Questions tagged [diophantine-equations]
Use for questions about finding integer or rational solutions to polynomial equations.
5,111
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Given n and m, Solve for x : $n ^ x = 1\mod m$ [duplicate]
Given positive integers $n$ and $m$, Find the smallest positive integer $x$ such that $$n ^ x = 1\mod m$$
$m$ is not necessarily prime, So if $m = p_{1}^{a_{1}}p_{2}^{a_{2}}p_{3}^{a_{3}}...p_{k}^{a_{k}...
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Analytic Number Theory - distribution of $x^2$ vs. the distribution of $x^2 - 2y^2$
My question originates from Rational Points on Elliptic Curves, (Silverman & Tate), though has little to do with elliptic curves.
In chapter $V$: Integer Points on Cubic Curves, section $3$ it ...
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47
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Is there a general method for solving a parabola in integers? [duplicate]
A parabolic equation such as the following:
$$538445x + 75816 = y^2 $$
with general form:
$$ax + b = y^2 $$
By a brute-force search, one Diophantine solution is $$ x = 2312 $$
Is there a smarter, ...
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2
answers
40
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diophantine equations over $Q_p$
I need to show that there is no trivial solution to the equation $3x^2+2y^2-z^2=0$ in $Q_3$
So can I look at solutions in $F_3$ and if I didn't find any then assume that there is no solution in $Q_3$?
...
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3
answers
106
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How can I generate solutions to the Diophantine equation: $a^2 + b^2 + 2c^2 = d^2$ [closed]
Is there a way to reduce or quickly find integer solutions to the equation $a^2 + b^2 + 2c^2 = d^2$ (where $a, b, c$ and $d$ are distinct natural numbers) ?
Sorry I’m really bad at Diophantine ...
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Parametrization of integer solutions of the equation $a^2+b^2=c^2+d^2=2x^2$
I need the general form of integer solutions to this equation $$a^2+b^2=c^2+d^2=2x^2$$ Here is my incomplete attempt:-
The parametrization of the integer solutions of the equation $$p^2+q^2=2y^2$$ is ...
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0
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The Diophantine equation $x^3+y^3=z^3+w^3$ and the Ramanujan number 1729.
The Diophantine equation $x^3+y^3=z^3+w^3$ and the Ramanujan number $1729$.
Can you please not only tell me, but also show me how to find solutions to such a Diophantine equation, for example, ...
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Strengthening of FLT [duplicate]
FLT is that there are no positive nontrivial integer solutions to the Diophantine equation $a^n + b^n = c^n$ for $n>2$. What about the conjecture that there are no nontrivial solutions to the DE $a^...
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Prove the existence of the number $n$ that satisfies a property.
Problem
Prove that there is a $n \in$ $ \mathbb{N} $ such that the equation has at least 2022 solutions.
$x+y+\sqrt{xy}=n$ ,with $x,y\in$ $ \mathbb{N}$
Attempt
My first attempt was to fix $x$ to $0, ...
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Who can prove that $7|4x^2+5x+2+d $ following solutions I found are correct?
Who can prove that $7|4x^2+5x+2+d $ following solutions I found are correct (complete)? Where $d$ is integer constant (not relation with $x$), find $d$ which make $7|4x^2+5x+2+d $ have solution, and ...
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Index of generating set to $E(\mathbb{Q} ) /2E(\mathbb{Q} )$ can be arbitrarly large
I'm trying to prove question 2.12.25 from "Elliptic Curves, Modular Forms, and Their L-Functions
by élvaro Lozano-Robledo".
Let $E$ be an elliptic curve, such that the image of the points $...
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1
answer
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Rational solutions to $\{x^2+y^2=2,x^2+z^2=4\}$
I am working through some examples from Silverman's The Arithmetic of Elliptic Curves and I've run into the following system of equations, $$\left\{x^2+y^2=2,x^2+z^2=4\right\}$$ I am quite confident ...
