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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Is 36 a sum of 4 rational fourth powers?
Hasse principle is known to hold for homogeneous quadratic equations, but fail for some 3- and 4-variable cubics, such as $5x^3+4y^3+3z^3=0$ or $15x^3+10y^3+4z^3+3t^3=0$. These counterexamples are ...
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Diophantine equation: Egyptian fraction representations of 1
According to the OEIS (A002966) there are 294314 solutions in positive integers to the equation
$$\sum_{i=1}^7\frac{1}{x_i}=1$$ assuming $x_1\leq x_2\leq\cdots\leq x_7$.
Similarly for 8 summands there ...
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Can you "slice" a triangular number into three equal slices?
Problem statement:
Does there exist positive integers $a<b<c$ such that
$$1 + 2 + \dots + (a-1) = (a+1) + \dots + (b-1) = (b+1) + \dots + c?$$
(Note that $a$ and $b$ are not in the sums.)
...
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What are the rational solutions to $y^4=x^3+x+1$?
What are the rational solutions to $y^4=x^3+x+1$?
This equation is interesting because it has substitution $y^2=z$ that reduces it to elliptic curve $z^2=x^3+x+1$. Sometimes, the existence of such ...
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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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On Generalizations of Fermat's Conjecture
We know the following facts:
(1) For all $1\leq n\leq 2$ the equation $x_{1}^{n}+x_{2}^{n}=x_{3}^{n}$ has a solution in $\mathbb{N}$.
(2) For all $3\leq n$ the equation $x_{1}^{n}+x_{2}^{n}=x_{3}^{n}...
user36136
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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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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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Infinitely many integer solutions to $X^4+Y^4-18Z^4= -16$
We found infinitely many integer solutions to
$$X^4+Y^4-18Z^4= -16 \qquad (1)$$.
The interesting part in this diophantine equation is the sum of
the reciprocals of the degrees is $3/4 < 1$, which ...
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Effect of abc conjecture on Fermat's Last Theorem
A website ( http://www.math.unicaen.fr/~nitaj/abc.html#Consequences ) says that the $abc$ conjecture implies that there are only finitely many solutions to the equation $x^n+y^n=z^n$ with $\gcd(x,y,z)=...
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Can we extend the proof of Catalan's conjecture?
What is it, in Mihailescu's proof of Catalan conjecture, that uses explicitly the fact that there is a 1 on the right hand side of $x^p - y^q = 1$? In other words, why can't we extend his argument to ...
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On the equation $9x^3+y^3=z^2+3$
The question is whether there exist integers $x,y,z$ such that
$$
9x^3+y^3=z^2+3.
$$
This is one of the nicest (if not the nicest one!) cubic equations for which I do not know whether integer ...
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Solving the quartic equation $r^4 + 4r^3s - 6r^2s^2 - 4rs^3 + s^4 = 1$
I'm working on solving the quartic Diophantine equation in the title. Calculations in maxima imply that the only integer solutions are
\begin{equation}
(r,s) \in \{(-3, -2), (-2, 3), (-1, 0), (0, -1),...
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On cubic reciprocity for $x^3+y^3+z^3 = 996$?
I. The Diophantine equation,
$$x^3+y^3+z^3 = 3w^3\tag1$$
with $x\geq y \geq z$ and $w=1$ has only two known solutions, namely $1,1,1$ and $4,4,-5$. Are there larger ones? As Noam Elkies points out ...
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On Euler's elliptic curve for $A^4+B^4 = C^4+D^4$?
To solve,
$$A^4+B^4 = C^4+D^4$$
we use Euler's method. Let,
$$(p+q)^4+(r-s)^4=(p-q)^4+(r+s)^4$$
and define $p = (a^3 - b),\, q = a y,\, r = b (a^3 - b),\, s = y.\,$ The equation above transforms to ...
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The number of representations of an integer as the inner product of integral lattice points
I was looking through some old notes of mine and I came across a couple lemmas/identities I wrote down in regards to a question I asked about four years ago. In particular I wrote that for an ...
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Integers $d$ for which the negative Pell equation is soluble for both $d$ and $2d$?
Let $\text{NPE}_d$ denote the negative Pell equation:
$$ x^2-dy^2=-1$$
Where $d$ is a given positive nonsquare integer and integer solutions are sought for x and y.
we know that (in this paper
archive)...
