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 can be said about the cube-free part of $x^3 -3xy^2 +y^3$?

For $x$ and $y$ in $\mathbf{Z}$, not both zero, let $cfp(x,y)$ be the cube-free part of $x^3 -3xy^2 + y^3$ (normalized to be $> 0$). One sees: (#) $cfp(x,y)$ is either a product of primes $p$, with ...
paul Monsky's user avatar
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1 vote
2 answers
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Describe all integer/rational solutions to $x^3+y^3+z^3+t^3+s^3=0$

The question is in the title. Equation $\sum_{i=1}^n x_i^3 = 0$ has no non-trivial integer solutions for $n=3$. For $n=4$, there are known descriptions of all integer/rational solutions, see Choudhry, ...
Bogdan Grechuk's user avatar
6 votes
0 answers
284 views

A generalization of the Diophantine $m$-tuple problem

Are there distinct positive integers $a_1,a_2,a_3,b_1,b_2,b_3$ such that $a_ib_j+1$ is a perfect square number for each $i,j$ ($1\leq i,j\leq3$)? I asked the following question in a group, and ...
Rogerhu's user avatar
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4 votes
0 answers
259 views

Equations involving sum of fourth powers

Do there exist rational numbers $x,y,z$ such that $$ \quad \quad z^3 - 1 = x^4+y^4 \neq 0 \tag{$a$} \quad ? $$ Also, do there exist rational numbers $x,y,z$ such that $$ \quad \quad z^3 - z = x^4+y^4 \...
Bogdan Grechuk's user avatar
1 vote
0 answers
150 views

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 ...
Bogdan Grechuk's user avatar
1 vote
2 answers
124 views

Number of integers $x \leq B$ such that $f(x)\mid g(x)$ for coprime polynomials $f,g$

Let $f, g \in \mathbb{Z}[x]$ be coprime polynomials. I am interested in an upper bound for $$ N(B) = \# \{ x \in [-B, B] \cap \mathbb{Z}: f(x)\mid g(x) \}. $$ I assume there must be something known ...
Johnny T.'s user avatar
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0 votes
2 answers
216 views

$y^3=x^4+x$, and computing all rational points on rank $0$ Picard curves

What are the rational solutions to the equation $$ y^3 = x^4 + x, $$ in particular, are there any (finite) solutions other than $(x,y)=(0,0)$ and $(-1,0)$? Context: This is the simplest-looking ...
Bogdan Grechuk's user avatar
0 votes
0 answers
75 views

Number of solution to homogeneous linear Diophantine equations

Let $T,M\in\mathbb{N}$ be fixed. Consider a linear Diophantine equation of the form $a_1 x_1 + a_2 x_2 + … + a_n x_n = 0 $ with $a_i \in [-T,T] \subset \mathbb{Z}$. Is there an asymptotic formula to ...
Gotham17's user avatar
3 votes
0 answers
499 views

Regarding the Challenge Problem in 3Blue1Brown's most recent video: Will $\binom{x}{4}+\binom{x}{2}+1=2^k$ for $x>10$? [duplicate]

Link to the video here with timestamp In deriving the formula for regions of Moser's Circle Problem, it observed that the formula $$ F(x)=\binom{x}{4}+\binom{x}{2}+1 $$ achieves values that are equal ...
wjmccann's user avatar
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5 votes
0 answers
168 views

Existence of large integer solution for a simple-looking equation

Is it true that for every $k>0$ Diophantine equation $$ y^2 + x^2y + z^2x + 1 = 0 $$ has an integer solution $(x,y,z)$ such that $\min\{|x|,|y|,|z|\}\geq k$? Motivation: this equation arises in the ...
Bogdan Grechuk's user avatar
1 vote
1 answer
185 views

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 $...
Bogdan Grechuk's user avatar
2 votes
1 answer
292 views

On the equation $4y^p= x^2 + 3$

Do there exist some non-zero rational numbers $x, y$ such that $x \neq \pm y$ and $$4y^p = x^2 + 3 \tag{1}$$ for some odd prime $p$? If one lets $y=a/b$ and $x=c/d$ where $a, b, c, d$ are non-zero ...
Spotify's user avatar
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2 votes
1 answer
250 views

An arithmetic problem involving a system of equations

Fix a positive integer $r$. Describe the solutions to the system of equations given by: $$\begin{equation}\sum_{1\leq i\leq r}X_i^2\equiv0\pmod{X_k}(1\leq k\leq r)\end{equation}$$ Example: In the case ...
semisimpleton's user avatar
1 vote
0 answers
29 views

Does $2$ variable linear Diophantine equation in $NC$ imply $2$ dimensional shortest vector is in $NC$?

