Questions on more advanced topics of number theory. Consider first if (elementary-number-theory) might be a more appropriate tag before adding this tag.

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Find all the primes p for which 11 āˆˆ Qp.

Find all the primes p for which 11 āˆˆ Qp. Attempt: If p=1 mod(4) then: (11/p)=(p/11)=1 if pāˆˆ Q11 which p=1,3,4,5,9 mod(11) and (p/11)=-1 if p=2,6,7,8,10. If p=3 mod(4) then: (11/p)=-(p/11)=-1 if ...
-1
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0answers
13 views

Show that (Z4;+4) and (Z5*,.5) are Isomorphic groups

Given groups (Z4;+4) and (Z5*,.5). Show that these groups are isomorphic by exhibiting a one-to-one correspondence alpha between their elements such that a+b = c (mod 4) iff alpha(a).alpha(b) = ...
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0answers
10 views

I need help in Diffie-Hellman (number theory )and RSA orogram maxima?

I) Modify the RSA program in Maxima to create an RSA example where p and q have 50 digits. Submit the entire run of the program (ignore the portions where the results are written to a file). II) For ...
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0answers
6 views

How to done RSA program in Maxima

Modify the RSA program in Maxima to create an RSA example where p and q have 50 digits. Submit the entire run of the program (ignore the portions where the results are written to a file).
1
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1answer
31 views

Last digits, numbers

Can anyone please help me? 1) Find the last digit of 7^(12345) 2) Find the last 2 digits of 3^[(3)^(2014)]. Attempt: 1) By just setting the powers of 7 we have 7^(1) = 7, 7^(2) = 49, 7^(3) = 343, ...
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0answers
23 views

Irreducibility of a polynomial in two variables

Anyone knows how to verify that the polynomial $$(ax)^n+a_{n-1}x^{n-1}+\cdots +a_1x+a_0-y^n$$ is irreducible? Where $n\geq 2$, $a,a_i\in\mathbb{Z}$, $a\neq 0$, and $(ax)^n+a_{n-1}x^{n-1}+\cdots ...
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5answers
56 views

prove that $3$ does not divide $n^2+1$

How do I prove that $3$ does not divide $n^2+1$, for all $n\in\mathbb{Z}$, thought of in separate cases, but did not get, induction also was unable to ....
2
votes
1answer
35 views

Find all integer solutions of $x^4 + 2x^3 + 2x^2 + 2x +5 = y^2$

Find all integer solutions to $x^4 + 2x^3 + 2x^2 + 2x +5 = y^2$. I'm in a dead end. I've transformed the expression in the following state: $(x^2+1)(x+1)^2 = y^2 -4$ I couldn't see anyway in ...
1
vote
0answers
17 views

Almost perfect numbers

A positive integer $n$ is called almost perfect if the sum of its divisors smaller than $n$ is $n-1$. What are all almost perfect numbers $n$ such that some power $n^k$ is also almost perfect for at ...
3
votes
1answer
36 views

Squares modulo 2^n [on hold]

How many squares are there modulo $2^n$? If we would deal with $p^n$, where p an odd prime, then we could use Hensel's Lemma, which clearly doesn't work with $2^n$.
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0answers
18 views

Diffie-Hellman Constructing a secret-sharing scheme

I) For the mod 29 system, g = 2 is a primitive root. If Alice chooses a secret a = 5, and Bob chooses a secret b = 11, what is the Diffie-Hellman shared secret they will create? II) Consider the ...
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1answer
36 views

What is the shared secret? in Diffie-Hellman [on hold]

Consider the elliptic curve $E : y^2 = x^3 + 11x + 19 \pmod{167}$. The point $P = (2,7)$ is on $E$. Suppose this $E$ and $P$ are used in a Diffie-Hellman key exchange, where Alice chooses the secret ...
1
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3answers
61 views

Prove that $7^{100}+3^{10}=8^{100}$ or $7^{100}+3^{10}<8^{100}$ [on hold]

Prove that $7^{100}+3^{10}=8^{100}$ or $7^{100}+3^{10}<8^{100}$ I tried using some theorems of divisibility, to show that one divides the other, and the other also divides the first, but could ...
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1answer
26 views

what is the Diffie-Hellman

For the mod 29 system, g = 2 is a primitive root. If Alice chooses a secret a = 5, and Bob chooses a secret b = 11, what is the Diffie-Hellman shared secret they will create?
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0answers
22 views

Riemann Zeta and Monotonicity

The second paragraph of Wolfram Mathworld Riemann Zeta Function states: The plot above shows the "ridges" of $|\zeta(x+\imath y)|$ for $0<x<1$ and $1<y<100.$ The fact that the ridges ...
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1answer
24 views

How to find a primitive root for mod n arithmetic [on hold]

Find a primitive root for mod n arithmetic (i.e. in $\mathbb Z / n\mathbb Z$) where $n =$ a. 17 b. 173 c. 1733
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0answers
29 views

Find all the primes $p$ for which $x^2\equiv13\pmod p$ has a solution.