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Criteria for a linear system of equations to have integer solutions
Suppose we have a linear system of equations given in matrix form as
$$
\begin{bmatrix}
a & b & c \newline
d & e & f \newline
g & h & i
\end{bmatrix}
\begin{bmatrix}
x \...
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Solving the Diophantine system of equations $w + x + y - z = a, wx + yz = b$
If we have a Diophantine system of equations of the form
$$
\begin{align*}
x + y & = a, \\
xy & = b
\end{align*} \tag{1}
$$
with $x,y$ unknown and $a,b$ known, we could solve it using the ...
- 2,182
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0
answers
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Find maximum value of $y$ for which $y^2xz^3 + yx^2$ is a perfect square
Problem:
Given $y,x,z$ are positive integer variables and $N$ is a given integer constant and $x < z \le N $ and $y$ is square-free and $y \ne x$,
find the maximum value of $y$ (in terms of $x$ or $...
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No Rational Solutions To Elliptic Curve System [closed]
I was asked to show that the following system has no rational solutions:
$y^2 = 17 + 2x^2
\\
y^2 = 34 +z^2
$
That is to say, that there are no $x,y,z \in \mathbb{Q}$ such that the above relations hold....
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-1
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1
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Problem related to narcissistic numbers [duplicate]
Prove that there is an infinite number of positive integers $a, b, c$ that satisfy the following condition:
$a^3+b^3+c^3=abc$ (the number formed by the integers $a,b,c$ in base $10$)
Examples:
$1^3+5^...
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5
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481
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Gap between two Pythagorean triples
Let $s$ and $t$ be two positive integers. I am interested in the following Diophantine system :
$$\mathscr{S}(s,t) :
\left\{
\begin{array}{l}
x^2+y^2=a^2\\
(x-s)^2+(y-t)^2=b^2
\end{array}
\right.$$
...
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3
votes
3
answers
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The diophantine equations $x^2-y^5=\pm2$
Update: It only remains to handle Equation $x^2-y^5=2$ (see edit below).
Question 1. What are the integer solutions to the equations $x^2-y^5=\pm2$?
I know that these diophantine equations have only ...
- 19.1k
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1
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On the elliptic curve $u(u+2186^2)(u+2188^2)=v^2$ and 7th powers?
I. Elliptic curve
Given some constant integer $m$, a solution to the elliptic curve,
$$E:=u\big(u+(m^7−1)^2\big)\big(u+(m^7+1)^2\big)=v^2$$
which is not a torsion point implies a polynomial identity ...
- 49.7k
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Is there a general algorithm to decide whether an integer value is attainable by a quadratic form?
I'm not sure if I phrased the question correctly, but let's say that due to a result called 15 theorem that "if a positive definite quadratic form with integer matrix represents all positive ...
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1
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Weighted sum of squares
Suppose $A,B,C$ are integers. When does the equation
$$ Ax^2+By^2+Cz^2=0$$ have a nontrivial solution in integer $x,y,z$?
I don't need all solutions, or even one solution, I just want to know if a ...
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Rational solutions of the equation $Y^3=X^3-5 X^2+8 X-4$
Source: Exercise from algebraic structures class.
The question is:
a) Find all the rational solutions of the equation
$$
Y^3=X^3-5 X^2+8 X-4
$$
b) Find all the integer solutions of that equation.
I'...
-3
votes
1
answer
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Determine a value, where quadratic equation will generate the perfect square
In order to find RSA factors ($pq = N$), we have to solve a quadratic equation $x^2+Bx+C=y^2$, where:
$x_1 < x_2$,
$x_1, x_2$ and $y_1, y_2$ are (positive) integer numbers,
$x_1$ is the smallest ...
3
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4
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Solve over natural numbers: $m^3=2n^3+6n^2$. Functional equation gives rise to a diophantine equation!
My question is basically to find all natural numbers $(m,n)$ such that
$m^3=2n^3+6n^2$
First for some background (this is not really that relevant but anyways):
I was trying to solve an olympiad ...
- 1,517
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A number theory question about diophantine equations [duplicate]
Determine the side lengths of a right triangle if they are integers and the product of the legs’ lengths equals three times the perimeter.
