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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1answer
92 views

For any positive integer n, find the range of the product 1∗3∗5∗7∗9∗…∗(2n−1) in terms of n. [closed]

For any positive integer $n$, find the range of the product $1*3*5*7*9* \ldots *(2n-1)$ in terms of $n$. I have posted my answer...
1
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2answers
50 views

Primitive Root question

Question: Show that if $m$ is a positive integer and $a$ is an integer relatively prime to $m$ such that $ord_{m}a = m-1$, then $m$ is prime. So if you could give me guidance and explanations of ...
11
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2answers
508 views

Solving the diophantine equation $y^{2}=x^{3}-2$

It is known that the diophantine equation $y^{2}=x^{3}-2$ has only one positive integer solution $(x,y)=(3,5)$. The proof of it can be read from the book "About Indeterminate Equation" (in Chinese, by ...
12
votes
2answers
234 views

How prove this $\sqrt[5]{1782+\sqrt[3]{35+15\sqrt{6}}+\cdots}$ is positive integer numbers.

Prove that $$\sqrt[5]{1782+405\sqrt[3]{35+15\sqrt{6}}+405\sqrt[3]{35-15\sqrt{6}}}-\sqrt[3]{35+15\sqrt{6}}-\sqrt[3]{35-15\sqrt{6}}\in N$$ This problem from this My try: let ...
1
vote
5answers
792 views

Actual math problem re: Number theory regarding divisibility rules

Here is the problem that supposedly has a solution and it is an extra credit problem for my 6th grade godchild. None of us can figure it out :( Q: For the number ABC, each distinct letter represents ...
3
votes
1answer
55 views

Approximating the square root of two with fractions

I would like to prove that there exist only finitely many $m, n \in \mathbb{N}$ satisfying $$\left | \sqrt{2} - \frac{m}{n} \right | < \frac{1}{4n^2}.$$ Any thoughts? Thank you for your help.
1
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1answer
50 views

If $p,q$ distinct primes with $p,q \equiv 1 \bmod 4$, show that $x^2 \equiv -1 \pmod {pq}$ is solveable

I can't seem to get anywhere with this problem. Any hints would be much appreciated: Suppose that $p$ and $q$ are distinct primes satisfying $p, q \equiv 1 \bmod{4}$. Show that the congruence $x^2 ...
1
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2answers
69 views

About sum of three squares

I am trying to find those $k$ for which the expression $1+(10k+4)^2 +(10m+8)^2$ is never a square number for any $m$. Thank you!
0
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1answer
58 views

Help with intro to number theory question

Let $a$, $b$, $m$ and $n$ be integers with $m > 0$ and $n > 0$. If $(n,m)\mid(a−b)$, then the system $$\begin{cases} x\equiv a\pmod m \\ x\equiv b\pmod m \end{cases} $$ has a solution. ...
0
votes
1answer
91 views

Numbers with integer multiples using only digits $2$ and $6$ (Austria Mathematical Olympiad 2006)

Let $N$ be a positive integer. How many non-negative integers $n ≤ N$ are there that have an integer multiple, that only uses the digits $2$ and $6$ in decimal representation? Obviously, $n$ can't be ...
2
votes
1answer
98 views

How find this minimum of the value of $n$( 2013 china Mathematical olympiad simulation test )

Let $S=\{1,2,3,\cdots,n\}$. Find the minimum of positive integer $n$, such that for any partition of $S$ into $A$ and $B$, $$A\cap B =\emptyset,A\cup B=S$$ then at least one of the subsets $A$ ...
0
votes
1answer
568 views

Prove there are infinitely many *primitive* solutions to $x^2 + y^2 = z^4$ [duplicate]

For x, y, z $\in \mathbb N $ where $\gcd(x,y) = 1$ These solutions must also be primitive. If we let $ w = z^2 $ so that $ x^2 + y^2 = w^2$ I have that for r, s $\in \mathbb N$ where $\gcd(r, s) ...
0
votes
1answer
107 views

Points on elliptic curves

I am learning elliptic curves theorem and I have read in more papers that for two distinct points $P$ and $Q$ there is always point $R$ such that $P+Q+R = \infty$. I know that this point should be ...
3
votes
1answer
95 views

How find this continued fraction

Question: let $x$ such this Continued fraction $$x=[0;1,3,5,7,9,11,13,\cdots]$$ How find the vaule of $x$. (can see:http://en.wikipedia.org/wiki/Continued_fraction) My try: I know this ...
16
votes
1answer
467 views

If $n^c\in\mathbb N$ for every $n\in\mathbb N$, then $c$ is a non-negative integer?