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Integer Solutions of $x+y^n = y + x^m$ for $n < m$
I found 8 of them and believe there is no more:
$$2+3^2=3+2^3$$
$$2+6^2=6+2^5$$
$$6+15^2=15+6^3$$
$$3+16^2=16+3^5$$
$$3+13^3=13+3^7$$
$$2+91^2=91+2^{13}$$
$$5+280^2=280+5^7$$
$$30+4930^2=4930+30^5$$
...
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On the equation $a^6+b^6+c^6=d^2$
I have been studying the equation $a^6+b^6+c^6=d^2$, trying to find rational solutions. I know it is a K3 surface, with high Picard rank, so there should be rational or elliptic curves on it.
When ...
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Effective proofs of Siegel's theorem using arithmetic geometry
This is a speculation and perhaps naive. The theorem of Siegel that
There exist only finitely many integral points on a curve of genus $\geq 1$ over a number ring $\mathcal O_{K, S}$ where $S$ is a ...
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Can repunits be perfect cubes?
Is it true that the equation $10^{n}-9m^{3}=1$ has only one positive integer solution, namely $n=m=1$? I can't find the answer. This has an equivalent description that the repunits $R_n = 11\dots1$ ...
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Six consecutive positive integers with certain shape
Are there 6 consecutive positive integers, where each of them is a square or the product of a prime and a square ?
If they exist, one of those six integers A will be the product of 2 and a square of ...
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What is the rank of the Mordell equation $y^2 = x^3 - 2$?
The mordell equation $E$ defined by $y^2 = x^3 - 2$ over $\mathbb{Q}$ is known to have only one non-trivial integer solution $P = (3,5)$ from here. However, the rank of Mordell-Weil group $E(\mathbb{Q}...
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Are there any solutions to the diophantine equation $x^n-2y^n=1$ with $x>1$ and $n>2$?
This problem arose when considering storage of cannonballs in n-dimensional pirate ships, as explained in this MSE post. This MO question can also be reduced to the $n=3$ case. If $x,y$ is a solution ...
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Failing of heuristics from circle method
The heuristic from circle method for integral points on diagonal cubic surfaces $x^3+y^3+z^3=a$ ($a$ is a cubic-free integer) seems to fit well with numerical computations by ANDREAS-STEPHAN ELSENHANS ...
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A diophantine equation in $\mathbb{N}$
While I was working on a paper on graph theory, I encountered a problem which I think is a number-theory-problem. I don't know if there are any tools to answer the question.
Find all natural numbers $...
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rational points of a hyperelliptic curve
I have the following hyperelliptic curve of genus $2$:
$$
y^2 = 561 x^6 - 41904 x^5 + 627264 x^4 + 11860992 x^3 - 197074944 x^2 + 124416^2
$$
I need to find all the rational points on this curve. ...
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General solution of the quartic $a^4+b^4=c^4+d^4$?
The background to the question:
$$a^4+b^4=c^4+d^4 \label{1}\tag 1 $$
Tito Piezas, Tomita & others have recently given some parametric solutions on Math stack exchange & Math overflow. In math ...
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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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Why does genus control the number of points
Often number theorists can bound the number of solutions to a diophantine equation based on the size of the points and the size of the coefficients. But this, as I understand it, can be a bit of a red ...
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A sequence based on Catalan–Mihăilescu problem
It was conjectured by Catalan in 1844 that the only solutions of the equation $x^a-y^b=1$ over variables $a,b,x,y\in\mathbb{N^+}$ are trivial ones: $3^1-2^1=1$ and $3^2-2^3=1$. The conjecture was ...
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Sets of integers represented by degree zero rational functions
Suppose $f(x_1,x_2,\dots)=\frac{P}{Q}$, where $P,Q$ are polynomials in several variables with integer coefficients that have the same degree. Let's denote by $S(f)$ the set of integers $n$ for which $...
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(Non-)Existence of curves of low degree on affine and projective varieties
I am interested in papers that investigates the existence or non-existence of curves of low degree (relative to the degree of the ambient variety). The starting example is that of surfaces and ...
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Fermat last theorem : proof of a criterion by Cauchy
In 13 Lectures on Fermat's Last Theorem, Ribenboim states the following theorem (on page 7) attributed to Cauchy:
If the first case of Fermat's theorem fails for the exponent $p$, then the sum:
$$ 1^{...
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Basic prerequisite (topics) to read current research in Diophantine equation for an independent researcher
I have completed studying Galois theory, Fermat's Last Theorem for Regular prime and some number theoretic complex analysis (prime number theorem), and basic linear forms in logarithm.
