It is not known whether $\Pi$ contains each natural number. On the other hand, no number is known not to ...
5
votes
3answers
192 views
What would be the immediate implications of a formula for prime numbers?
What would be the immediate implications for Math (or sciences as a general) if someone developed a formula capable of generating every prime number progressively and perfectly, also able to prove (or ...
2
votes
0answers
51 views
About linear recurrence sequences
Let $\{a_n\}_{n=0}^\infty$,$\{b_n\}_{n=0}^\infty$,$\{c_n\}_{n=0}^\infty$ be three complex sequences and satisfy
\begin{eqnarray*}
&&\sum_{k=0}^2\alpha_ka_{n+k}=0,\\
...
5
votes
2answers
53 views
About the infinite solutions of a Diophantine equation
Consider the following problem:
$$\sum_{k=1}^N k^2=q^2$$
where q is an integer number. This can be written as:
$$\frac{1}{3}N^3+\frac{1}{2}N^2+\frac{1}{6}N=q^2$$
In the same way we can write:
...
1
vote
1answer
29 views
Lengendre symbol calculation
I'm trying to calculate the lengendre symbol of (3/383) without using the Quadratic Reciprocity Law, and with not much success.
I've thought about checking if 2^191 is congroent to 1 modulo 383 but it ...
2
votes
1answer
32 views
Solving a congruence equation
Let $\pi$ be a primitive $p$-th root of unity, $p$ some prime and
$$ x= \sqrt{ \left(\frac{ \pi^k -1}{\pi-1}\right) \overline{\left(\frac{ \pi^k -1}{\pi-1}\right)} } \in \mathbb{Z}[\pi]$$
for some ...
0
votes
2answers
29 views
finding numbers from LCM and HCF
Let two numbers be x,y and their LCM and HCF be L and H respectively.Then,
$7H=L$ and
$x+y=392$
How can I find x and y?I have tried to use $7 (\gcd)^2=xy$ but nothing came out of it.Any hints?
1
vote
0answers
28 views
How to show this space is complete?
Let $\mathbb F_q$ be the finite field of $q$ elements. We let $K_{\infty}$ to be the field of formal power series in $x^{-1}$ over $\mathbb F_q$. If
$$
\alpha = \sum_{- \infty}^{t} a_i x^i,
$$
then ...
3
votes
1answer
66 views
effective way to get the integer sequence A181392 from oeis
the sequence A181392 are perfect squares and any digit in the sequence says
"I am part of an integer in which you'll find d digits "d"" (see A108571, How can we call them? "digit-valid"?)
How to get ...
2
votes
0answers
114 views
primes of the form $p=8k+1, 8k+3$ can be expressed as $p=a^2+2b^2$
I have trouble showing that primes of the form $p=8k+1, 8k+3$ can be expressed as $p=a^2+2b^2$.
Thanks in advance.
1
vote
0answers
21 views
Modified Arithmetic-Geometric Mean
Let $\{x_n\}$ and $\{y_n\}$ be defined iteratively, $x_0:=\beta >1, \ y_0:= 1$ and $x_{n+1}= \frac{x_n+y_n}{2}$, $y_{n+1} = (x_n.y_n)^{\frac{1}{2}}$; i.e. they are respectively the arithmetic and ...
1
vote
2answers
45 views
On the no trivial $3$-tuples $(p, q, \alpha) \in \mathbb{N}^3$ such that $\sum_{k = 1}^{n}k^p =\Big [\sum_{k=1}^{n}k^q\Big]^\alpha $.
It is well known that $\sum_{k = 1}^{n}k^3 =\Big [\sum_{k=1}^{n}k^1\Big]^2$. My question is very simple.
There are $3$-tuples $(p, q, \alpha) \in
\mathbb{N}\times\mathbb{N}\times\mathbb{N}$, ...
4
votes
1answer
86 views
Binary vs. Ternary Goldbach Conjecture
Is there an "understandable" explanation of why the ternary Goldbach conjecture is tractable with current methods, while the binary Goldbach conjecture seems to be out of scope with current ...
