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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What about $n^{\frac{1}{x}+\frac{1}{y}}+n^{\frac{1}{y}+\frac{1}{z}}=n^{\frac{1}{z}+\frac{1}{x}}$ over positive integers?
I've considered the following equation for positive integers $x,y,z\geq 1$, and for positive integers $n\geq 2$
$$n^{\frac{1}{x}+\frac{1}{y}}+n^{\frac{1}{y}+\frac{1}{z}}=n^{\frac{1}{z}+\frac{1}{x}},\...
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What about an alternative formulation for different prime constellations in the spirit of Suzuki's theorem for twin primes?
It is known that the twin prime conjecture is a special case of the $k$-tuple conjecture. See if you want the article with title k-Tuple Conjecture from the encyclopedia Wolfram MathWorld.
On the ...
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Large radical of an integer and three AB conjectures
In this Note, We propose a new definition called "large radical of an integer". Using this definition, three very useful $AB$ conjecture are given.
1. Large counter examples of the ABC conjecture
...
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enumerate line partitions of points in the plane
Let $S$ be a nonempty subset of $\mathbb{R}^2$ and $l$ a line in $\mathbb{R}^2$ disjoint from $S$. Then $l$ partitions $S$ into two disjoint sets $S = S_1 \cup S_2$ in the obvious way. Note that, at ...
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Is the conjecture $min(A,B) \le rad(ABC)$ new and correct? [closed]
$\DeclareMathOperator\rad{rad}$Conjecture: If $A, B, C$ are positive integers with $\gcd(A, B)=1$, $\gcd(B, C)=1$, and $\gcd(C, A)=1$, and if $A+B=C$, then $\min(A,B) \le \rad(ABC)$.
If the ...
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On the values of $\prod_{k=1}^{(p-1)/2}(e^{2\pi i/12}-e^{2\pi i k^2/p})$ for primes $p>3$
In a recent preprint, I investigated
$$S_p(x):=\prod_{k=1}^{(p-1)/2}(x-e^{2\pi ik^2/p}),$$
where $p$ is an odd prime and $x$ is a root of unity.
Motivated by Question 337879 and Question 338325, ...
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Is the diophantine equation $3x^2+1=py^2$ always solvable for each prime $p\equiv 13\pmod{24}$?
In Question 337879, I conjectured that for any prime $p\equiv3\pmod4$ the equation $$3x^2+4\left(\frac p3\right)=py^2\tag{1}$$
always has integer solutions, where $(\frac p3)$ is the Legendre symbol. ...
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Diophantine equations $ax^4+by^2=c$ in rational numbers
Are there general ways for given rational coefficients $a,b,c$ (I am particularly interested in $a=3,b=1,c=8076$, but in general case too) to answer whether this equation has a rational solution or ...
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Is Multilinear Hilbert's tenth problem version undecidable?
A multilinear polynomial $f\in\mathbb Z[x_1,\dots,x_t]$ has terms only of form $$b\prod_{i=1}^tx_i^{a_i}$$ where $a_i\in\{0,1\}$ and $b\in\mathbb Z$.
Is there no general purpose algorithm for ...
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Find solutions and get a first statement for these diophantine equations inspired in certain figurate numbers
Yesterday I was thinking in speculative relationships between certain figurate numbers, please see if you need the tables, and references from the article of the
encyclopedia MathWorld Figurate Number....
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How many solutions of x^3 +y^3 = z^3+3 are known? [duplicate]
I've been told that there is reason to think that the equation
$x^3 + y^3 = z^3 + 3$ has solutions in positive integers other than
$$4^3 + 4^3 = 5^3 +3.$$
Can someone tell me the current status of ...
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On the product $\prod_{k=1}^{(p-1)/2}(x-e^{2\pi i k^2/p})$ with $x$ a root of unity
Let $p$ be an odd prime. Dirichlet's class number formula for quadratic fields essentially determines the value of the product $\prod_{k=1}^{(p-1)/2}(1-e^{2\pi ik^2/p})$. I think it is interesting to ...
