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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Diophantine equation over Z[i]
I'm trying to generate the set of solutions of a specific diophantine equation over Z[i].
The equation is the following:
$$ z_1^2 + z_2^2 + z_1*z_2 + 39 = 0$$
with $ z_1$ (resp $z_2$) such that $\...
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Efficient counting of Egyptian fractions with bounded denominators
I was amazed to discover that sequence http://oeis.org/A020473 in the OEIS has almost four hundred terms computed.
I wonder how one can get that far? E.g., how one can compute A020473(100)?
P.S. ...
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A hard diophantine equation: $m!+27=n^3$
I would like prove that the following diophantine equation is unsolvable: $m!+27=n^3$.
Thanks in advance.
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Symmetric functions on three parameters being perfect squares
Is it possible for $x+y+z, xy+yz+zx$, and $xyz$ to be perfect squares at the same time for positive integer values of $x,y,z$?
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0,1 solution to system of linear integer equations
I have the following problem:
$A x = b$
where $A, b$ - $m \times n$-matrix and $m$-vector of nonnegative integers (respectively).
$x \in \{0,1\}^n $ - vector of binary variables, which need to be ...
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Growth of $n=n(k)$ for which there's a non-trivial solution to $x_1^k+\cdots+x_n^k=y^k$?
Walter Hayman just asked me the following question. What, if anything, is known about the growth of the function $n(k)$, where $k\geq1$ is an integer, and $n=n(k)\geq2$ is the smallest integer for ...
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A problem on cubic Diophatine equations
What is the best algorithm to find all the integer points (X,Y) on this curve
$X^3+aX-bY^3=m,a,b,m\in\mathbb{Z}$(a>0,b>0,b is not a cubic number)?
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a family of Pellian equations
I have a question concering the family of Pellian equations
$$x^2 - (k^2+1)y^2 = k^2. \qquad (*)$$
For an integer $k\geq 2$, the equation (*) has at least three classes of solutions
in integers, ...
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Infinite solutions of a diophantine equation [closed]
Given the Diophantine equation$$ax^2+bxy+cy^2+dx+ey+f=0$$
if the coefficients $(a,b,c,d,e,f)$ are chosen among all the prime numbers, we have infinite equations. Is it possible to prove that the ...
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Hurwitz integers and $F_4$
The Hurwitz integers are
$$
\mathcal H=
\{a+bi+cj+dk:a,b,c,d\in\mathbb Z\;\text{ or } \;a,b,c,d\in \tfrac12+\mathbb Z\}.
$$
I want to know if there is a formula, for $m\in\mathbb Z$, for the number ...
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how many consecutive integers $x$ can make $ax^2+bx+c$ square ?
The following problem was raised in a Mathlinks thread:
If $a,b,c\in\mathbb Z$ such that $a\ne0$ and $b^2-4ac\ne 0$, for how many consecutive integers $x$ can $ax^2+bx+c$ ba a perfect square ?
The ...
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Special arithmetic progressions involving perfect squares
Prove that there are infinitely many positive integers $a$, $b$, $c$ that are consecutive terms of an arithmetic progression and also satisfy the condition that $ab+1$, $bc+1$, $ca+1$ are all perfect ...
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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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On a remark of Tait on FLT for the exponent 3
This is one of those recreational questions that aren't really about research. I found a curious remark in an old volume of American Mathematical Monthly (1922) which I'll quote below:
In the ...
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Impossible Heronian Triangles (Ratio of 2 Sides)
There is no Heronian triangle (or simply consider triangles on an integer lattice
which also have integer side lengths) for which one side is half the length of
another side. What other "side-side ...
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is exponential diophantine over Qp
Thanks to Matiyasevic, we all know that exponential is diophantine over the integers. Also, thanks to transcendental number theory, we know that exponential is not diophantine over the rationals. So ...
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Can the difference of two distinct Fibonacci numbers be a square infinitely often?
Can the difference of two distinct Fibonacci numbers be a square infinitely often?
There are few solutions with indices $<10^{4}$ the largest two being $F_{14}-F_{13}=12^2$ and $F_{13}-F_{11}=12^2$...
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Explicit solutions of C(n,2)=x^2 ? [closed]
"On a Diophantine Equation" paper of Erdös, at some point it is said that it is well known that $C(n,2)=x^2$ has infinitely many integer solutions. I am just wondering the formula generating all ...
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Determining the exceptional set in the theorem of Ax & Kochen
Ax & Kochen [1] proved that for every $d\in\mathbb{N}$ there exists a finite set $A(d)$ such that for every prime $p\not\in A(d),$ every homogeneous polynomial of degree $d$ over $\mathbb{Q}_p$ in ...