Supposing that a real number $c$ is given, is the following true? "If $n^c$ is a natural number for every natural number $n$, then $c$ is a non-negative integer." Though this seems true, I can't ...
1
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1answer
80 views

Questions about $(\mathbb{Z}/2^n\mathbb{Z})^{\times}$

We have $(\mathbb{Z}/2^n\mathbb{Z})^{\times}$ is a group with $\varphi(2^n)=2^{n-1}$ elements. Prove that $x^2=1$ has exactly four solutions in $\mathbb{Z}/2^n\mathbb{Z}$. Moreover, can we show that ...
1
vote
1answer
57 views

Finding the Nth number in a generated list

I am generating numbers as follows: Let the first digit range from 1 to 2 inclusive. Let the second digit range from 1 to 3 inclusive. Let the last digit range from 1 to 2 inclusive. I am then ...
1
vote
2answers
74 views

Finding integer solutions to this equation

$p^{\; \left\lfloor \sqrt{p} \right\rfloor}\; -\; q^{\; \left\lfloor \sqrt{q} \right\rfloor}\; =\; 999$ How do you find positive integer solutions to this equation?
4
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1answer
420 views

log(log(123456789101112131415…)))

How would you fin the integer closest to log(log(1234567891011121314...2013)) where the number is the concatenation of numbers 1 through 2013 inclusive. log() in this case is log base 10. Also, how ...
1
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1answer
27 views

Interesting continued fraction problem $|r_i-u_0/u_1|\le\frac1{k_ik_{i+1}}$

Let $u_0/u_1$ be a rational number in lowest terms, and write $u_0/u_1=\langle a_0, a_1,...,a_n\rangle$ in standard continued fraction notation. Show that if $0\le i<n$, then ...
0
votes
0answers
77 views

System of congruences that do not satisfy CRT assumptions (via algorithm)

Let $x_i,a_i\!\in\!\mathbb{Z}$. The following procedure solves a system of congruences $$x \equiv x_i\pmod{a_i}\;\;\text{ for }i\!=\!1,\ldots,n$$ when $a_i$ are pairwise coprime. Assume that ...
3
votes
1answer
154 views

Bernoulli numbers and the sum of the $m$-th powers of the first $n$ integers [closed]

Let $S_m(x)$ denote the following polynomial in $x$ $$S_m(x) = \sum_{k=0}^m \frac1{k+1}\cdot\binom mk\cdot B_{m-k}\cdot x^{k+1}$$ Prove that $$S_m(x+1)-S_m(x) = x^m$$ for all $m >0$. ...
1
vote
1answer
132 views

Representation of prime number by binary quadratic form

For wich prime numbers $p$ there exist integers $x,y$ such that $x^2+5y^2=p$? For cases $x^2+y^2$ or $x^2+2y^2$ this condition is equivalent to discriminant of the form is quadratic residue modulo p, ...
1
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0answers
87 views

Attempt at finding zeta zeros by recurrence of zeta function.

The following Mathematica program converges to most of the riemann zeta zeros, by using an approximation as a starting point. ...
3
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2answers
102 views

On the equation $(1-x)^2/x + (1-y)^2/y + (1-z)^2/z + 4 = 0$

The problem is to solve the equation, $$\frac{(1-x)^2}{x} + \frac{(1-y)^2}{y} + \frac{(1-z)^2}{z} + 4 = 0\tag{1}$$ in the rationals. Treating this as an equation in $z$, easy solutions would involve ...
0
votes
0answers
104 views

Ring of integers of unramified extension

Let $L/K$ be unramified extension of local fields, and $k,l$ - their residue fields, $l=k(\overline \alpha)$. Is it true that $\mathcal O_L=\mathcal O_K[\alpha]$? And can it be proved if it's true.
2
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0answers
107 views

Need to find a better algorithm to solve a project euler problem dealing with coprime pairs.

I've been working on this for a while and found several solutions so far, but none are fast enough to find the necessary $S(10^7)$. Here is the question: For an integer $M$, we define $R(M)$ as ...
0
votes
1answer
33 views

How to calculate the rest of $2^{p^r-p^{r-1}+1}$ divided by $p^r$

I have the next problem: $p$ is an odd prime number and $r$ is a natural number, $r>1$. How can I calculate the rest of the division of $2^{p^r-p^{r-1}+1}$ by $p^r$ ?
3
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0answers
69 views

$ \mathcal O_K/(p) \cong \mathbb F_p[T]/(T^n)$?

Let $K$ be a totally ramified extension of $\mathbb Q_p$ of degree $n$. Then $$ \mathcal O_K/(p) \cong \mathbb F_p[T]/(T^n) .$$ What is this isomorphism?
0
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1answer
73 views

If $(n^2-n) $ is divisible by m and I have given the value of m , how can I find the value of n where n<m?