I tried this question for 3+ hours and I get 6 and 0 every ...
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Finding a rational (or integral) parametrization for a system of (homogeneous?) quadratic forms?
The solution of the $3 \times 3$ magic square of squares involves finding $8$ arithmetic progressions of squares (APS) using $9$ integers. I'm looking at a much smaller portion of the problem: $3$ ...
- 2,506
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Can the coefficients in this equation be found?
I have the following equation:
$$
Ω = -2 A^4 - 2 B^4 + 2 C^4 - D^4 + E^4 + F^4 - G^4 - H^4 + I^4 + J^4 - K^4 + 12 A B L P
$$
where
$C = B + A$
$E = D + A$
$I = H + G - F$
$J = H + G - D - A$
$K = H + ...
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Do we have $n$ such that $a^n\equiv b\mod p$? [duplicate]
Given fixed $a,b$ and fixed prime $p$ such that $(a,p)=1,(b,p)=1,a \not\equiv b\mod p$. Do we have $n$ such that $a^n\equiv b\mod p$?
Progress: This obviously is not true for all $a$. In fact if $a$ ...
1
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0
answers
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views
What is the relevance of studying the Markov Diophantine Equation And its Generalized Forms.
The relevance of the Markov equation and its Generalized forms are still a mystery to me. I cannot find anywhere online about the rational or relevance for studying the topic outside of pure curiosity....
1
vote
1
answer
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Find rational solutions to $x^2 + y^2 = 6$
This question comes from Rational Points on Elliptic Curves (Silverman & Tate) Exercise $1.7$ (a).
Find rational solutions (if any) to $x^2 + y^2 = 6$
I think there exist no solutions and here ...
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1
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How many integer solutions of $2^x+3^y-7n=0,~n \in \mathbb Z$?
Consider the equation $2^x+3^y=7n,~n \in \mathbb Z$.
How many integer (positive) solutions are there for this equation? Can it have infinitely many positive integer solutions?
What is an example of a ...
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Number of positive integer solutions of diophantine inequality.
Given a Diophantine inequality of the form $$a_{1}x_{1}+a_{2}x_{2}+\dots+a_{n}x_{n}\leq N $$ How many positive integer solutions does it have? Here all $a_{i}\geq0$ and $x_{i}\geq 0$.
I was able to ...
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votes
2
answers
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Writing a square as a sum of three non-zero squares in geometric progression
Let $k$ be a given positive integer. I want to solve the following system of Diophantine equations: $$\begin{cases} a^2 + b^2 + c^2 = k^2 \\ b^2 = ac \end{cases}$$
where $a, b, c \in \mathbb{N}$ are ...
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3
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Solving the diophantine equation $1/x+1/y+1/z=1/2$
I need help finding all positive integer solutions to the following Diophantine equation:
$$\frac1x+\frac1y+\frac1z=\frac12$$
What I figured out so far was that we essentially need to find $3$ ...
- 2,606
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votes
1
answer
100
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Is it possible to further generalise Brahmagupta's identity?
Is it possible to generalise Brahmagupta's identity
further, by which I mean, for different n, for example take the equations:
$$1357 = 37^2 - 3\times2^2$$
$$1357 = 38^2 - 87\times1^2$$
$$1357^2 = ...
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votes
4
answers
168
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All natural number solutions to the equation $a^2+b^2=c^2+d^2=2x^2$
Yesterday, I posted this question, and got that if $a$, $b$ and $c$ are in the form $$a=k(m^2-n^2+2mn)$$
$$b=k(n^2-m^2+2mn)$$
$$c=k(m^2+n^2)$$ where $m$ and $n$ are natural numbers, $a$, $b$ and $c$ ...
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votes
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Fast algorithm for solving diophantine equation $x^4=a^4+b^4+c^4+d^4$
This problem was posed to an acquaintance of mine (at their university) and piqued my interest so I tried to solve it. The description goes as follows:
Write a program that finds a solution to the ...