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The "universal" diophantine equation
There is a diophantine equation in some number (I think the minimum is now 9) of variables, that can be used to represent
All other diophantine equations (could be wrong on this)
Any particular set ...
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non negative integer solutions : Diophantine Equations [closed]
I want to know the exact number of non-negative integer solutions of $a_1 + 2a_2 + \ldots + k \cdot a_k = n$.
I know that it is the co-efficient of $x^n$ in $(1 - x^{a_1})^{-1} \cdot (1 - x^{a_2})^{-...
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Do the solutions to the unit equation lie dense in the complex numbers
Let $S\subset \overline{\mathbf{Q}}\subset \mathbf{C}$ be the set of solutions to the unit equation, i.e., $S$ consists of algebraic integers $a$ such that $a$ and $1-a$ are units in the ring of ...
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Which Diophantine equations can be solved using continued fractions?
Pell equations can be solved using continued fractions. I have heard that some elliptic curves can be "solved" using continued fractions. Is this true?
Which Diophantine equations other than Pell ...
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Existence of a non-trivial zero (in the rational cyclotomic field) of a form
It is well known that if a field K is quasi-algebraically closed (i.e. all forms with coefficients in K of degree d in n > d variables have a non-trivial zero in K) then it has no central divison ...
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For any $n$, does there exist a number field with at least $n$ solutions to the unit equation
Let $n$ be a positive integer.
Does there exist a number field $K$ such that the number of solutions of the unit equation $$a+b =1, \quad a,b\in O_{K}^\ast$$ is at least $n$? Can we write down such a ...
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Linear diophantine equation in n variables
Let n>3. Is there any way to generate all integer solutions of linear diophantine equation in n variables, or at least to determine number of such solutions?
Thanks in advance.
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Diophantine $x^p+y^q=(x+y)^r$
Is the equation:
$$x^p+y^q=(x+y)^r$$
in integers $x,y,z,p,q,r$ with $p \geq 2,q \geq 2, r \geq 2$ complete solved?
For $(p,q,r)=(n,n,n+1)$ a parametrization is $t=1-s$ and $ t(s^n+t^n),s(s^n+t^n),s^...
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Diophantine equation $2(x - 1/x) = y - 1/y$
Does the Diophantine equation $2(x - \frac{1}{x}) = y - \frac{1}{y}$ have only trivial rational solutions, i.e, $x=\pm1, y = \pm1$?
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Is there a section disjoint from 0, 1 and infinity on the projective line
Let $K$ be a number field with ring of integers $O_K$. Is there a section of $\mathbf{P}^1_{O_K}$ over $O_K$ whose image is disjoint from $0$, $1$ and $\infty$? If $K=\mathbf{Q}$ this is not possible ...
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Is there a two-variable prime-representing polynomial (in the sense of Jones-Sato-Wada-Wiens)?
In the math.se question Proof of no prime-representing polynomial in 2 variables, Alon Amit asks if Ribenboim's claim that a prime-representing polynomial (a Diophantine polynomial in which the ...
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Torsion points of abelian varieties in the perfect closure of a function field
The following is a problem, which was recently brought to my attention by H. Esnault and A. Langer.
Let $K$ be the function field of a smooth curve over the algebraic closure $k$ of the finite field ...
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Reduction from factoring to solving Pell equation
The paper Polynomial-Time Quantum Algorithms for Pell's Equation and the Principal Ideal Problem claims
There are reductions from factoring to solving Pell’s equation, and from solving Pell’s
...
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The NP version of Matiyasevich's theorem
By Matiyasevich, for every recursively enumerable set $A$ of natural numbers there exists a polynomial $f(x_1,...,x_n)$ with integer coefficients such that for every $p\ge 0$, $f(x_1,...,x_n)=p$ has ...
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Bilinear system of Diophantine Equations
$\forall i,j \in$ $\{$$1, \cdots,n$$\},$ let $x_{i},y_{i}$ be unknowns and $n_{ij} \in \mathbb{Z}$ with $i \le j$ be the knowns.
Consider the following $\frac{n(n+1)}{2}$ with $n > 2$ ...
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System of Diophantine equations
$p + p' = m$
$q - q' = n$
$pp' = qq'$
$(m^{2} + n^{2})\equiv1\pmod 4$ and $n^{2}\equiv0\pmod 4$.
Only $m,n$ are known in the above. Are there any known techniques to guess the values of $p$ and $q$...