If $(n^2-n) $ is divisible by m and I have given the value of m , how can I find the value of n ? Here $ n<m . $ My idea is : $ n^2-n = mk \\$ $ n^2-n-mk=0 \\$ n can be rational if the ...
1
vote
0answers
176 views

galois norm and trace of field extensions

Let $K \subset L \subset E$ and let Nm$_{E/K}(x)$ and Tr$_{E/K}(x)$ be its norm and trace, the determinant and trace of $x$ acting by multiplication on $E$. How can one show that $$ ...
2
votes
1answer
38 views

Number theory Legendre problem please [closed]

How to solve Legendre symbol. $\left(\frac{-2}{59}\right)$ I know that $\left(\frac{a}{p}\right)$ Can anyone briefly tell me how to solve it? Thank you
5
votes
1answer
242 views

Korea Math Olympiad 1993

An integer which is the area of a right-angled triangle with integer sides is called Pythagorean. Prove that for every positive integer n > 12 there exists a Pythagorean number between n and 2n.
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2answers
75 views

The value of two integers $a$ and $b$

I want to find two integers $ a $ and $ b $ which satisfies the following constraint . $\gcd(a, b) = 1$ $0 < a/b < 1$ $a * b = (n!) ^ {n!}$. I will be given the value of $n$ . I have to ...
3
votes
1answer
88 views

Finding every solution such that ${kk\cdots k}^2+ll\cdots l=mm\cdots m$ such as ${333}^2+222=111111$

Let $[k,n]$ be a $n$-digit integer such that $kk\cdots kk$ in the decimal system where $k=1,2,\cdots,9$. For example, $[8,2]=88, [8,5]=88888$. Then, here is my question. Question 1 : Can we find ...
1
vote
0answers
125 views

Hensel's lemma in $\mathbb Z_2$

Can you give me a concrete example for a quadratic form $$ f(x,y)=ax^2+bxy+cy^2 \in \mathbb Z_2[x,y] $$ which has a primitive solution $(x^*,y^*) \in \mathbb Z_2 \times \mathbb Z_2$ (mod 4) with the ...
1
vote
1answer
143 views

Goldbach's conjecture

Just wanted to know if anybody has come close to solving the Goldbach's conjecture problem ? I have seen some videos on youtube where interesting geometric patterns have been found. Anybody here ...
3
votes
2answers
182 views

The number 3211000 is 7-special

Define a positive integer $k$ to be $n$-special if it satisfies the following properties: It has $n$ digits (0, 1, ..., 9) The 1st digit is equal to the number of 0's in the decimal representation ...
2
votes
0answers
275 views

In the Hunt for Kaprekar's Constants for more than 4 digits.

Kaprekar's constant is $6174$ . Take any four digit number with at least two different digits; create two four digit numbers by writing the digits in descending order and in ascending order; subtract ...
2
votes
1answer
71 views

How show $8|[(\sqrt[3]{n}+\sqrt[3]{n+2})^3]+1$

let $n$ be postive integer numbers,and such $n>2$,show that $$ 8\,\,\left.\right\vert\,\,\left\lfloor% \left(\vphantom{\Large A}\sqrt[3]{n\,} + \sqrt[3]{n + 2\,}\,\right)^{3}% \right\rfloor + 1 ...
0
votes
1answer
210 views

collatz conjecture $\mod 2^n$ stopping distance

an interesting book Old and new unsolved problems in plane geometry and number theory by Klee/Wagon (1991) includes the Collatz conjecture. on p225 they consider iterates $\mod 2^n$ and state that ...
0
votes
1answer
100 views

Clarification on solution to a problem:

The problem is as follows: How many zeroes do we write when we write all the integers from 1 to 243 in base 3? The given solution starts as follows: The 1-digit numbers don't have any zeroes. The ...
0
votes
1answer
155 views
0
votes
1answer
74 views

Order of subgroup on elliptic curve over $Z_p$

I should determine the order of subgroup on elliptic curve over $\mathbb{Z}_p$ where $p$ is prime, and point $X$ is generator of some subgroup. While generating the subgroup by points addition I found ...
2
votes
1answer
48 views

Question about counting within a congruence

Say I have quantity $z$ times $(z-1)$ and I know that $x$ divides it. For how many values of $z$ is this true?
1
vote
1answer
77 views

Find the number of 3 x 3 matrices with elements in F_p such that determinant is non zero?

My question is: How do I find the number of $3 \times 3$ matrices $A$ with elements in $F_p$ such that the determinant is non-zero? I don't really now how to go at it. I have a feeling that maybe ...
3
votes
1answer
194 views

Proving statements using Euclidean division

I have a series of statements that are proved based on the equation for Euclidean division, this is: Given two integers $a$ and $b$, with $b ≠ 0$, there exist unique integers $q$ and $r$ such ...
3
votes
1answer
104 views

Number Theory question

Show that in any set of 51 positive integers, there are 11 integers $d_{1} < d_{2}\ <  ---  < d_{11}$ with the property that the sum $5^{d_1} + 5^{d_2} +    + 5^{d_{11}}$is an ...
2
votes
0answers
59 views

Global maximum for $\log x - \frac{x}{\pi(x)}$

$\log x - \frac{x}{\pi(x)}$ hits a global maximum at $x=24,137$ with a value of $1.11196\dots$ Is there any documentation about this anywhere? I couldn't find any. Apologies if there is and it is ...
2
votes
3answers
185 views

Proofs for $0^0 =1$? [duplicate]

Everyone knows the following: $$0^x = 0 \quad \wedge \quad x^0 = 1 , \quad\forall x \in R^*$$ One morning, I wake up asking myself the question "$\text{What is $0^0$, then?}$". So, I did what any ...