0
votes
2answers
80 views

Find all non-trivial triplets $(a,b,c)$ such that $ a+b^{d}\equiv0\pmod c $ for all $d>0$

Let $a,b,c$ be co-prime integers $>2$ . Find all non-trivial triplets $(a,b,c)$ such that $ a+b^{d}\equiv0\pmod c $ for all $d>0$.
1
vote
3answers
96 views

Find all $n\in\mathbb N$, $n>3$ such that $p(n)=2n+16$, where $p(n)$ is the product of all the prime numbers less than $n$.

Find all $n\in\mathbb N$, $n>3$ such that $p(n)=2n+16$, where $p(n)$ is the product of all the prime numbers less than $n$. E.g. $p(7)=2\cdot3\cdot5$ (and $n=7$ is a solution). Let ...
3
votes
1answer
74 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 ...
0
votes
1answer
45 views

Reducing radical congruence to polynomial congruence

I am trying to find a way to describe all integer values of $x$ for which the following holds true: $\sqrt[2]{(1/2) * x * (x - 1) + (1/4)} + (1/2)\in \mathbb{Z}$ Noting that this can be equivalently ...
2
votes
0answers
61 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?
4
votes
0answers
62 views

Why has $3^x+4^y=5^z$ has only one solution (2,2,2) in positive integers? [duplicate]

First, do we have to exclude the cases, where $(x,y,z)$ are not all even or odd and then show the only possibility ? or is there a geometric solution maybe ?
0
votes
0answers
29 views

Is there a way to solve system of homogenous polynomial equations of degree 2 mod composite n?

I have quite specific system of equations, that I need to solve, each of the equations in the system is of following form: $$ \sum_{j=1}^{n}b_{ij}u_{j}^{(k)} = c_i^{(k)} $$ with both $i,k$ live in ...
0
votes
1answer
220 views

How to Solve an equation with mod for a variable?

I have following equation to be solved, but I am having some trouble in making an understanding and doing so. (d * e) % v = 1 e and v are known. How to solve this ...
0
votes
1answer
32 views

For what $x$ is $\sum_{k=1}^{n-1} (x+k)^n \equiv 0 \pmod n$ dependend on $n$? (so far only *odd* n)

(This is a detail in my attempted answer of this MSE question) We look at $$f_n(x) = \sum_{k=1}^{n-1} (x+k)^n $$ I came to the following observation - for odd $n$ at the moment -, but do not see how ...
1
vote
1answer
52 views

Implications of solubility of equations modulo all natural numbers

Let $P(x_1,x_2,...,x_n)=0$ be a given polynomial Diophantine equation in $n$ variables with integer coefficients (for example $x_1^2+3x_2-10+x_1x_2^4=0$). Suppose further that this equation has a ...
1
vote
1answer
31 views

how comes $s=4$ and $t=3$ for $4=7s-8t$

i am given these two problems: $x\equiv 1 (\bmod 7 )$ and $x \equiv 5( \bmod 18)$ I tried this way: $x\equiv 1 (\bmod 7 )$ is basically $x = 1 + 7s$ and $x\equiv 5 (\bmod 18 )$ is $x=5+18t$ then ...
1
vote
1answer
60 views

A Modular Diophantine Equation

$a = (N \bmod c)\bmod d$ $b = (N \bmod d)\bmod c$ That is $a$ and $c$ is remainder of $N$ when divided by $c$ and $d$ in different order. What can we say about $N$ if $a,b,c,d$ are known and $N ...
4
votes
2answers
114 views

Diophantine equation $x^2-dy^2=k$ in $\mathbb{Z}_n$

Does anyone know when $x^2-dy^2=k$ is resoluble in $\mathbb{Z}_n$ with $(n,k)=1$ and $(n,d)=1$ ? I'm interested in the case $n=p^t$
4
votes
2answers
176 views

Solving Pell's equation(or any other diophantine equation) through modular arithmetic.

Let us take a solution of Pell's equation ($x^2 - my^2 = 1$) and take any prime $p$. Then we have found a solution of the Pell's equation mod $p$. Now, conversely, for any prime $p$, we can find a ...
3
votes
1answer
69 views

Find all integer solutions of $35x^{31} + 33x^{25} + 19x^{21} \equiv 1 \pmod{ 55}$

Find set of all integers x for which the following holds: $35x^{31} + 33x^{25} + 19x^{21} \equiv 1 \pmod {55}$ Since $55 = 5\cdot 11$, simultaneous congruences: $35x^{31} + 33x^{25} + 19x^{21} ...
2
votes
1answer
70 views

How many solutions to prime = $2 b^2 c^2 + 2 c^2 a^2 + 2 a^2 b^2 - a^4 - b^4 - c^4$

Let $a,b,c$ be integers, no sign restriction. Let $p$ be a given prime. How to find the number of solutions to $p = 2 b^2 c^2 + 2 c^2 a^2 + 2 a^2 b^2 - a^4 - b^4 - c^4$ ? Note, from Heron's ...
3
votes
3answers
156 views

How many solutions to prime = $a^3+b^3+c^3 - 3abc$

Let $a,b,c$ be integers. Let $p$ be a given prime. How to find the number of solutions to $p = a^3+b^3+c^3 - 3abc$ ? Another question is ; let $w$ be a positive integer. Let $f(w)$ be the number of ...
1
vote
2answers
92 views

How many solutions to prime = $(d^2-2ad+b^2-2ab+2a^2)(d^2-2cd+2c^2-2bc+b^2)$?

Let $a,b,c,d$ be integers $>-1$. Let $p$ be a given prime. How to find the number of solutions to $p = (d^2-2ad+b^2-2ab+2a^2)(d^2-2cd+2c^2-2bc+b^2)$ ? I assumed that this polynomial above does not ...
0
votes
1answer
196 views

How to solve $ x^2+4x+2 \equiv 0 \pmod{49}$

How to decompose equations below and then solve 1)$$ 2x^3 + 7x - 4 \equiv 0 \pmod{25} $$ 2)$$ x^2+4x+2 \equiv 0 \pmod{49}$$ Thank you.
1
vote
2answers
209 views

Solving an equation in modular arithmetic

Given $A, B, C$ positive integers, $B < C,$ I would like some thoughts about (possibly efficient) ways to find the smallest integer $X$, where $0 < X < C$, such that: $$A X + B \pmod{C - ...
4
votes
2answers
91 views

Deciding if a univariate quartic has a solution mod p

I have an equation in $x$ and I would like to determine if it has any solutions modulo a large prime $p$. Suppose $p$ is large enough that I can factor numbers up to $p$, but I cannot test all values ...
1
vote
1answer
97 views

Solving $a + b x = c y$ in the integer domain for general $a$

I have the following equation: $\frac{a + b x}{c} \in \mathbb{N}$ where $a,b,c,x \in \mathbb{N}$. and I want to find all x that satisfy these requirements. This should be the same as: $a + b x = c ...
2
votes
0answers
32 views

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 ...
1
vote
2answers
73 views

Enumerating all $x$ such that $b^n$ divides $x^2-x$

Given $b$ and $n$, I need to efficiently enumerate all integers $x$ (and $k$) such that: $$x^2-x=kb^{n}$$ $$x∈⟦b^{n-1},b^n-1⟧$$ I simply don't know how to proceed. I tried the naive quadratic ...
6
votes
4answers
947 views

solve $100x - 23y = -19$

I need help with this equation $$100x - 23y = -19.$$ When I plug this into Wolfram|Alpha, one of the integer solutions is $x = 23n + 12$ where $n$ is a subset of all the integers, but I can't seem to ...