Modular arithmetic (clock arithmetic) is a system of integer arithmetic based on the congruence relation $a \equiv b \pmod{n}$ which means that $n$ divides $b-a$.

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Is every model of modular arithmetic either even or odd?

Modular Arithmetic (MA) has the same axioms as first order Peano Axioms (PA) except $\forall x (Sx \ne 0)$ is replaced with $\exists x(Sx = 0)$. ...
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Constructing pairs of units $(x,y)$ which solve $x^2 + y^2 \equiv -1 \pmod{N}$

A classic result on the way to the Lagrange Four Squares theorem — for instance proven by Theorem 87 of Hardy & Wright, as noted by this remark on the Four Squares theorem — is that ...
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Statement about Woodal primes.

A Woodal number is an integer of the form $n 2^{n}-1$. A Woodal prime is an integer that is both a prime and a Woodal number. Let $p$ be a prime of the form 1 mod 4. Then $p 2^{p} -1$ is never a ( ...
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Subsets of cyclic group with distinct pairwise differences

Given a number $m\in\mathbb N$, let $\mathbb Z_m=\{0,1,\dots,m-1\}$ denote the ring of integers modulo $m$ (although we won't need multiplication, so any cyclic group of order $m$ will do). Given a ...
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Solving $x^e\equiv a\pmod n$

I am trying to put together the techniques involved in solving $$x^e\equiv a\pmod n, \text{ for known } n\in\mathbb N^*, e\in\mathbb N^*, a\in\mathbb Z_n, \text{ unknown }x\in\mathbb Z_n$$ I'm ...
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Calculations in the field $\mathbb{F}_{13}$

The text i am reading claims that $\frac{246}{14}=1$ in the field $\mathbb{F}_{13}$. However, i cannot figure out why this is correct. Since $13 \cdot 18 = 234 \Rightarrow 246 = 12$ and $14 = 1$. How ...
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72 views

Composite $n$ such that $\sigma(n) \equiv n+1 \pmod{\phi(n)}$

I'm looking for composite $n$ such that $$\sigma(n)\equiv n+1\pmod{\varphi(n)}$$ Are there only finitely many? Can this be proved? This is Sloane's A070037 but there's not much information in the ...
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Find n's for which $P_n$ is prime.

Consider the numbers $P_n=(3^n-1)/2$. Find $n$'s for which $P_n$ is prime. Conjecture: If $n \equiv 1 \mod 6$, and $n$ is prime, then $P_n$ is prime. I have tried proving this by contradiction but ...
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62 views

Find all $x,y\in\mathbb{Z}$ s.t $2x^3-7y^3=3$

Find all $$x,y\in\mathbb{Z}$$ such that $$2x^3-7y^3=3$$ Solution: We consider first $$2x^3-7y^3\equiv3 \pmod 2$$ $$5y^3\equiv 1 \pmod 2$$ $$y^3\equiv 1 \pmod2$$ which has solution $y\equiv 1 ...
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Fast way to find the smallest root of modular polynomial

Suppose you're given a polynomial with integer coefficients: $$ P(x) = \sum_{i=0}^{n}{a_i x^i} $$ Is there a fast way to find the smallest root modulo $M$, where $M$ is some composite number with ...
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191 views

How to solve congruence $x^y = a \pmod p$?

I'm having trouble solving this congruence: $$x^{114} \equiv 13 \pmod {29}.$$ I thought that it made sense to try to solve it using this idea: "Suppose you want to solve the congruence $ x^y \equiv a ...
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86 views

Equivalence classes of triplets satisfying $x^2+y^2+z^2=0$ over $\mathbb{F}_p$

The affirmative answer to this question illustrates that the equation $$x^2+y^2+z^2=0$$ has $p^2-1$ nontrivial solutions over $\mathbb{F}_p$ (solutions that are not $(0,0,0)$). If $(x,y,z)$ is a ...
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64 views

An intuitive interpretation of Montgomery pair corrlation function vs. prime divisibility?