What else ...
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How many Pythagorean triples are there in which every member is triangular?
How many Pythagorean triples $(a,b,c)$ are there such that $a, b$ and $c$ are triangular?
Any two solutions with only $a$ and $b$ interchanged are considered equivalent.
The question of existence ...
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Why are solutions to $\sqrt[k]{x_1^k+x_2^k+x_3^k+x_4^k}$ for $k=2,3$ curiously smooth?
Given an integer solution $s_m$ to the system,
$$x_1^2+x_2^2+\dots+x_n^2 = y^2$$
$$x_1^3+x_2^3+\dots+x_n^3 = z^3$$
and define the function,
$$F(s_m) = x_1+x_2+\dots+x_n$$
For $n\geq3$, using an ...
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$3^n - 2^m = \pm 41$ is not possible. How to prove it?
$3^n - 2^m = \pm 41$ is not possible for integers $n$ and $m$. How to prove it?
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Diophantine equation $3^a+1=3^b+5^c$
This is not a research problem, but challenging enough that I've decided to post it in here:
Determine all triples $(a,b,c)$ of non-negative integers, satisfying
$$
1+3^a = 3^b+5^c.
$$
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How many solutions does $\frac{1}{x_1}+\frac{1}{x_2}+\cdots +\frac{1}{x_n}=1$ have?
It is well known that $\frac{1}{2}+\frac{1}{3}+\frac{1}{6}=1$ and this is the only solution to $\frac{1}{x_1}+\frac{1}{x_2}+\frac{1}{x_3}=1$
with $2\leq x_1<x_2<x_3$.
My question is:
Let $...
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Sum of consecutive cubes
I'm investigating when the sum of $n$ consecutive cubes equals a cube, i.e., for which $n$ does
$$\sum_{i=0}^{n-1} (k+i)^3 = k^3 + (k+1)^3 + \cdots + (k+n-1)^3 = Y^3 $$
have nontrivial solutions $(k,...
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Analysis of a quadratic diophantine equation
Hi! This is my first post on Math Overflow. I have two equations: $a(3a-1) + b(3b-1) = c(3c-1)$ and $a(3a-1) - b(3b-1) = d(3d-1)$. I'm trying to find properties of $a$ and $b$ that lead to solutions, ...
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Symmetric functions on three parameters being perfect squares
Is it possible for $x+y+z, xy+yz+zx$, and $xyz$ to be perfect squares at the same time for positive integer values of $x,y,z$?
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hard diophantine equation: $x^3 + y^5 = z^7$
Does the equation $x^3+y^5=z^7$ have a solution $(x,y,z)$ with $x,y,z$ positive integers and $(x,y)=1$? In his book H. Cohen (Number theory,2007) said "[...] seems presently out of reach".
I couldn't ...
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Prove that $1$ is the sum of three tetrahedral numbers infinitely many different ways
It's well known that $1$ is the sum of three cubes infinitely many different ways but is it true for perhaps the tetrahedral numbers as well? Let $T_n = (1/6)n(n+1)(n+2)$. Then the following are the ...
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Equation $x^2=y^p + 1$
can you help me please for solving this diophantine equation : $x^2=y^p+1$
and if you can give me a general method to studying such equation : $x^2=y^p+t$
Thanks
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a family of Pellian equations
I have a question concering the family of Pellian equations
$$x^2 - (k^2+1)y^2 = k^2. \qquad (*)$$
For an integer $k\geq 2$, the equation (*) has at least three classes of solutions
in integers, ...
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how many consecutive integers $x$ can make $ax^2+bx+c$ square ?
The following problem was raised in a Mathlinks thread:
If $a,b,c\in\mathbb Z$ such that $a\ne0$ and $b^2-4ac\ne 0$, for how many consecutive integers $x$ can $ax^2+bx+c$ ba a perfect square ?
The ...
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"strange" diophantine and parity of the partition function
Let $\{x_i\}:=\{x_1=5, x_2=13, x_3=29, x_4=37, x_5=45, \dots \}$
be the sequence of those positive integers of the form
$$
p^{4\alpha+1}n^2$$
in increasing order where $p\equiv 5\pmod 8$ is prime ...
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Is there any formula to find number of Pythagorean triplets between two integers 2 and j, j>2?
Given $j \geq 5$, is there a formula for the number of Pythagorean triplets $(a, b, c)$ satisfying the constraint that $a, b, c \leq j$?
There exists at least one Pythagorean triplet for $j\geq5$; ...