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Algorithm for solving systems of linear Diophantine inequalities
So, I posted on StackOverflow looking for a reasonably fast algorithm to solve systems of linear Diophantine inequalities and was pointed to this article by Cheng-Zhi Gao and Yu-Lin Dong. The problem ...
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Coprime integer solutions to $ \frac{x^n \pm y^n}{x \pm y}=z^m $ with $n>5 , m>1$
Are there coprime integer solutions to:
$$ \frac{x^n \pm y^n}{x \pm y}=z^m $$
with $n>5 , m>1$ and excluding $z=0$?
I suppose the abc conjecture implies finitely many solutions.
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Are there any solutions to $\frac{3^n - 2^n}{2^k-3^n} = N$
Are there any solutions to $\frac{3^n - 2^n}{2^k-3^n} = N$ for $n$, $k$, $N$ $\in\mathbb{N}$, greater than 2.
This is related to a previous answered question: Are there any solutions to $2^n-3^m=1$
...
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Integral roots to degree $d$-forms in four variables inside a box
Hi,
After the response to the following question, Rational roots to quadratic forms in 4 variables, I am now considering the following question.
Let $d \geq 2$ be a positive integer, and suppose ...
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Rational roots to quadratic forms in 4 variables
Hi,
I am interested in the following question. Let $F(x_1, x_2, x_3, x_4)$ be a quadratic form in four variables with integer coefficients. Let $B > 0$ be a parameter. Define $N_1(F,B)$ to be the ...
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Quartic case of a theorem of Bombieri and Pila
I am interested in the ternary case of a theorem of Bombieri and Pila, in E. Bombieri and J. Pila, "The number of integral points on arcs and ovals", Duke Mathematical Journal., 59 (1989), 337-357. ...
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A heuristic for the density of solutions to Diophantine equations
Let $f\in\mathbb{Z}[X_1,\ldots,X_n]$ be a Diophantine equation which, for the purposes of this question, I will assume is homogeneous and nonsingular on $\mathbb{R}^n\setminus\{0\}$ (so that $\nabla f\...
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Exponential Diophantine $\prod x_i^{e_{x_i}} + \prod y_i^{e_{y_i}} = \prod z_i^{e_{z_i}}$ ,$e_{x_i}>3,e_{y_i}>3,e_{z_i}>3$
Does the exponential diophantine equation
$$ \prod x_i^{e_{x_i}} + \prod y_i^{e_{y_i}} = \prod z_i^{e_{z_i}}$$
with $x_i, y_i,z_i$ coprime and $e_{x_i}>3,e_{y_i}>3,e_{z_i}>3$ have solutions?...
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A question on M. Mignotte's Paper: "Petho's Cubics"
I have been reading the paper "Petho's Cubics" by M. Mignotte (appeared in Publ. Math. Debrecen, 56/3-4 (2000)) unfortunately I don't understand section 4 of the paper. I am hoping someone could help ...
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Why can Diophantine equations represent exponential growth?
The wikipedia page on Matiyasevich's theorem challenges:
Unfortunately there seems to be as yet no short intuitive explanation as to why Diophantine equations can represent exponential growth only ...
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General integer solution for $x^2+y^2-z^2=\pm 1$
How to find general solution (in terms of parameters) for diophantine equations
$x^2+y^2-z^2=1$ and $x^2+y^2-z^2=-1$?
It's easy to find such solutions for $x^2+y^2-z^2=0$ or $x^2+y^2-z^2-w^2=0$ or $x^...
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Prove a parametrization function is surjective
As a starting note, I would like to say that I haven't (yet) taken courses in Set Theory, so some higher-level notation may be lost on me (and I may not write everything conventionally), but I'll do ...
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Does the following Diophantine equation have nontrivial rational solutions?
Are there any solutions to the equation $s^{2}(1+t^{2})^{2}+t^{2}(1+s^{2})^{2}=u^2$ where $s,t,u\in \mathbb{Q}$ and $0 < s,t<1$? If so, is there a simple way to parametrize them all?
If I am ...
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Lower bounds on the easier Waring problem
The easier Waring problem asks for the least number $v=v(k)$ such that every every integer is a sum of $v$ $k$'th powers with signs, i.e. every $n\in \mathbb{N}$ is of the form $$n=x_1^k\pm x_2^k\pm\...
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Pythagorean 5-tuples
What is the solution of the equation $x^2+y^2+z^2+t^2=w^2$ in polynomials over C ("Pythagorean 5-tuples")?
There are simple formulas describing Pythagorean n-tuples for n=3,4,6:
n=3. The formula ...