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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2
votes
1answer
11 views

Finding the maximal order in a number field

Finding the maximal order in a number field Suppose the number field is $K:=\mathbb Q(\alpha)$, then $\mathbb Z[\alpha]$ is not necessarily the integral closure of $\mathbb Z$ in $K$. I know the ...
0
votes
1answer
13 views

A question in Number Theory - prove there exist m>2010 s.t f(m) is not prime

Let $$f(x)=\sum_{i=0}^n a_nx^n$$ be a polynomial with $$a_n \in Z,n>0,a_n\neq0$$ Prove that there exists some natural number $$m>2010$$ such that $$|f(m)|$$ is not a prime number. I tried to ...
-5
votes
0answers
43 views

Around Fermat Last Theorem [on hold]

HINT.- According to an old and still unproven conjecture of V. Bouniakowsky, P(n) is infinitely often a rational prime. Here we need just one prime to arrive to the conclusion.
0
votes
1answer
33 views

A question about primes, number theory [duplicate]

I tried to solve this question but without a success: Let $p$ be a prime number,and $p^2+2$ is also prime, prove that $p=3$. I tried to show $p^2+2$ as a product of numbers and then to show that ...
2
votes
1answer
18 views

Find smallest discrete logarithm, knowing some discrete logarithm.

Discrete logarithm is a value $x$ that satisfy the equality $$a^x \mod m = b$$ Is there an easy way to find the smallest possible discrete logarithm, knowing some other discrete log. Basically if I ...
0
votes
0answers
22 views

Hyperbolic curves and elliptic curves

I am so sorry, if I am very wrong. I know some what hyperbolic functions/curves, and elliptic curves as well. Now my question is that; 'Is there hyperbolic elliptic curves?. If yes, what are the ...
1
vote
1answer
10 views

Converting Unbalanced Ternary Numbers to Balanced Ternary Number

Can someone please provide a step by step algorithm for converting unbalanced ternary to balanced? for instance: (Base 10) 501 = (Base 3 Unbalanced) 200120 I've done some research on this conversion ...
0
votes
0answers
19 views

genus of an algebraic curve [on hold]

I would like to draw the better answer to my question and I believe that, math-stack will help me out. How and why to find GENUS of an algebraic curve? Is there any relation between genus and ...
-3
votes
0answers
44 views

some amazing properties of combinatorial numbers [on hold]

I want to prove $$ C_{2^{i+1}-k-1}^k=\frac{(2^{i+1}-k-1)(2^{i+1}-k-2)\cdots(2^{i+1}-k-(k-1))(2^{i+1}-2k)}{k(k-1)\cdots 2\cdot 1} $$ is even, for all $k=1,2,3,\cdots, 2^i-1$. Here $i\geq 1$. How to ...
2
votes
1answer
13 views

How are Digit Extraction Formulas Special?

There are hundreds of similar looking formulas to the BBP that I've seen on the internet, but those are termed as spigot algorithms only. Why is it that none of those other pi formulas can be used ...
0
votes
2answers
36 views

Prove that if $n \equiv 7 \pmod 8$, then $n$ cannot be expressed as the sum of three squares.

I begin by contradiction. Assume that $n$ can be expressed as the sum of three squares. That is $n = a^2 + b^2 + c^2$. Now since $n \equiv 7 \pmod 8$ then $8 \mid n - 7$ so $8 \mid a^2 + b^2 + c^2 - ...
3
votes
1answer
46 views

Negative Pell's Equation: Prove that $k=3$.

I made this problem (while solving another problem) but I haven't been able to prove it. Let $x,y,k\in \mathbb{Z}^+$. Prove that if $x^2-(k^2-4)y^2=-1$ then $k=3$. Any pointers are appreciated, but ...
7
votes
2answers
111 views

$\forall x\in\mathbb{R},,$ if $x^{2}$ is rational, then $x$ is rational.

This is my attempt at this question. Is this correct? $\forall x\in\mathbb{R},,$ if $x^{2}$ is rational, then $x$ is rational. This statement is false. Using counterexample, let $x=\sqrt{2}$. Since ...
3
votes
3answers
78 views

What is the remainder of $314^{164}$ divided by 165?