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A generalization of Lander, Parkin, and Selfridge conjecture
My question: Are the conjectures as follows correct?
Given a positive integer $P>1$, let its prime factorization be written
$$P=p_1^{a_1}p_2^{a_2}p_3^{a_3}...p_k^{a_k}$$.
Define the ...
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Solutions of $y^2=\binom{x}{0}+2\binom{x}{1}+4\binom{x}{2}+8\binom{x}{3}$ for positive integers $x$ and $y$
I was interested in create and solve a Diophantine equation similar than was proposed in the section D3 of [1]. I would like to know what theorems or
techniques can be applied to prove or refute that ...
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How many integers below $n$ can be expressed as a sum of $k$ $m$th powers?
For $m,k \geq 2$, let $C_{m,k}(n)$ denote the number of positive integers less than or equal to $n$ which can be expressed as a sum of $k$ $m$th powers.
I am interested in the asymptotic behavior of ...
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Is there a error/typo in the proof related to Goormaghtigh equation in Yann Bugeaud's paper?
I found the following theorem in a paper by Yann Bugeaud (page 12) ,
the theorem was not written in detail, to be specific,following two lines on page 13 were not understandable-
I think this ...
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Diophantine applications of Paramodularity
I’ve asked this question to quite a few people in person and so far haven’t seen a good answer... but I believe one should exist, so here goes!
Ok, we all know how to (roughly) prove Fermat’s Last ...
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Diophantine equation: $n^2=c(4ab-a-b)-b$?
I asked the following question here, but I did not get a full answer, so I put it here that may be some help.
Let $n$ be a positive integer. The Diophantine equation
$$
n^2=c(4ab-a-b)-b,\qquad (a,b,...
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Two equations and a question related to a well-known conjecture from number theory
On the Wikipedia page, the Beal´s conjecture is stated as:
If $A^x+B^y=C^z$, where $A,B,C,x,y,z$ are positive integers with $x,y,z>2$, then $A$,$B$, and $C$ have a common prime factor.
I think ...
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On the Diophantine equation $x^{4}+y^{4}= z^n$
I'm told that Ellenberg and Bruin proved that the Diophantine equation $x^{4} + y^{4} = z^n$ has no primitive solutions in which $xyz\neq 0$ and $n\geq 4$. Does the proof use the same methods that ...
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All the integer solutions of a certain semi-algebraic system
I would like to find all integer solutions of the following system:
$$a+b+c+ab+ac+bc=-2,$$
$$a,b,c\le a+b+c-1.$$
One solution is $2,2,-2$. Is it possible to describe all others?
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pari/gp "bnfisintnorm" as poor man (quadratic) Thue equations solver?
For simplicity explaining only the quadratic case.
Given integers $n,m$, pari/gp "bnfisintnorm" finds $X,Y$
such that $X^2+n Y^2=m$ working in the number field
with defining polynomial $x^2+...
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Limit of the real part of a geometric sequence
I came across the following problem, which turned out to be surprisingly hard:
Show that $\underset{n\rightarrow \infty}{\lim} \left| \mathrm{Re}((\frac{1+i\sqrt{7}}{2})^n)\right| = \infty.$
...
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Explicit bivariate quadratic polynomials where Coppersmith is better than standard solver?
http://www.numbertheory.org/pdfs/general_quadratic_solution.pdf gives a general method to solve quadratic bivariate diophantine equation while Coppersmith introduced a method to solve bivariate ...
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Representing integers efficiently with quadratic polynomials
For every large enough $T$ given four integers $a,b,c,d$ with absolute value less than $T^2$ are there integers $w_1,x_1,y_1,z_1,w_2,x_2,y_2,z_2$ with absolute value less than $T$ such that
$$w_1x_1+...
