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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Proving the Uniformization Theorem for Elliptic Curves (An Exercise from Silverman's Advanced Topics on Elliptic Curves )

In Silverman's Advanced Topics in the Arithmetic of Elliptic Curves there are two demonstrations of the Uniformization Theorem for the Elliptic Curves (It says that, given an Elliptic Curve $E$, ...
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5 views

Combinations of a sequence with any adjacent product no more than a given number

I was wondering whether there is an analytic solution or recursive solution of the combination calculation of natural number sequence with length X, where the multiplications of any adjacent numbers ...
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2answers
140 views

Is $\ \Large\pi^e$ rational?

Is the number $\ \Large\pi^e$ rational?
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3answers
39 views

5 digit number $a6a41$ divisible by 9

In the 5-digit number $a6a41$ each of the a's represent the same number. If the number is divisible by 9, what is the digit represented by $a$? I first approached this by saying $$2a + 11$$ since ...
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1answer
13 views

What are numbers which are reverse digits of each other when divided by each other gives finite numbers after decimal point (non periodics)

Are there numbers which are reverse digits of each other when divided by each other gives a finite numbers after decimal point(non periodic)? For example (xyz are digits) xy/yx=abc.abcfinite and ...
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76 views

Showing that $38^n+31$ is prime

I was reading a question in one of the previous pages, in searching for a proof i stumble across what seem like a contradiction. All i want is for someone to provide the missing link in my argument. ...
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17 views

Extracting small factors from $\varphi(n)$ efficiently

Just like using the Euclidean algorithm and variants to compute the greatest common divisor of two (arbitrarily large) integers is considerably faster than factoring them and collecting their common ...
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1answer
16 views

Is Fermat's number composite for all $n>4$?

Does there exist any proof that the number $F_n=2^{2^n}+1$ is composite for every $n>4$?
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3answers
77 views

Maximize $\sqrt{2x + 13} + \sqrt[3]{3y+5} + \sqrt[4]{8z+12}$

Given three non-negative (as pointed out by Calvin Lin) real numbers $x+y+z = 3$, find the maximum value of $\sqrt{2x + 13} + \sqrt[3]{3y+5} + \sqrt[4]{8z+12}$. (Source : Singapore Math Olympiad ...
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5 views

Size of the “fixed” terms in the prime k-tuple conjecture

The prime $k$-tuple conjecture predicts that for $(a_{1}n + b_{1}), \ldots, (a_{k}n + b_{k})$ an "admissible" k-tuple, where the $a_{i}, b_{i}$'s are fixed, then there are $$ \sim c ...
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1answer
31 views

Prime number of Z and prime element of Z[i]

I am looking at the class note from graduate number theory: Let p be prime number in Z and r be prime element in Z[i]. If r is an associate of p, then p congruent 3 (mod 4). I spent hours trying to ...
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250 views

Simply put, what are the similarities between integers and polynomials?

The Princeton Companion to Mathematics mentions that polynomials (for instance, ones with rational coefficients) share similarities with integers, thus leading to the idea of a general structure of ...
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Algorithm to find primes up to $n$ in $O\left(\frac{n}{\log n}\right)$?

Consider the problem of given an integer $n$, generating a list of the primes not greater than $n$. An optimized version of the Sieve of Eratosthenes can do such task in $O(n)$, while the more modern ...
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The integers mod sqrt{d} [on hold]

Show that $\mathbb{Z}[\sqrt{d}]$ has a unit which is neither $+1$ or $-1$. Conclude that $\mathbb{Z}[\sqrt{d}]$ has an infinite number of units.
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1answer
15 views

integer solutions to bivariate polynomial of second degree

I am trying to determine if there is a way to quickly determine if an equation of the following type $$0 = axy+x-y-A$$ has integer solutions ($a,A$ are integers). If anyone knows how to do this or ...
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18 views

Solutions to $3\cdot 5 p_1 \pm 37^n p_2 =2\cdot 29^m p_3$

Let $p_k$ be either primes larger than $40$ or equal to $1$. $n,m$ are larger than $0$ and $b$ is either $1$ or $2$. I'm searching solutions for the following equation: $$ 3\cdot 5 p_1 \pm 37^n p_2 ...
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85 views

I managed to prove this … Can it be used for anything?

I managed to show: $$ x = \sum_{k=1}^{\infty} \sum_{r=1}^{\infty} \mu (k) x^{kr} $$ where $ \mu(k) $ is mobius function and $ x $ belongs from (-1,1) Can this be used for anything?
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1answer
127 views

Is Douglas Hofstadter's version of Godel's proof utter nonsense?

Is Douglas Hofstadter's version of Godel's proof, which he offers in his book Godel, Escher, Bach, utter nonsense? Hofstadter goes to great length to disguise the fact that there are two distinct ...
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12 views

Geographic representation of summations and floors

I am struggling to write this proof, and cannot figure out how to view this "geometrically:" Let $a$, $b\in\mathbb{N}$ be odd and relatively prime. Show that $$ ...
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23 views

Any rational as integer plus sum of $n$ reciprocals

Does there exist an integer $n$ with the following property? For any rational number $r$, there exist integers $a,b_1,\ldots,b_n$ such that $r=a+\sum_{i=1}^n\frac1{b_i}$.
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1answer
52 views

Any simpler proof of Catalan's conjecture?

visit "http://mathworld.wolfram.com/CatalansConjecture.html" Does there exist any simpler or different proof of Catalans conjecture?
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1answer
30 views

Help on a perfect square.

Consider a question, that xyxyxyxy cannot be a perfect square. How should i tackle this problem. All i use is it must be $0,1 ($mod $3,4)$ and then the math, are there any another beatiful ways ...
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2answers
24 views

Divisibility question regarding integer $n$ [on hold]

How many positive numbers $n$ for which $3n-6$ is divisible by $n-1$? How do you do this?
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1answer
27 views

Number theory question maximum possible difference between $a$ and $b$

$1287a 45b$ is a 8-Digit number, where $a$ and $b$ are not zero. The number is divisible by 18. What is the maximum possible difference between $a$ and $b$? My solution: I first said since it's ...
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Digits of $n$ factorial

With the notable exception of $0$, for large enough $n$, the digits in base $10$ for $n!$ seem pretty much uniformly distributed (I have also checked for other few bases $> 2$). Have anyone ...
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Find all the primes p for which 11 ∈ Qp.

Find all the primes p for which 11 ∈ Qp. Attempt: If p=1 mod(4) then: (11/p)=(p/11)=1 if p∈ 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 ...
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56 views

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

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

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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11 views

How to done RSA program in Maxima [on hold]

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).
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5answers
72 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 ....
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1answer
44 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 ...
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24 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 ...
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1answer
48 views

Squares modulo 2^n

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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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
47 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 ...
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3answers
65 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
32 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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29 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
28 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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36 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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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?
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21 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 ...
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0answers
18 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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53 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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19 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
14 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 ...
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1answer
26 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 ...
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1answer
53 views

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

Let $\mathfrak{ N }_S = (\mathbb{ N }; 0, S)$. 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 ...
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2answers
285 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 $a,b$.
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1answer
27 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}} ...