What is the remainder of $314^{164}$ divided by 165? Since 165 is not a prime, we cannot apply Fermat's Little Theorem directly. However since $165=3\times 5\times 11$, we could try to divide ...
1
vote
1answer
25 views

Does there exist a quadratic generalization of the continued fraction approximants?

Let $t$ be a real number and let $\frac{p_n}{q_n}$ be its continued fraction approximants. These have the property that $$ \left| t - \frac{p_n}{q_n} \right| < \frac{1}{q_n q_{n+1}} $$ In other ...
3
votes
0answers
24 views

Multiples of some set has density

Nathanson gives a proof in Elementary Methods in Number Theory (Theorem 7.14) that, if a set $S$ of positive integers has $$ \sum_{s\in S}\frac1s<+\infty $$ then the set of positive multiples of ...
0
votes
0answers
18 views

Is 1 a quatratic residue modulo any number? [on hold]

For any number n, is 1 always a quadratic residue mod n?
7
votes
0answers
66 views

prove that $x^2 + 5 =y^3$ has no solutions for $x,\ y \in \mathbb{Z}$ [duplicate]

So the question is completely stated by the title. My own thoughts: I can prove that $x^2 + 1 = y^3$ has no solutions for $x,y \in \mathbb{Z}$ by using the factorization: $$ y^3 = (x-i)(x+i) $$ in ...
0
votes
2answers
43 views

Integer solution to the equation

Does there exists an integer solution (for every integer $m\geq 1$) for the following equation? $$x_1x_2...x_n+(2y+1)z+y=4m+3$$ where, $1\leq x_1\leq x_2\leq...\leq x_n\leq l$,$0\leq y \leq ...
0
votes
3answers
14 views

system of conguences

I am trying to figure out how to solve: Find $x, y \in \mathbb{Z}$ such that $$2x+y\equiv 4\pmod{17}$$ and $$5x-5y\equiv 9\pmod{17}.$$ I know this means $2x+y-4$ and $5x-5y-9$ are multiples of $17$, ...
2
votes
1answer
17 views

Looking for different proofs for $-1$ is a quadratic residue of primes of the form $4k+1$ and related facts

Suppose $p$ is a prime of the form $4k+1$ , then $4|p-1=|\mathbb Z_p^*|$ , as $\mathbb Z^*_p$ is a cyclic group , so there is $\bar x \in \mathbb Z_p^*$ such that $o(\bar x)=4$ , then $o(\bar x^2)=2$ ...
0
votes
0answers
17 views

Question related to image of $[1,N]^n$ under a linear tranformation

I am reading an article and I am a bit confused about the following passage. I would appreciate any clarification. It goes as follows: Let $\bar{F}$ be a collection of $r$ linearly independent ...
0
votes
1answer
45 views

Does this compound interest problem coincide to the value of e by coincidence?

An account starts with €$1.00$ and pays $100\%$ interest per year. If the interest is credited once, at the end of the year, the value of the account at year-end will be €$2.00$ . What happens if the ...
2
votes
1answer
41 views

$3 X^3 + 4 Y^3 + 5 Z^3$ has roots in all $\mathbb{Q}_p$ and $\mathbb{R}$ but not in $\mathbb{Q}$

This is an exercise in my textbook in a chapter about the Hasse-Minkowski-theorem: Show that the polynomial $3 X^3 + 4 Y^3 + 5 Z^3$ has a non-trivial root in $\mathbb{R}$ and all $\mathbb{Q}_p$. ...
0
votes
0answers
25 views

Determine all $n \in \mathbb{N}$ such that $GCD(n,48)=6$, $14|n$ and $|Div^+(n)|=12$.

Determine all $n \in \mathbb{N}$ such that $\gcd(n,48)=6$, $14|n$ and $|Div^+(n)|=12$. What I did: $14|n$ then $2|n$ and $7|n$ so $n=2\cdot7\cdot q$, $q \in \mathbb{Z}$. Then $6|n$ implies $2|n$ and ...
4
votes
0answers
35 views

Is there anything known about the value where the Euler and Hadamard products for $\zeta(s)$ are equal?

