Questions tagged [diophantine-equations]
Diophantine equations are polynomial equations $F=0$, or systems of polynomial equations $F_1=\ldots=F_k=0$, where $F,F_1,\ldots,F_k$ are polynomials in either $\mathbb{Z}[X_1,\ldots,X_n]$ of $\mathbb{Q}[X_1,\ldots,X_n]$ of which it is asked to find solutions over $\mathbb{Z}$ or $\mathbb{Q}$. Topics: Pell equations, quadratic forms, elliptic curves, abelian varieties, hyperelliptic curves, Thue equations, normic forms, K3 surfaces ...
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A question regarding Goormaghtigh conjecture
I have a question regarding Goormaghtigh conjecture on the Diophantine equation
$$\frac{x^m-1}{x-1}=\frac{y^n-1}{y-1}.$$
Suppose that a positive integer $N$ is given. How many integer solutions are ...
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Minimal solution of simultaneous congruences
I would to determine the set of values $\lbrace a_1,a_2,a_3,\ldots,a_n \rbrace$ that minimizes the value of $x$ such that:
$$x\equiv a_1\mod p_1$$
$$\vdots$$
$$x\equiv a_n\mod p_n$$
where every ...
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Generalizing a pattern for the Diophantine $m$-tuples problem?
A set of $m$ non-zero rationals {$a_1, a_2, ... , a_m$} is called a rational Diophantine $m$-tuple if $a_i a_j+1$ is a square. It turns out an $m$-tuple can be extended to $m+2$ if it has certain ...
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FLT and integral points on elliptic curves
For integers $x,y,z,t,n$ define $S_n : xy(x+y)=t^n$.
For $ n > 2$, Fermat's Last Theorem implies there are no integral
solution on $S_n$ with $x,y$ coprime and $xy(x+y) \ne 0$ since $x,y,x+y$ are
...
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Solving $(u-x_1)^{1/3}+ (u-x_2)^{1/3}+ (u-x_3)^{1/3} = {v}^{1/3}$ with elliptic curves
Let $x_1$,$x_2$,$x_3$ be the roots of the cubic $x^3+px+q$ over $\mathbb Q$, the idea is that rational solutions $(u,v)$ of the equation
$$(u-x_1)^{1/3}+ (u-x_2)^{1/3}+ (u-x_3)^{1/3} = {v}^{1/3} \...
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Polynomial parametrization of the solutions to $yz=x^2+x\pm 1$
If a Diophantine equation has infinitely many integer solutions, how to describe them all? One standard approach is polynomial parametrization. For example, all integer solutions to the equation
$$
yz=...
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Parametrizing the solutions to a diophantine equation of degree four [closed]
Good evening,
Consider $x^4+y^4+z^4=2t^4$ where x,y,z,t integer.
Is it known how to find all parametrisation of this equation ?
If you have any parametrisation or reference of this equation, please ...
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How can one show that the hyperelliptic curve $y^2 = x^{p} + \frac{1}{4}$ has only one positive rational solution for every prime $p>3$?
Without applying Fermat's Last Theorem, how can one show that the hyperelliptic curve $y^2 = x^{p} + \frac{1}{4}$ has only one positive rational solution $(x,y) = (0, \frac{1}{2})$ for ever prime $p \...
user83236
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Concise formulation of set of equation systems
I have the following set of equation systems, and I would like to find a short, formal way to write it down. My main difficulty is that I cannot find a good way to write the indices of the variables $\...
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Help with this system of Diophantine equations
A couple hours ago, I'd posted a Diophantine equation question, but realized that I'd committed a rather preposterous blunder deriving it.
This is the actual question which I'm trying to solve:-
For ...
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Positive divisors of $P(x,n)=1+x+x^2+ \cdots + x^n$ that are congruent to $1$ modulo $x$
This is a follow-up question to Positive integer solutions to the diophantine equation $(xz+1)(yz+1)=z^4+z^3 +z^2 +z+1$
Let \begin{equation}
P(x,n)= 1+x+x^2+ \cdots + x^n, \end{equation}
\begin{...
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Find all rational solutions of this diophantine-equation?
Now, today, my friend tell me this problem was posted by American Mathematical Monthly (Vol. 111, No. 2 Feb., 2004), p. 165 by Wu wei Chao ,and It is said that this problem is unsolved, until now. ...
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Solve in positive integers $n!=m^2$
Is anybody know a solution of this problem?
(Sorry, correct question is here.)
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A generalization of Bernoulli's inequality and what does it application for?
Let $a_1 \ge a_2 \ge \cdots \ge a_n \ge 1$ or $0 \le a_1 \le a_2 \le \cdots \le a_n \le 1$ and $\alpha_1 \ge \alpha_2 \ge \cdots \ge \alpha_n \ge 1$ then
$$\left(\sum_{i=1}^{n}{\alpha_i} \right)\...