Consider the Linear Diophantine in known $a,b,c\in\mathbb Z$ $$ax+by=c.$$ Above can be solve by Extended Euclidean which is not in $NC$ as far as we know. It is clear if Extended Euclidean is in $NC$ ...
Turbo's user avatar
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0 votes
0 answers
70 views

Lowest asymptotic bound to $4^n - 2v_n^2$ where $v$ is an odd integer, $n$ fixed

The general problem is this. I try to find a positive integer $\delta_n$ such $qv_n^2 +\delta_n = p\cdot 4^n$. More precisely, I am looking for a lower bound (depending on $n$) for $\delta_n$ as $n\...
Vincent Granville's user avatar
10 votes
2 answers
1k views

Integer solutions of an exponential equation

How can I solve this equation? $$7^{x} +2=y^{2}$$ $x$ and $y$ must be natural numbers.
DADAS's user avatar
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2 votes
0 answers
90 views

Persistence of KAM tori as a function of dimension

I have tried posting this question in MSE, but I think it might be too technical so I'm trying again here. In KAM theory one tries to describe the persistence of quasi-periodic motion when an ...
QuantumBrick's user avatar
4 votes
1 answer
834 views

Does this conic have a rational point?

Consider the conic $$C = \{X^2+uY^2+vZ^2=0\}\subset\mathbb{P}^2_{\mathbb{Q}(u,v)}$$ over the function field $\mathbb{Q}(u,v)$. Does $C$ have a $\mathbb{Q}(u,v)$-rational point?
Puzzled's user avatar
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1 vote
1 answer
197 views

Rational points on genus 3 curves defined by short equations

(a) Find all pairs of rational numbers $(x,y)$ such that $$ y^3-y=x^4-x. $$ (b) Find all pairs of rational numbers $(x,y)$ such that $$ y^3+y=x^4+x. $$ If not a complete answer, I would be happy to ...
Bogdan Grechuk's user avatar
1 vote
0 answers
128 views

Is every even number greater than $44$ not divisible by $8$ of the form $x^2+y^2+z^4+t^4$?

Related to this question, where Bogdan Grechuk suggested this question. Q1 Is every even number greater than $44$ not divisible by $8$ of the form $x^2+y^2+z^4+t^4$...
joro's user avatar
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3 votes
0 answers
232 views

Is $16a+5$ always of the form $x^2+y^2+z^4$?

Working over the integers. For $a$ up to $10^7$, $16a+5$ is always of the form $x^2+y^2+z^4$. Q1 Is $16a+5$ always of the form $x^2+y^2+z^4$? Heuristic argument: For prime $p=4b+1$, both of $p$ and $...
joro's user avatar
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3 votes
0 answers
254 views

Are all odd integers greater than $599$ of the form $x^2+y^2+z^4+t^4$?

For $a \le 10^7$, the equation over integers $4a+1=x^2+y^2+z^4+t^4$ has solutions. Q1 Is it true that all integers of the form $4a+1$ are also of the form $x^2+y^2+z^4+t^4$? Heuristic argument: ...
joro's user avatar
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7 votes
3 answers
559 views

Question on a crucial lemma in Euler's approach to Fermat's Last Theorem for $n=3$

As many of you may know, the illustrious L. Euler put forward a proof of the case $n=3$ of Fermat's Last Theorem via infinite descent. The thing is that, at a certain point, he resorted to the ...
José Hdz. Stgo.'s user avatar
0 votes
1 answer
105 views

Is there a method to check if two sections of an elliptic surface are dependent over the endomorphism ring or not?

Let $E\!: y^2 = x^3 + a(t)x + b(t)$ be an elliptic curve over the function field $\mathbb{F}_{q}(t)$ over a finite field $\mathbb{F}_{q}$ of characteristic $5$ or greater. For simplicity, let $E$ be ...
Dimitri Koshelev's user avatar
6 votes
1 answer
626 views

Hilbert's tenth problem for equations with finitely many solutions

Is there a known example of a set $S$ of Diophantine equations such that $S$ is computable; it is a theorem that every equation in $S$ has (at most) finitely many solutions; the function that maps an ...
Timothy Chow's user avatar
1 vote
0 answers
97 views

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 $\...
joro's user avatar
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2 votes
0 answers
86 views

Complexity of finding solutions of trapdoored polynomial?