Take the Hadamard product for the Riemann $\xi$-function ($\rho$ is a non-trivial zero of $\zeta(s)$): $$\xi(s) =\frac12\, s\,(s-1) \,\pi^{-\frac{s}{2}}\, \Gamma\left(\frac{s}{2}\right)\, \zeta(s) ...
-1
votes
2answers
68 views

Squares numbers into squares

Solve the two related questions below in which lowercase letters are digits in base $10$, $a > 0$ and $N\in\Bbb N$. Find the values of $N$ in $(1)$ and prove or deny $(2):$ $$ ...
0
votes
0answers
22 views

the asymptotic approximation of a sum

$p_{n}$ and $p_{j}$ are two primes with $p_{n}<p_{j}$ where the $n$ and $j$ denotes the $n$th and the $j$th prime. I have this sum $$\sum \limits^{k=\frac{b-p^{2}_{n}p_{j}}{2p_{n}p_{j}} ...
1
vote
2answers
47 views

Are there arbitratily long runs of consecutive integers n that are NOT of the form $n = p^k$ or $n = 2p^k$ for some $k>0$ and $p$ an odd prime?

I know that there are arbitrarily long runs of consecutive non-primes. I am interested in this question because an integer $n>4$ has a primitive root if and only if it has the form $n = p^k$ or ...
0
votes
0answers
17 views

Generate a function that shuffles a number withing a given range which is reproducible

Lets say I have an array of numbers $1 2 3 4 5 6 7$. I want to shuffle these numbers in some order , $7 5 4 3 1 26$ . However , it should be revesrible. That is given the second array I must be able ...
0
votes
1answer
20 views

Exercise 2.13 from a computational introduction to number theory and algebra

This is an exercise from V. Shoup. A computational introduction to number theory and algebra. Let $p=2, e=3, a=b=1, c = 0$, then $p^{2e} = 64, z\in \{0,1,2,\cdots,63\}$, the conclusion is, there ...
1
vote
1answer
55 views

Show that $\limsup_{x \to \infty} \frac{\pi(x)}{x/ \log x} \geqslant 1. $

Show that $$\displaystyle\limsup_{x \to \infty} \dfrac{\pi(x)}{x/ \log x} \geqslant 1. $$ I've seen $\displaystyle\lim_{x \to \infty}$ operator, but I haven't seen $\displaystyle\limsup_{x \to ...
1
vote
2answers
47 views

What is the value of $1^2 + 2^2 + 3^2 + \cdots + (p-1)^2\pmod{p}$?

What is the value of $1^2 + 2^2 + 3^2 + \cdots + (p-1)^2\pmod{p}$? Let's try a several primes greater than 3... If $p=5$, then we have $1^2 + 2^2 + 3^2 + 4^2 = 30$, so that $30\pmod{5} = 0$ If ...
6
votes
2answers
78 views

Prove or disprove : $a^3\mid b^2 \Rightarrow a\mid b$

I think it's true, because I can't see counterexamples. Here's a proof that I am not sure of: Let $p_1,p_2,\ldots, p_n$ be the prime factors of $a$ or $b$ \begin{eqnarray} a&=& ...
1
vote
0answers
25 views

Prove that $a_{i,1}x_1 +a_{i,2}x_2 +···+a_{i,n}x_n ≤c_i, 1≤i≤n $ are all satisfied by a nonzero $n-tuple$ of integers.

My setting is that $c_1, · · · c_n$ are positive real numbers, and $A = [a_{i,j} ]$ is an $n × n$ non-singular matrix. Assume that $c_1 · · · c_n > | det(A)|.$ I want to prove that the n-linear ...
4
votes
2answers
54 views

Show that $\sum\limits_{p \leqslant x}1/p = \frac{\pi(x)}{x} + \int_2^x \frac{\pi(u)}{u^2} du.$

Show that $$\displaystyle\sum\limits_{p \leqslant x}1/p = \dfrac{\pi(x)}{x} + \int_2^x \dfrac{\pi(u)}{u^2} du.$$ In the equation above, $\pi(x)$ denotes the prime counting function. To get ...
1
vote
0answers
47 views
+50