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votes
3
answers
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All natural number solutions for the equation $a^2+b^2=2c^2$
$a$, $b$ and $c$ of all Pythagorean triplets can be written in the form
$$
\begin{split}
a &= 2mn\\
b &= m^2-n^2 \\
c &= m^2+n^2
\end{split}
$$
where $m$ and $n$ are natural numbers. For ...
- 95
1
vote
1
answer
150
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Solve $7^a+b!=13^c$ over $\mathbb{Z}^{+}$
Solve the following Diophantine equation $7^a+b!=13^c$ over positive
integers.
Clearly $7^1+3!=13^1$ and $7^2+5!=13^2$. Are there any more solutions? Here are some thoughts.
$b!$ must be even, so $b&...
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1
answer
110
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Count number of solution to Diophantine equation $k_1a^2+k_2ab+k_3b^2-k_4c^2=0$
I am looking to count number of solutions of diophantine equation $k_1a^2+k_2ab+k_3b^2-k_4c^2=0$.
such that $ 1 \le a, b, c \le N$ and $gcd(a, b) = 1$
and $k_1,k_2,k_3,k_4$ are positive constant ...
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1
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49
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diophantine equation with varying degrees of x and y
Solve
$$x^3y^2(2y-x)=x^2y^4-36$$ where $x$ and $y$ are integers.
Preparing for Math Olympiads and I am totally unable to come up with a method of solving this equation without factoring (if there is ...
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4
votes
1
answer
193
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Find efficient way to generate all solutions to Diophantine equation $a^2+5ab+3b^2-c^2=0$ under a given bound $N$
I am looking to solve Diophantine equation $a^2+5ab+3b^2-c^2=0$.
a, b, c are all positive.
Since the number of solutions are infinite. Lets say we are only interested in solutions till a limit N ie $1 ...
- 421
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votes
1
answer
60
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Proving no integer solution exists that makes a polynomial a perfect square
The context for this is the following coding problem on Hackerrank. I'm trying to understand why one of their sample inputs (Sample Input 4) has no solution.
After a bit of math, it comes down to ...
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-1
votes
2
answers
148
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Solve Diophantine equation $a^2+5ab+3b^2-c^2=0$
Solve Diophantine equation $a^2+5ab+3b^2-c^2=0$
My thoughts are to express it as $(pa+qb)^2 = c^2 $and then solve it as Pell's equation.
One solution is $(1,9,17)$.I don't know whether it is a ...
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0
answers
24
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Solve the diphantine equation: xy + 114x - 81y = 9245 [duplicate]
Im not very experienced in this type of maths as I have only solved simple diophantine equations with euclides, so please explain the whole process used to get to the solution.
The exercise which this ...
- 1
3
votes
1
answer
165
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Parametrizations of $ x^4+y^4+z^4=9t^2 $ integer solutions
I would like to derive all the parametrizations for the nontrivial solutions of this Diophantine equation:
$ x^4+y^4+z^4=9t^2 $
I already know that with the Fauquembergue's parametrization I can find ...
- 389
3
votes
0
answers
51
views
What is the quartic undecidable Diophantine equation with 58 unknowns that's undecidable?
I found in Jones' 1980 work "Undecidable Diophantine Equations" Theorem 4 that it is claimed that at least one 58 unknown quartic Diophantine equation is known to be undecidable. It is not ...
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votes
0
answers
19
views
Are the solution sets equal for diophantine equation with opposite general solution signs at parameter?
To my knowledge, this formula can be used to solve the general Diophantine equation
$x=x_0\color{blue}+\frac{b}{d}\cdot t\\y=y_0\color{red}-\frac{a}{d}\cdot t$
or this
$x=x_0\color{red}-\frac{b}{d}\...
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-1
votes
1
answer
49
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Wrong result with diophantine equation when there is subtraction
I have equation $966x-686y=70$ and I get the wrong solution every time when there is "-" in equation $ax\textbf{-by}=c$. I don't know where I am making a mistake.
And my solution seems like ...
- 56