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Question about parametric representations of solutions to $x^3+y^3+z^3=n \in \mathbb N$
There are such representations for $n=1,2$. However, by the Wikipedia article, it seems that there are no known parametric (polynomial) representations $P,Q,R$ such that $(P(m))^3+(Q(m))^3+(R(m))^3=3$....
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*Why* is Bombieri-Pila uniform?
I am about to give a couple of lectures on Bombieri-Pila/the determinant method. Bombieri-Pila gives a bound $$|C(\mathbb{Q}) \cap B \cap \mathbb{Z}^2| \ll_{d,\epsilon} N^{1/d+\epsilon}$$ for $C$ a ...
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Positive integers written as $\frac{a(a+1)}2+\frac{b(b+1)}2+4^c5^d$
Let $\mathbb N=\{0,1,2,\ldots\}$. Those
$T_n:=n(n+1)/2$ with $n\in\mathbb N$ are called triangular numbers. It is well known that
$$\{T_a+T_b+T_c:\ a,b,c\in\mathbb N\}=\mathbb N\tag{1}$$
which was ...
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Solving $(u-x_1)^{1/3}+ (u-x_2)^{1/3}+ (u-x_3)^{1/3} = {v}^{1/3}$ with elliptic curves
Let $x_1$,$x_2$,$x_3$ be the roots of the cubic $x^3+px+q$ over $\mathbb Q$, the idea is that rational solutions $(u,v)$ of the equation
$$(u-x_1)^{1/3}+ (u-x_2)^{1/3}+ (u-x_3)^{1/3} = {v}^{1/3} \...
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Quadratic diophantine equations and geometry of numbers
Let (for concreteness) $a = 2$, $b = \sqrt{5}$ and $\varphi = (\sqrt{5}+1)/2$. I am interested in solutions $(w,x,y,z) \in \mathbb{Z}[\varphi]^4$ of the system
$$
w^2 - ax^2 -by^2 + abz^2 = 1
$$
$$
\...
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Elementary constraints for the solutions of $z^{n-2}y(y+z)=x^n$?
Related to FLT and this question.
For natural $n > 4 $ define the curve $C_n : z^{n-2}y(y+z)=x^n$.
$C_n$ has the trivial points with $x=0$ for all $n$.
The answer in the linked question shows ...
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Strong Approximation for solutions to quadratic Diophantine equations
Can anyone either direct me to an relatively elementary proof in the literature--or show me why this (Conjecture 1 stated below) is true--or if I am mistaken and it is not true:
For any 4-tuple $\xi =...
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Is it true that $\{x^3+2y^3+3z^3:\ x,y,z\in\mathbb Z\}=\mathbb Z$?
It is easy to see that no integer congruent to $4$ or $-4$ modulo $9$ can be written as the sum of three integer cubes. In view of this and Question 331163, I proposed the following conjecture in ...
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Diophantine equation $3^a+1=3^b+5^c$
This is not a research problem, but challenging enough that I've decided to post it in here:
Determine all triples $(a,b,c)$ of non-negative integers, satisfying
$$
1+3^a = 3^b+5^c.
$$
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Squares in the set $\{\sum_{j=1}^m j^2: m\in\mathbb{N}\}$ [closed]
Are there infinitely many squares in the set $$\{\sum_{j=1}^m j^2: m\in\mathbb{N}\} ?$$
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Efficient solutions to general Bézout’s identity $a_1 b_1 + \dots + a_n b_n = 1$
Suppose I have integers $a_1, \dots, a_n$ which are coprime, meaning that
$$a_1 b_1 + \dots + a_n b_n = 1$$
has a solution in integers $b_1, \dots, b_n$.
I would like to get a bound saying ...
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Upper bound for a subset of $\mathbb{N}^2$
Question: Consider the set
$$ A(m) : = \{ (a, b) \in \mathbb{N}^2 : \; N \leq a \leq 2 N, \; a^2 - b^2 = m \}, $$
where $m \in \mathbb{Z}$ and $N \in \mathbb{N} \setminus \{ 0 \}$. Then
$$ \...