I found that for $p=3$, we have $x^2\equiv2^2\equiv4\equiv13\pmod 3\equiv-9\equiv0\pmod 3$. But how do I find out all the primes such that this holds?
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0answers
7 views

Let n be a natural number. Any set, {a1,a2,…,an}, of n integers for which no two are congruent modulo n is a complete residue system modulo n.

I'm not very good at number theory but it seems to me that this can just simply proved by the definition of a complete residue system. Is it that simple?
2
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0answers
19 views

Question on Fermat Numbers Factorization

Let $F_{n}=2^{2^n}+1$ be a Fermat number. A classic idea using orders and Fermat's Little Theorem shows that a prime divisor $p$ of $F_{n}$ must be of the form $p=k .2^{n+1}+1$. Furthermore, using the ...
2
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0answers
16 views

Minimum degree of polynomial assuming exactly k prime values

Dirichlet's theorem states that there are infinitely many primes of the form $an+b$ for coprime integers $a$ and $b$. This implies that The minimum degree of a polynomial $f \in \mathbb{Z}[X]$ ...
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1answer
49 views

prove that the number $38^n+31$ is composite [duplicate]

Prove that for every positive integer $n$, $38^n+31$ is a composite number. for example $38+31=69$ is composite. $38^2+31=1475$ is also composite. I have tried modulo but it didn't work.
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3answers
17 views

$p$ divides the sum of the quadratic residues $\bmod p$

Could you help me at the following exercise? Show that, if $p>3$ is a prime,then $p$ divides the sum of the quadratic residues $\bmod p$.
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1answer
13 views

Bernoulli numbers identity with binomial coefficient

The generating function for the Bernoulli numbers $B_k$ is given by $f(z) = \frac{z}{e^z -1}= \sum_{k=0}^{\infty} \frac{B_k}{k!} z^k$. Applying the identity $$1 = \frac{e^z -1}{z} \cdot ...
2
votes
1answer
24 views

About the implicit funtion in a holomorphic situation.

Let $f(x,y)$ be a polonomial with integral coefficients which has a zero $(a,b)\in \mathbb{R}^2$ such that the partial derivative respect to $y$ at this point is nonzero. Then by the implicit function ...
0
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0answers
22 views

Why is $\mathfrak{ N }_S$ not finitely axiomatizable?

Let $\mathfrak{ N }_S = (\mathbb{ N }, S)$. With axioms ($A_S$): 1: $\forall x (Sx \neq 0)$ 2: $\forall x \forall y (Sx = Sy \rightarrow x = y)$ 3: $\forall y (y \neq 0 \rightarrow \exists x (y = ...
5
votes
2answers
284 views

GCD of two large integers

For two random $d$ digit integers $a,b$, what is the probability $\gcd(a,b)<B$? Here $B$ is much much smaller than $a,b$.
2
votes
1answer
25 views

How to prove the transformation formula for Jacobi classic theta function

How to prove the following transformation formula: $$ \theta(x)=\frac{1}{\sqrt{x}} \theta\left(\frac{1}{x}\right), $$ where $\theta$ is the Jacobi theta function $\theta(x)=\sum_{n\in \mathbb{Z}} ...
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0answers
33 views

Proof of the inequality $2\uparrow^n 4 < 3\uparrow^n 3 < 2\uparrow^n 5$

I tried to prove the inequality $$2\uparrow^n 4 < 3\uparrow^n 3 < 2\uparrow^n 5$$ for all natural numbers $n\ge 1$ For n = 1 , the claim is true because of 16 < 27 < 32. The left ...
2
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0answers
26 views

Elements of a Dedekind domain can be chosen to have valuation $1$ with respect to one prime, $0$ everywhere else

I noticed this is true for $\mathbb{Z}$, but I was wondering whether it was true in general. Let $R$ be a Dedekind domain and $P_1, ... , P_s$ maximal ideals. The localized ring $R_{P_i}$ is a ...
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0answers
11 views

Can the General Number Field Sieve be used to factor in any unique factorization domain?

Related slightly to my question about factoring in quadratic rings, can you use the general number field sieve to factor in any unique factorization domain? Can you use it in any UFD that isn't the ...
4
votes
1answer
419 views

Fermat's Equation

Can somebody help me with this. I am trying to prove something from Fermat's equation. Fermat's Equation $x^n + y^n = z^n$, where $x,y,z$ and $n$ are positive integers. His last theorem states that ...
8
votes
1answer
202 views

Cool Integral = $\pi/2$ !!

I am trying to calculate the integral $$ I_n=\int \limits_0^\infty \prod_{k=1}^n \frac{\sin \frac{x}{2k-1}}{\frac{x}{2k-1}}dx. $$ (I have literature on this, if people want). Note, we can write the ...
0
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0answers
15 views

Numbers ending at some point.

Numbers, at some point, are required to end but we still have the perspective on numbers that they go on forever. For example, If I have a meter stick and on it is a drawn line. And on a wall near me ...
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0answers
18 views

Are roots of unity in hypercomplex algebras well defined?

