# Tagged Questions

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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### Proof that expression is integer [duplicate]

hi guys can you help me with this? Proof that expression is integer $$\frac{(2n)!}{2^nn!}$$
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### Find the number of natural number solutions of $a+2b+c=100$

Find the number of natural number solutions of $a+2b+c=100$ I remember something like stars and bars if the equation I change to $a+b_{1}+b_{2}+c=100$ then i get $\dbinom{99}{3}$ ways. If the ...
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### Number theory: Is this argument correct and cube question [duplicate]

First of all, is this argument correct? Suppose $m < n$ are integers. Then for every $k \in \mathbb{N}$ $$m + k < n + k.$$ What I did: Suppose that $m + k \ge n + k$, the by the cancelation ...
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### Scope of expressions which determine whether an integer can be resolved into 2 triangles [closed]

It is straightforward to show that for any prime P of the form 4n+1, (P-1)/4 +kP can be resolved into the sum of two triangles if and, only if, k can be, and similarly for all primes of the form 4n+3, ...
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### What is the least prime $p$, such that $[p-1000,p+1000]$ does not contain a prime $\ne p$?

I am looking for the least prime number $p$, such that the interval $[p-1000,p+1000]$ contains no prime except $p$. In other words, the prime nearest to $p$ has a distance $>1000$ to $p$. I found ...
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### What's the condition for (x+kp) and pq being coprime?

Suppose $p$ and $q$ are large primes and $N=pq$. $x$ is an arbitrary integer in $\mathbb{Z}_p$ and $k$ is a random integer. Then what is the condition for $k$ (suppose $x$ is fixed) such that ...
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### Do UF+PNT+SMO+GRH imply SOC?

The title may sound esoteric, but let's make it explicit. Suppose that the conjunction of unique factorization (UF, still open), prime number theorem (PNT, proved by Yoshikatsu Yashiro), Strong ...
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### Is an algorithm to find all primes up to $n$ that runs in $O(n)$ time fast?

I kindly ask you if it is useful or fast for a prime number generator to run in $O(n/3)$ time? I believe I have a way to generate all $P$ primes up to $n$, quickly and neatly, in $P$ comparisons and ...
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### Calculate cycle length

let $a, n, m \in \mathbb{Z}$ and $i\in\mathbb{N}$ and $$(a+in) \mod m$$ Is there a closed way to tell for what $i$ the congruence begins to cycle? Thanks
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### Solvability of $a \equiv x^2 \mod b$

Suppose you want to prove that $\exists x \in \mathbb{Z}$ with $a \equiv x^2 \mod b$. Write $b = \prod_{i = 1}^{k} p_i^{e_i}$, the prime factorisation of $b$. Why is the equivalent with finding ...
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### Sum of powers of a matrix with primitive polynomial modulo $2^{r}$

I need to prove an statement in the matrix form, which leads to the following equality modulo $2^{r}$. Which I couldn't prove but with computer simulation for lots of primitive polynomial, it seems to ...
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### Avoiding range of a bivariate integer function or diophantine function

I'm trying to find a function or sequence (of integers) which avoids all the range values of the following integer function where $x,y \in \{0,1,2,...\}$ and $f(x,y)=5+23*x+7*y+30*x*y$. Does anyone ...
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### Considering only integers, answer the following questions:

Considering only integers, answer the following questions: (i) A number N = 21P53Q4. The number of ordered pairs (P,Q) such that the number āNā is divisible by 44 is? (ii) A number N = 73P4961Q0. ...
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### Product of a Finite Number of Matrices with a Cosine Entry

Does any one know how to prove the following identity? $$\mathop{\mathrm{Tr}}\left(\prod_{j=0}^{n-1}\begin{pmatrix} 2\cos\frac{2j\pi}{n} & a \\ b & 0 \end{pmatrix}\right)=2$$ when $n$ is ...
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### Proving $x^2 - x = y^5 - y$ is a hyperelliptic curve

Greetings to one an all! How can we prove the curve "$x^2 -x = y^5-y$" is a hyperelliptic curve? Is a hyperelliptic curve the same as a hyperbolic elliptic curve or are there any differences?
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### Is there a method to calculate large number modulo?

Is there a (number theoretic or algebraic) trick to find a large nunber modulo some number? Say I have the number $123456789123$ and I want to find its value modulo some other number, say, ...
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### Application of Legendre, Jacobi and Kronecker Symbols

Legendre, Jacobi and Kronecker Symbols are powerful multiplicative functions in ...
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### Elementary proof that $\sum_{k=1}^{\infty}{\frac{1}{m^{k^2}}}$ is irrational (for any integer $m > 1$)

I used similar technique as Fourier's proof of irrationality of $e$ https://en.wikipedia.org/wiki/Proof_that_e_is_irrational to show that this series is indeed an irrational number but I was wondering ...
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### are these two continued fractions equivalent?