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Reference request: Markoff type equations
Consider the equation $x^2+ay^2+bz^2=(1+a+b)xyz.$ If there are infinitely many integral solutions, then up to permutations $(a,b)=(1,1),(1,2),(2,3).$ I have found a presentation by Waldschmidt (https:/...
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Can we "invert" Diophantine equations such as $x^3+y^3+z^3=k$ in to halting probabilities for some universal Turing machine?
Following Poonen [1], Davis[2], Chaitin [3], and Ord and Kieu [4]:
Is it possible that there is a polynomial $P$ of degree $d\le 4$, along with a prefix-free universal Turing machine $T$, such that ...
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Upper bound for the number of solutions of a Diophantine equation
Consider the Diophantine equation
$$k^2 + k - \sigma (\ell^2 + \ell) = m,$$
where $N \leq k \leq 2 N$, $L \leq \ell \leq 2 L$, $m \in \mathbb{Z}$ and $\sigma \in \mathbb{R}$.
For which values ...
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Extension of Erdos-Selfridge Theorem
Erdos and Selfridge open their paper "The Product of Consecutive Integers is Never a Power" (1974) with the theorem
$\text{Theorem 1:}$ The product of two or more consecutive positive integers is ...
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Solutions to Diophantine equation for Ramanujan graph construction
I am working with so-called Ramanujan graphs which have the property that a lower bound on their girth can be stated. I am reading about those in paper On the Construction of Turbo Code Interleavers ...
3
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Fermat's cubic equation in quadratic extension of $\mathbb{Q}$
Is still relevant or interesting be capable to bring a criteria in order to classifly quadratic extensions of $\mathbb{Q}$ based on the existence or not existence of non-trivial solutions of Fermat's ...
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Probability of small solutions to an uniform random linear diophantine equation?
Take the set $$T(c_1,\dots,c_t)=\{(x_1,\dots,x_t)\in\mathbb Z^t\backslash\{(0,\dots,0)\}:\sum_{i=1}^tc_ix_i\equiv0\bmod q\}$$ where $c_1,\dots,c_t\in(-q/2,q/2)\cap\mathbb Z$.
What is probability ...
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How many roots of polynomial in $\mathbb Z[x]$ and $\mathbb Q[x]$ are integers on average?
Given $d,B>0$ the number of polynomials in $\mathbb Z[x]$ of degree $d$ and coefficient size at most $B$ have at least one integer roots should be $B^{O(d)}f(d)$ at some function $f$ (from Random ...
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Universality of $y^4-x^3$ mod $p$
For pedagogical reasons, I got interested in the equation $y^4-x^3=a$ over $\mathbf F_p$.
To my surprise (maybe I'm naive), there is only one couple $(p,a)=(13,7)$ for which there is no solution, at ...
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3
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Numbers that can be written as a sum of three cubes in exactly one way (a^3 + b^3 + c^3)
Based on online info, it seems that most of these numbers have many solutions. Are there any that have only 1 known solution or only a few solutions?
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Counting primitive solutions to a diophantine inequality
This is a refinement (perhaps a simpler version) of a question I asked here before and couldn't get an answer for.
Fix $\alpha \in (0,1]$ and a small constant $c>0$. For $x \in [0,1]$ and $N\in\...
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Find the positive integers $x^3+y^3=3z^3$ [closed]
By Fermat Last theorem, I don't know if that's been discussed.
Find all positive integers $x,y,z$ such
$$x^3+y^3=3z^3$$
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trivial solutions for Diophantine equations
Let $K$ be an odd degree number field. Consider the Diophantine equation:
$$
X^4 + bY^4 =Z^2
$$
where $b\neq 0$.
Say we know that the above equation has only trivial roots in $K$ (for some fixed ...
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