4
votes
2answers
83 views
Find the greatest integer $k$ for which $1991^k$ divides $1990^{{1991}^{1992}}+1992^{{1991}^{1990}}$
Find the greatest integer $k$ for which $1991^k$ divides $$1990^{{1991}^{1992}}+1992^{{1991}^{1990}}$$
It is easy to see that $k \geq 1$ as $1990 \equiv -1$ and $1992 \equiv 1 \pmod{1991}$
Also, I ...
1
vote
2answers
37 views
power of a number in a factorial
what is the largest power of 24 in 150! ?
HINT : answer is 48
I need to know the method for solving such questions when the ...
1
vote
1answer
60 views
Number array divided into several parts, genelize $a>b>c>d>0$ so $ab+cd>ac+bd$ to more numbers
Now, we have an original number array: $$a_1 > a_2 > a_3 > ... > a_{mn} > 0$$, I wonder whether the following inequality is the truth, if so, could you give me the proof or some ...
2
votes
1answer
46 views
Prove that $n^2 \leq \frac{c^n+c^{-n}-2}{c+c^{-1}-2}$.
Let $c \not= 1$ be a real positive number, and let $n$ be a positive integer.
Prove that $$n^2 \leq \frac{c^n+c^{-n}-2}{c+c^{-1}-2}.$$
My initial thought was to try and induct on $n$, but the ...
7
votes
3answers
270 views
Perfect squares using 20 1's, 20 2's and 20 3's.
How many perfect squares can be formed using 20 1's, 20 2's and 20 3's. This is a recent exam question, which I had no clue how to solve?
There is some kind of trick here, since time allotted to ...
1
vote
0answers
28 views
Could a determinstic primality test specialized to this form of prime exist?
Is it possible there could be an "efficient" deterministic primality test for prime numbers of the form
$$(2^n + 1)^2 - 2$$
or
$$(2^n - 1)^2 - 2$$
in the same vein as the Lucas-Lehmer test for ...
6
votes
1answer
128 views
All number fields with absolute value of discriminant $\le 20$
I need to find all number fields with absolute value of discriminant $\le 20$.
Using Minkovsky's theorem I understood that it should be quadric or cubic extension. The case of quadric is very ...
0
votes
2answers
38 views
divisibility observation in particular patterns
Please look at the following observation made by trial and error method.
Let us take some 2-digit numbers like 12, 15, 24,...
12 = 1 * 2 = 2 => 2|12 (2 divides 12)
15 = 1 * 5 = 5 => 5|15
for 3 ...
1
vote
0answers
17 views
Synthetic Division with mods
$x^4+x+1$/
$2x^2$+1
In $F_5$ (means mod 5)
I said let the leading coefficient be 2.
Since $3$ $*$ $2$ - $5$ $*$ $1$ $=$ $1$, choose 3 to multiply
$3$($2x^2$ $+$ $1$)= $x^2$ + $3$ (this is ...
1
vote
1answer
58 views
Does this number belong to the set of real numbers?
Suppose we build a number in this way:
we put the natural numbers one after the other. For example, for the first 5 numbers:
$n_1=1,n_2=2,n_3=3,n_4=4,n_5=5$ we obtaine a new number $N_5=12345$. For ...
1
vote
0answers
43 views
Is zero a cluster point of $n\sin n$?
I have seen somewhere that if $0\le \alpha<1$, then zero is a cluster point of the sequence $n^\alpha \sin n, n=1,2,\cdots$.
My question is what if $\alpha=1$? Or $\alpha>1$?
1
vote
2answers
30 views
GgT, (polynomial) division and finite fields…
Exercise: Let $f,g \in \mathbb{Z}_2[x]$ be the polynomials $f = x^6 + x^5 + x^4 + 1$ and $g = x^5 + x^4 + x^3 + 1$. Has the diophantic equation $f u + g u = x^4 + 1$ solutions $u,v \in \mathbb{Z}[x]$? ...
1
vote
1answer
40 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 ...
0
votes
1answer
34 views
Find quotient space on $\mathbb{N} $
On $\mathbb{N}$ is given equivalence relation R with $nRm \iff 4|n-m$. Topology on $\mathbb{N}$ is defined with $\tau=\{\emptyset\}\cup\{U\subseteq\mathbb{N}|n\in U \wedge m|n \implies m\in U\}$.
I ...