I would like to pose the following conjecture.Given $$\phi(q) =\cfrac{1}{1-q+\cfrac{q(1-q)^2}{1-q^3+\cfrac{q^3(1-q^2)^2}{1-q^5+\cfrac{q^5(1-q^3)^2}{1-q^7+\ddots}}}}$$ and ...
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### Does number 1 really exist? [closed]

1) As per decimal system when we start numbering we can start from 0.000000.....1 or the number before 1 ie .99999999999......9 so since .00000... can be infinity we dont even start with the first ...
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### Finding the missing digits of $23!$

It is given $23!=2585201xy38884976640000$. Now it is required to find the value of $x$ and $y$. I know I could find it by using divisibility rules and solving simultaneous equations. Is there any ...
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### The form of the zeta function of an elliptic curve over a finite field

I seek a (very) elementary proof that the zeta function of an elliptic curve $E$ over $\mathbb{F}_q$ has the form $$Z(T)=\frac{1-aT+qT^2}{(1-T)(1-qT)}.$$ Something tedious and computational making use ...
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### Non trivial non prime or maximal ideals [on hold]

Let R = $\mathbb{Z}$ x $\mathbb{Z}_{80}$ A) Determine a maximal ideal of R B) Determine a prime ideal that is not maximal c) Determine a nontrivial ideal of R that is neither prime or maximal ...
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### Are the Bernoulli denominators always divisible by these corresponding primes?

I was wondering whether it has been proven/disproven yet or at least conjectured that the bernoulli denominator of $B_{2n}$ is divisible by $2n+1$ if and only if $2n+1$ is prime? If not, must the ...
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### Estimates for $1/\zeta(s)$

Recently I am reading Stein's Complex Analysis, and he is going to prove the prime number theorem after estimating the value $1/\zeta(s)$. However, I don't understand the technical details of the ...
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### Finding primes using the Fibonacci sequence in modular form

I was wondering if the following is already a known result in mathematics. I have tested it and it seems to work every single time. If I write the Fibonacci sequence in $\bmod (a)$ form and it ...
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### How to solve special type of Diophantine equation

I am so excited to learn finding integer solutions of the equation $x^2 -y^5 = x-y$. I just found few solutions by plugging various integers in place of $x$ and $y$. But, I need a permanent method or ...
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### Minimize given LCM

Find the smallest possible value of $n_1+n_2+\cdots+n_k$ such that $LCM(n_1,n_2,\ldots,n_k)=(2^2)(3^3)(5^5)$. Note that $k$ is not fixed. I know the answer should be $k=3$, $n_1=2^2$, $n_2=3^3$, and ...
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### regularization of sum $n \ln(n)$

I was testing out a few summation using my previous descriped methodes when i found an error in my reasoning. I'm really hoping someone could help me out. The function which i was evaluating was ...
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### Induction Proof - Primes and Euclid's Lemma

I'm having some trouble with this proof. Here's the question: Use mathematical induction and Euclid's Lemma to prove that for all positive integers $s$, if $p$ and $q_1, q_2, \dotsc, q_s$ are prime ...
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### Which power of an integer matrix is identity modulo $p^\alpha$?

I've read this question about identity power of an integer matrix. But how about power of a matrix modulo $p^\alpha$. $$A^m \equiv I \pmod{p^\alpha}$$ How can I find the minimal $m$ that the above ...
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### Decomposing an integer into primes raised to different powers

The number $711000000$ can be written as $79^1 \times 2^6 \times 3^2 \times 5^6$. How are these numbers found? I guess the more general question is - given $n \in \mathbb Z$, how can you ...
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### Solving the equation $x^2-7y^2=-3$ over integers

I'd like to solve the following Pell equation: $$x^2-7y^2=-3$$ Where $x$ and $y$ are integers. I applied the usual procedure, which avoids continued fractions: The two minimal positive integer ...
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### Ordering of natural numbers

Show that it is possible to arrange the numbers 1, 2, . . . , n in a row so that the average of any two of these numbers never appears between them. Hint: Show that it suffices to prove this fact ...
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### What inconsistencies/ fallacies do techniques used in assigning -1/12 to the infinite sum “1+2+3+4+…” contain? [closed]

It would be helpful if the downvoters could communicate the reason for downvoting the question. I think I have clarified the question adequately below. If I am missing something, requesting you to ...
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### Fi Binary Number [closed]

A Fi-binary number is a number that contains only 0 and 1. It does not contain any leading 0. And also it does not contain 2 consecutive 1. The first few such number are 1, 10, 100, 101, 1000, 1001, ...
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### Arithmetic Derivatives: Arithmetic Logarithmic Derivative Problem

In Calculus, whenever we see a constant and want to take the derivative of it, it always is 0. However in Number Theory, we have something called the arithmetic derivative in which we can ...
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### Orbits of the $\text{SL}(n,\mathcal{O}_K)$-action on $\mathbb{P}^{n-1}(K)$ for a number field $K$.

I was reading some notes of Keith Conrad where he proves that the number of orbits of the $\text{SL}(2,\mathcal{O}_K)$-action on $\mathbb{P}^{1}(K)$ for a number field $K$ is precisely the class ...
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### Chiese Remainder Therorem : No solutions

I have asked a similar question before on Chinese Remainder Theorem. Now concepts are getting clear. Thinking of a possible case where there are no solutions. Suppose the question is ...