Theorem - If the Riemann hypothesis would be true, and the Montgomery pair correlation conjecture (see linked article page 183-184) true too; let $p \in \Bbb P$ prime, $n \in \Bbb N$ and $$\mathcal ...
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627 views

Curious 123456789 x 8 + 9 = 987654321

I was recently asked why the following results holds \begin{align} 1 \times 8 + 1 &= 9 \\ 12 \times 8 + 2 & = 98 \\ 123 \times 8 + 3 & = 987 \\ 1234 \times 8 + 4 & = 9876 \\ 12345 ...
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146 views

Number of elements in projective special linear group over $\mathbb{Z}/n\mathbb{Z}$

While reading a paper about the modular group $\Gamma = PSL_2(\mathbb{Z})$, I came upon the following sentence ($\Gamma(N)$ is the kernel of the canonical map $PSL_2(\mathbb{Z}) \rightarrow ...
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79 views

Fast check If the remainder is 1

Is there any fast method to 'say' that $R = (A \mod B)$ is $1$ or $R > 1$ or $R \neq1$ or $R > k>1$ ( where $k$ is a small integer on $32$ bits) without to actually calculate the real value ...
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57 views

Find all integers n which satisfies $1^n+9^n+10^n=5^n+6^n+11^n$

Find all $n\in\mathbb Z$ which satisfies $1^n+9^n+10^n=5^n+6^n+11^n$ for $n=2\ or\ n=4$ it is equal but are there other numbers?
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32 views

Congruence of polynomials

Prove that $y^i \equiv y^{i \pmod 4} \pmod {y^4 + 1}$ for all $i \ge 0$. Any hints on how to solve this? I don't even know where to start.
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323 views

How can I solve system of linear equations over finite fields in WolframAlpha?

Is it possible to solve system of linear equations over finite fields using Wolfram Alpha? If yes, how can I do that? Let us take a system $x+y+z=0$, $2x+y+2z=0$, $x+3y+z=0$. If I want to solve this ...
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102 views

When quadratic formula $ax^2 + bx +c = 0$ (mod p) is solvable modulo prime p?

I am learning about quadratic congruences and I don't now how to decide, for which a,b,c and p there is a solution of the congruence. It is sufficient if the determinant $\sqrt{b^2-4ac}$ has a ...
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54 views

How to find connection between two modulo statements

Given $x\mod n=a$ and $ x\mod m=b$, (here, $n=1000000007$ while $m$ can be any number less than $n$) is there any meathod to find $b$ in terms of $(n,a,m)$...The answer should not contain $x$. I tried ...
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Least value of a multi-residue CRT

Given coprime moduli $m_1,\ldots,m_n$ with $k\ge2$ residues in each modulus, what is the least nonnegative value congruent to one of the specified residues to each modulus? Obviously it must be less ...
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35 views

Is there any relation between the modular inverse of the same integer under different modulus?

I mean, suppose $$ab \equiv 1 \mod{m}$$ $$ac \equiv 1 \mod{n}$$ I wonder if there is any relation between $b$ and $c$? Could we compute one from another? Thanks in advance!
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35 views

For which minimal $k$ true is that ${4}^{k}\cdot n\leq \displaystyle\sum^{n}_{i=1}{a}_{i}^{k}\leq {5}^{k}\cdot n$, ${a}_{i}\in {1,2,3,4,5,6}$?

I've got the following inequality, which bounds Minkowski distance. ${4}^{k}\cdot n\leq \displaystyle\sum^{n}_{i=1}{a}_{i}^{k}\leq {5}^{k}\cdot n$ and values of ${a}_{i}\in {1,2,3,4,5,6}$ We know all ...
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27 views

Name for summation that returns GCD(k,n)

Define $$ H(k,n)=2\sum_{i=1}^{n-1}\left\lfloor\frac{ki}{n}\right\rfloor\;. $$ We can prove that $H(k,n)=nk-k-n+\gcd(k,n)$. Does this $H$ carry some known name?
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eliminate very kth element in mod n… save a given element for last

we are given 1....n numbers. lets say we are to save a given number element k for last in elimination. We start eliminating them in the following manner. I eliminate 1 at first. Then eliminate the ...
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How much information do I gain from each modular inequality?