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How difficult is to find rational points on these genus 3 curves:
How difficult is to find all rational points on these genus 3 curves:
$$
(a) \quad \quad x^3 + y^3 x +y^2 - y = 0
$$
$$
(b) \quad \quad x^4 - y^3 + x y + x = 0
$$
Short motivation. Consider the ...
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Hardness of solving $0=\sum_{i=1}^k \operatorname{linear}_i(x_1,\ldots,x_n)^D$ over the rationals
This is related to cryptography and this question
and another question.
In short, we are asking about decomposing multivariate polynomial
as sum of perfect powers of linear polynomials.
Working over $\...
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Solutions of a linear diophantine equation
Let $N(h)$ be the number of solutions of the following linear diophantine equation:
\begin{equation}
x_1 + 2x_2 + 3x_3 + \dots + (h-1)x_{h-1} = 6h-6;
\end{equation}
where $h\geq 2$ and solution means ...
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On the equation $a^4+b^4+c^4=2d^4$ in natural numbers with $a<b<c<d$
I asked a simillar question with the weaker restriction:
On the equation $a^4+b^4+c^4=2d^4$ in positive integers $a\lt b\lt c$ such that $a+b\ne c$
.
I couldn't find any solution to this equation. ...
user178594
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Representing integers as sums of three powers
A famous open question, discussed several times on MathOverFlow, asks Which integers can be expressed as a sum of three cubes in infinitely many ways?. This is open even for $n=3$, that is, we do not ...
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Hyperelliptic curves imply FLT-like results
Probably this is known, but doesn't show in searches.
If a certain hyperelliptic curve has only trivial rational points,
FLT-like curve also has only trivial rationals points for fixed $n$.
Working ...
- 25.4k
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On $x^3-y^2=1728 \text{ unit}$ in number fields
Consider solution of
$$x^3-y^2=1728 \text{ unit} \qquad (1)$$
in a number field.
This is related to the discriminant of elliptic curve
in terms of $c_4,c_6$.
Via elliptic curves it might have ...
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For what integer $n$ are there infinitely many $-a+nb+c = -d+ne+f$ where $a^6+b^6+c^6 = d^6+e^6+f^6$?
(Much revised for clarity.) I was considering the system of equations,
$$-a+nb+c = -d+ne+f\tag1$$
$$a+b+c = d+e+f\tag2$$
$$a^2+b^2+c^2 = d^2+e^2+f^2\tag3$$
$$a^6+b^6+c^6 = d^6+e^6+f^6\tag4$$
...
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A good introduction to S unit equations
I was looking up some stuff when I stumbled across S unit equations. It seems to me that they are quite helpful in number theory, as given in this paper.
http://faculty.nps.edu/pstanica/research/...
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The security of one-time digital signatures from a solution to a diophantine equations
I wonder how well arbitrary Diophantine equations can be used to make one time digital signature schemes.
For our one-time digital signature scheme, the public key is a collection of polynomials $f_1(...
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System of diophantine equations related to Ozanam's problem
Could you please help with finding of general solution of diophantine system for rational a, b, c, d
$(a^2+b^2)(c^2+d^2)=A^2$
$(a^2-b^2)(c^2-d^2)=B^2$
for some rational A and B.
This is related ...
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Rational solutions to $P(x,y)=0$ for $P$ reducible over ${\mathbb C}$
There are facts in Mathematics that are so "obvious" and "well-known" that no-one includes a proper proof. An example is:
Theorem: If polynomial $P(x,y)$ with rational coefficients ...
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Quadratic diophantine equation in $\mathbb C[T]$
I am trying to solve the following quadratic diophantine equation in $\mathbb C[T]$, but I did not manage. I hope someone could give some hints or solutions to my problem. Here is the equation
$$\...
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A system of nonlinear Diophantine equations whose positive solutions are not coprime
Consider the following system of Diophantine equations:
$$v_1k_1=k_1^3-k_2^3+k_3^3 \\
v_2k_2=k_1^3+k_2^3-k_3^3 \\
v_3k_3=-k_1^3+k_2^3+k_3^3 \tag{1}$$
where $v_1,v_2,v_3$ and $k_1,k_2,k_3$ are integer-...
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Solvability of two-variable quadratic equations with a parameter
(a) Prove that there exist infinitely many values of integer parameter $a$ such that equation
$$
2 x^2+a x y+y^2+1 = 0
$$
is solvable in integers $(x,y)$.
(b) The same question for a similar equation
$...
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Elementary constraints for the solutions of $z^{n-2}y(y+z)=x^n$?
Related to FLT and this question.
For natural $n > 4 $ define the curve $C_n : z^{n-2}y(y+z)=x^n$.
$C_n$ has the trivial points with $x=0$ for all $n$.
The answer in the linked question shows ...
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