While playing around with cyclotomic fields, I started to wonder about taking the roots of unity in higher dimensional analogues of the complex plane. Are the roots of unity well defined in the ...
2
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2answers
67 views

Logic and number theory books

I've recently decided to start preparing for uni, so I figured I need to learn logic and some number theory. I picked up Burton's Elementary Number Theory and wasn't quite comfortable with it, seemed ...
2
votes
2answers
79 views

for what integers $x$ do there exist $x$ consecutives integers, the sum of whose squares is prime?

for what integers $x$ do there exist $x$ consecutives integers, the sum of whose squares is prime? I tried use $$1^2+2^2+...+n^2=\frac {n(n+1)(2n+1)}{6}$$
2
votes
2answers
34 views

Determine if $-42$ is a quadratic residue of $mod(61)$

This is what I have so far: Using Legendre symbol, we have $(\frac{-42}{61})\equiv(\frac{19}{61}).$ Since $gcd(19,61)=1,$ $(\frac{19}{61})\equiv1.$ Is this correct?
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0answers
26 views

Looking for references

I am looking for reference on the following problem. Let $S= \{ ax_1+bx_2 \mid x_1 \in X_1 , x_2 \in X_2, \}$ where $X_1,X_2\subseteq \mathbb{R}^n$ and $a,b \in \mathbb{R}$. Note that $a$ and $b$ ...
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0answers
22 views

When can a congruence relation be transformed into quadratic reciprocity expressions?

When can a congruence relation $$p \equiv c_1, c_2, \ldots, c_r \mod{N}$$ be transformed back into quadratic reciprocity expressions $$\left (\frac{d_1}{p} \right) = \left (\frac{d_1}{p} \right) = ...
2
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1answer
20 views

Euclidean Algorithm for Modular Inverse, with negative numbers

I might be on to something quite simple which I'm failing to see, while calculating modular inverses. For example, calculating 7x = 5 (mod 12) Which is the same as saying: 7x - 5 = 12k Which ...
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0answers
28 views

Every prime of the form 4n+1 can be written as sum of two squares which are unique for each 4n+1 prime [on hold]

Prove that every prime of the form 4n+1 can be written as sum of two squares and the choice of the two squares, one even and one odd is unique for each 4n+1 prime.
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1answer
34 views

Find the natural numbers so that n=2*a^2 ,n=3*b^3 ,n=5*c^5.Number theory problem.

Well here it is i spend almost 3 hours on this one!! Find the general form of the natural numbers that are twice a square ,tripple of a cube and 5 times a 5-ith power.Who is the smaller of them?.What ...
7
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2answers
726 views

Is $\text{rational}^{\text{irrational}}$ rational or irrational?

Is the number $\text{rational}^{\text{irrational}}$ rational or irrational? For example $2^{\sqrt{2}}$: is it rational or irrational? I tried using a logarithm but it didn't work. It seems by ...
3
votes
1answer
59 views

Positive integer solutions to $x^2+y^2+x+y+1=xyz$

The question asks for positive integer solutions to $x^2+y^2+x+y+1=xyz$ . We at first note that $x|y^2+y+1$. Now,let there exist positive integers $x,y$ that satisfy the given equation.Then ...
0
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1answer
21 views

Extension of an arithmetic function like Gamma function

As you know Gamma function is an extension of the factorial function. Is there any other function which is the extension of an arithmetic function? For example extension of the Euler's totient ...
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2answers
38 views

integer solutions to $x^2+y^2+z^2+t^2 = w^2$

Is there a way to find all integer primitive solutions to the equation $x^2+y^2+z^2+t^2 = w^2$? i.e., is there a parametrization which covers all the possible solutions?
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1answer
16 views

$n$ is some natural number. Let $x$ be the integer part of $\sqrt n$ and $y$ be the decimal part. If $x^2 - y^2 = 1+4y$ what is $y^x$?

$n$ is some natural number. Let $x$ be the integer part of $\sqrt n$ and $y$ be the decimal part. If $x^2 - y^2 = 1+4y$ what is $y^x$? This is some high school problem but I can't solve it. Any help? ...
0
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0answers
14 views

Group of numbers with common euler's totient function result [duplicate]

I was asked to find the group of integers, which share the result of euler's function of 84. To be clear: which numbers, when applying eulers function on them, result 84. By calculating I found that ...
2
votes
1answer
39 views

Rewriting $\tau(p)\Delta(\tau)$ when $p$ is prime

$p$ is a prime, and $\tau$ is Ramanujan's tau function: $$p^{11}\Delta(p\tau)+\frac{1}{p}\sum_{k=0}^{p-1}\Delta\bigg(\frac{\tau + ...
1
vote
3answers
37 views

If $\gcd(a,b)=\gcd(a,c)=1$ then $\gcd(a,bc)=1$

I need help in this. If $a,b,c \in \mathbb{Z}$ and $\gcd(a,b)=\gcd(a,c)=1$ then $\gcd(a,bc)=1$.