Problem details: Let $p = 2^{48}$, $s\leftarrow\{0,1\}^{48}$, and $a,b$ known constants. Furthermore let $f(x) = a x + b \pmod{p}$ and let the value $r_k$ be defined by the first-order recurrence ...
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A system of linear equations in integer squares - solvable?

I consider this more a "recreational math" problem, possibly lacking a solution (because it stems from the question of "magic square of squares") or simply intractable with reasonable effort. ...
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190 views

Baby-step giant-step algorithm

I have equation in form $a^x = p (\textrm{mod q})$, knowing $p, q, a$. I have to use Shank's algorithm (Baby-step giant-step). I found some exercise and explanation on ...
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Multiplication structure for finite abelian rings of order $p^2$.

Consider an odd prime $p$ and a positive integer $a$ as close to 2 as possible that is not a quadratic residue $mod$ $p$. If we extend the ring $mod$ $p$ with the element $b = a^{\frac{1}{2}}$ then ...
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Solving $key=(\sum_{K=0}^n\frac{1}{a^K})\mod m$ with High limits

I was solving this equation:- $$key=(\sum_{K=0}^n\frac{1}{a^K})\mod m$$ Given $$ 1,000,000,000 < a, n, m \; < 5,000,000,000 $$ $$ a, m \; are \;coprime $$ I solved it bruteforcely but it ...
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$a^{(b^c)} \mod m$ where $c$ can be very very large

I am trying to solve the following problem. I need to find the value of $$ a^{(b^x)} \bmod m $$ where $a,b$ are integers and $$ x = \pmatrix{n\\0}^2 + \pmatrix{n\\1}^2 + ... + \pmatrix{n\\n}^2 ...
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114 views

Determining maximum value of $a^b\bmod N$ when $\gcd(a,b)$ is known

Suppose we know the greatest common divisor, $\gcd(A,B)$, of two numbers $A$ and $B$. Is there a way that we can find the maximum value of $a^b \bmod N$ where $N$ is any number? We have a finite ...
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26 views

Mistake involving Chinese remainder theorem

I ran into a snag in attempting to solve the following problem. $$x\equiv r\pmod p$$ $$x\equiv0\pmod q$$ I did the following $$x=k_1q=k_2p+r$$ $$k_1q\equiv r\pmod p$$ $$k_1\equiv rq^{-1}\pmod p$$ ...
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Primality of $P_{q,j}=(2^{q^{j+1}}-1)/(2^{q^j}-1)$ , when $q$ is prime, $j\ge0$?

Let $P_{q,j}=(2^{q^{j+1}}-1)/(2^{q^j}-1)$ , $q$ prime and $j\ge0$. $P_{2,j}$ is a Fermat number, $P_{q,0}$ is a Mersenne number. Apart from Fermat primes and Mersenne primes, and apart from ...
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36 views

Why doesn't this base 10 number x mod 2^y work for converting base 10 to binary

Okay I tried to convert 1 million to binary by dividing by a power of 2 and taking the remainder and dividing that by a power of 2 and so on and I got this: 1111010000100100000 Google says 1 million ...
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Modular arithmetic - Suggestions to begin

I've always wanted to start studying modular arithmetic to try to solve problems like: $$\text{find } n \in \mathbb{N} : 4n^2 \equiv 1 ~(\text{mod }{10^4})$$ I've good basis of mathematical analysis ...
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Solving $a = b^x \mod c$ where $a$, $b$, and $c$ are large integers

I am trying to solve this problem : Given $a$, $b$, $c$ integers (that could be very large), find $x$ integer such as : $a=b^x \mod c$ I tried by coding in C a loop checking the equality and ...
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Group of invertible elements, isomorphic to $Z_4$?