Product of Stirling Numbers of the first kind

I have been messing around with coefficients of various polynomials and was wondering if there was a way to reduce the following stuff. Let polynomial, ...
0
votes
0answers
35 views

Prime - number theory [duplicate]

Why is the digit 1 is not a prime number? 1 can be devided by 1 and itself. I think it's because we can express like 1= 1x1x1 ... is it true or not?
2
votes
0answers
32 views

Notation Question in proof of Kronecker-Weber theorem

I am trying to understand this proof of the Kronecker-Weber theorem by Franz Lemmermeyer http://arxiv.org/pdf/1108.5671.pdf. The set-up is this: $K/\mathbb{Q}$ is a cyclic extension of prime degree ...
2
votes
1answer
50 views

Why is $\mathbb{Q}_\infty = \mathbb{R}$?

Why, in the context of p-adic numbers, do we have the convention $$\mathbb{Q}_\infty = \mathbb{R} \quad$$ ? It must have something to do with the generalization of the Legendre-symbol for ...
1
vote
4answers
88 views

If $a$ and $b$ are positive integers and $4ab-1 \mid 4a^2-1$ then $ a=b$.

Prove that if $a$ and $b$ are positive integers and $$(4ab-1) \mid (4a^2-1)$$ then $a=b$. I am stuck with question, no idea. Is there any way to prove this using Polynomial Division Algorithm? Would ...
1
vote
0answers
33 views

Sums of triangles

If $P$ is any odd prime, is there a proof that working Mod P, each number from $0$ to $P-1$ except $\frac{(P^{2} -1)}{4}$ can be formed as the sum of the triangle of a number <(P+1)/2 and the ...
-2
votes
2answers
32 views

How do you show that$ ∏j ≡1 $(mod p) where j is $1 \le j\le p-1$ and $\frac{j}{p}=1$

Also, $P$ is a prime of the form $4k+3$ and $k$ is an element of natural numbers including $0$.($\frac{j}{p}$) denotes a legendre symbol.
-4
votes
0answers
19 views

Summation of a series with a A.P [on hold]

what will be the summation of this series n-r+1C2 + n-2*r+1C2 + n-3*r+1C2+..... 1C2; where n and r are natural numbers.Can we derive a formula from this
-1
votes
0answers
19 views

is find magic square from even degree?

please tell me is there any journal that find magic square to degree of even? I find a good way to solve it but not for all degree actually for a particular degree of it.for example degree ...
1
vote
2answers
12 views

Prove the order of the group homomorphism of an element divides the order of the element.

Let $\phi : G \rightarrow H$ be a group homomorphism. Prove $\forall g \in G$, the order of $\phi(g)$ divides $g$. I've gotten to the point where I've shown that if, $ord(\phi(g)) < ord(g)$ then ...
2
votes
1answer
25 views

How many pairs of polynomials $(U,V)\in \Bbb Z[x]^2$ such that $P=U^2+V^2$ for a given polynomial with integer coefficients?

This question is no more than curiosity question. For integers we know that a positive integer $n$ is a sum of two squares if and only if for any prime $p$ such that $p\equiv 3 \mod 4$ we have ...
4
votes
0answers
75 views

Solve $(x+1)^n-x^n=p^m$ in positive integers

Solve in positive integers: $$(x+1)^n-x^n=p^m$$ $p$ is prime, $n\ge 2$. Seemingly Zsigmondy's Theorem and LTE won't work here. Though you can tell (as suggested by user barto), using ...
1
vote
0answers
19 views

Power series in p-adic integers

How can we show that for $x \in \mathbb{Z}_p$, $\log_p(1+x)$ converges in $\mathbb{Z}_p$ when $|x|_p < 1$? To clarify, $\log_p(1+x)$ is the power series: $$\sum_{n=1}^\infty ...
0
votes
0answers
22 views

Showing Zp is isomorphic to the completion of Z with the p-adic norm using Cauchy sequences

Following on from James' question, here: Showing Zp is isomorphic to the completion of Z, when Z is considered as a p-adic metric space I understand that every element a = a1,a2,a3,... of Zp can thus ...