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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7 views

some amazing properties of combinatorial numbers

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 ...
1
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1answer
12 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
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2answers
30 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
40 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
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2answers
104 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 ...
2
votes
2answers
66 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
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1answer
22 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
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0answers
23 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
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0answers
18 views

Is 1 a quatratic residue modulo any number?

For any number n, is 1 always a quadratic residue mod n?
7
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0answers
64 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
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1answer
23 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
13 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
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1answer
16 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
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0answers
16 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
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1answer
43 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 ...
1
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1answer
36 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
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0answers
33 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) ...
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2answers
67 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
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0answers
21 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
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2answers
46 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
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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
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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
54 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
76 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
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0answers
24 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
53 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
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0answers
41 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
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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
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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
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4answers
87 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
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0answers
32 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 ...
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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.
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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
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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
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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
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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 ...
0
votes
1answer
45 views

Number Theory - Order of $p^2$ [duplicate]

How can you show that if $p$ is prime then the numbers with maximum possible order modulo $p^2$ is $\phi(\phi(p^2))$. I tried finding order(a) modulo 9, and obtained the following: $1$ is $1$, $2$ is ...
1
vote
1answer
46 views

What is behind these series?

I just found out (I am an amateur) that if I have the following series I get the following answers for the a nth number . (each series is the sum of the previous one) $$1 ,1 , 1, \dots, 1 $$ ...
3
votes
1answer
35 views

Primes of the form $x^2 + 3y^2$

Im trying to prove that a prime $p\neq 3$ is of the form $p=x^2 + 3y^2$ if $p \equiv 1 \pmod{3}$. I have think in a prove as follows: As we know that $-3$ is a quadratic residue mod p, we know that ...
0
votes
1answer
20 views

Why $n^{d(n)/2}$ is not getting satisfied?

Respected all. We know that the product of all positive divisors of $n\in \mathbb N$ is $n^{d(n)/2}$ where $d(n)$ is the number of positive divisors on $n$. What will happen if $d(n)=odd$ say we ...
2
votes
1answer
18 views

Discriminant of n algebraic numbers equals $0$ iff the algebraic numbers linearly dependent

Let $K \subset L$ be two number fields with $[L:K] = n$. Let $\{\alpha_i:1 \leq i \leq n\} \subset L$. Then $\operatorname{disc}(\alpha_1 \dots \alpha_n) = 0 \iff \alpha_i$ are linearly dependent ...
0
votes
1answer
16 views

Showing Zp is isomorphic to the completion of Z, when Z is considered as a p-adic metric space

Question: Show that Zp is isomorphic to the p-adic completion of Z; that is, the completion of Z when Z is considered a metric space via the p-adic metric. I'm stuck. If we take an element a in Zp, ...
2
votes
0answers
30 views

Show that the Euclidean algorithm works for Gaussian integers. [on hold]

Here is a question from introductory level number theory at college: Prove or show that the Euclidean algorithm works for Gaussian integers. Thank you very much for your valuable time and help!
2
votes
0answers
28 views

Poles and zeroes

If $f$ be the function defined by $$f(x)=2sin\frac{x}{2}\prod_{k=1}^{\infty}\frac{(1-e^{ix}q^k)(1-e^{-ix}q^k)}{(1-q^k)^{2}}$$ where $q = e^{2\pi it}$ $h(x)=\frac{f'(x)}{f(x)}; \quad \quad ...