The group $f(8)$ of invertible elements in the ring $Z_{10}$ has four elements, $f(10) = \{ [1,], [3], [7], [9]\}$. Is this group isomorphic to $Z_4$ or to the symmetry group of the rectangle? ...
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41 views

Finding modular variable from some information about multiplications

I have got a very specific mathematic/algorithmic problem. In fact, it is a an automata theory problem(buchi automaton), but this problem seems to be naturaly translated into a modular arithmetic ...
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35 views

Identity based encryption

I am implementing ID based encryption in c# right now i am having problem at the following mathematics $$H_1: \{0,1 \}^n \times \{0,1 \}^n \to \mathbb{Z}^*_p, \text{ what does this expression ...
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Is the modulo operation a standard mathematical operation which actually means Euclidean division?

I was having this long discussion with a colleague as to whether the Modulo operation is a standard mathematical operation or not. He insisted the programming languages don't have a defined standard ...
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49 views

Check proof of Fermat's Little Theorem

I wrote an informal proof of Fermat's Little Theorem. Can someone check to see if the reasoning is valid, and if so how I could formalize it: Theorem: $n^{p-1} \equiv 1 \pmod p$ $ n \times 2n ...
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Sufficient condition for an equivalence

What is a sufficient condition for the equivalence $$ a_1 \uparrow a_2 \uparrow ... \uparrow a_n \equiv a_1 \uparrow a_2 \uparrow ... \uparrow a_n \uparrow a_{n+1}\ mod(\ m)\ ? $$ In a closely ...
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34 views

Lucas Number Equivalent of the Pisano Period?

I'm doing some work with the Pisano Period, and it's leading to also talk about the Lucas numbers $mod \ m$, and their period. Is there a specific notation already designated? Sticking with using a ...
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59 views

Question about tetration modulus a prime $p>100$

Define $x§y$ as the power tower : $x^{x^x...}$ where $...$ means $y$ times. For instance $2§1=2,2§2=4,2§3=16,2§4=2^{16}$. See : http://en.wikipedia.org/wiki/Tetration Let $p$ be a prime larger than ...
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47 views

Could Someone Just Verify This Proof for Me? (Euler's Theorem)

I came up with this proof for my number theory class. Is it valid? Proposition: $u\in U_m \Rightarrow u^{\varphi(m)}=1$ (Where $U_m$ is the multiplicative group of integers modulo $m$) Attempted ...
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25 views

Finding the totient functionlike function for an irrational number like (a+b*sqrt(5)) where a and b are whole numbers mod M where M is a whole number.

I need to find if a value $T$ exists for irrational number of the form $(a+b\cdot \sqrt{5})$ such that $(a+b\cdot \sqrt{5})^T = 1 \pmod M$. Also ,is it possible to find out upper bound for T .
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65 views

Finding $x^{k}\mod n$ quickly- find algorithm using $x^{2l}=x^{l} \cdot x^{l}$ and $x^{2l+1}=x \cdot x^{2l}$

Finding $x^{k}\mod n$ quickly- find algorithm using $x^{2l}=x^{l} \cdot x^{l}$ and $x^{2l+1}=x \cdot x^{2l}$. Here's my simple algorithm: We first check if $k=1$ or $k=2l$ or $k=2l+1$ for some $l ...
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212 views

How many ring homomorphism are there from $\mathbb{Z} / 12\mathbb{Z}$ to $\mathbb{Z} / 20\mathbb{Z}$? Find the image and kernel of them.

How many ring homomorphism are there from $\mathbb{Z} / 12\mathbb{Z}$ to $\mathbb{Z} / 20\mathbb{Z}$? Find the image and kernel of them. The above is the question, this is my attempt at an ...