Related to this question Cryptography signature scheme based on hardness of finding points on varieties. Working over $K=\mathbb{Q}[x_1,...,x_n,y_1,...y_m]$. By abuse of notation, for polynomial $f$, ...
joro's user avatar
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2 votes
0 answers
94 views

Cryptography signature scheme based on hardness of finding points on varieties?

Related to this question Complexity of finding solutions of trapdoored polynomial. I am trying to build signature scheme based on hardness of finding points on varieties. Let $K$ be field and $M=K[x_1,...
joro's user avatar
  • 24.2k
4 votes
0 answers
126 views

Rational points on a cubic surface with small coefficients

Do there exists integers $(x,y,z,t)\neq (0,0,0,0)$ such that $$ 2x^3+2y^3+z^3+t^3+2x^2y-2z^2x-y^2z-z^2t = 0 ? $$ A short motivation: there are many known counterexamples to the Hasse principle for ...
Bogdan Grechuk's user avatar
8 votes
2 answers
575 views

Existence of rational points on a generalized Fermat quartic

Question: Do there exist integers $(x,y,z)\neq (0,0,0)$ such that $$ 13x^4+11y^4=8z^4 ? $$ Some motivation: This is currently the smallest (in a sense defined here On the smallest open Diophantine ...
Bogdan Grechuk's user avatar
0 votes
1 answer
167 views

How can we solve the following number theory problem? [closed]

Let $m$ and $n$ be positive integers less than $2000$ which satisfies the equation $(m^2-mn-n^2)^2=1$. How can we determine the largest possible value of the expression $m^2+n^2$?
Subodh Khanal's user avatar
2 votes
2 answers
405 views

Existence of rational points on generalized Fermat quintics

Do there exist integers $(x,y,z)\neq (0,0,0)$ such that $$ (a) \quad 2x^5+3y^5=6z^5 $$ $$ (b) \quad x^5+3y^5=7z^5 $$ Here is a short motivation. Equation $ax^d + by^d=cz^d$ is trivial for $d=1$, ...
Bogdan Grechuk's user avatar
5 votes
1 answer
440 views

Existence of rational points on some genus 3 curves

Do there exist a pair of rational numbers $(x,y)$ such that $$ (a) \quad x^4+x^3+y^4+y-1=0 $$ $$ (b) \quad x^4+x^3+y^4+y^2-1=0 $$ Magma function IsLocallySoluble returns that both equations are ...
Bogdan Grechuk's user avatar
0 votes
0 answers
151 views

A question on the Hilbert-Kamke problem

The Hilbert-Kamke problem consists in studying the integral solutions of the Diophantine system $$ x_1^i + \dots + x_s^i = n_i \text{ for } 1\leq i\leq k $$ with $x_i\geq 0$ for $i = 1,\dots,k$. I am ...
Puzzled's user avatar
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1 vote
0 answers
312 views

How would one go about solving this conjecture concerning exponential Diophantine equations?

I’ve been working on the Collatz Conjecture, and I believe I’ve reduced it to a more tractable problem. Unless there are some errors I’ve overlooked, I have managed to reduce the Collatz Conjecture to ...
John Eaton's user avatar
1 vote
1 answer
256 views

On the Diophantine equation $a^3 + b^3 = c^3 + d^3$

Let $a, b, c$ and $d$ be positive integers. What are the conditions that $a, b, c$ and $d$ should satisfy for the equality $$a^3 + b^3 = c^3 + d^3$$ to hold? In particular, can $a, b, c$ and $d$ be ...
user501735's user avatar
-3 votes
2 answers
160 views

Non-vanishing of this ternary quadratic expression [closed]

I'm dealing with the expression $x^2+y^2+6z^2+8xy+4x+4y−6xz−6yz$. I want to show that this expression is always non-zero whenever $x,y$ and $z$ are positive integers. How does one do this? (Note that ...
Benjamin L. Warren's user avatar
1 vote
0 answers
158 views

On quadratic Diophantine equations with n variables

Consider the following problem. Given a quadratic equation $$ \sum_{i,j=1}^n a_{i,j} x_ix_j + \sum_{k=1}^n d_{k} x_k + e = 0, \qquad a_{i,j},d_k,e\in\mathbb{Z}$$ if it exists, find (at least) a ...
Gotham17's user avatar
2 votes
1 answer
235 views

System of two linear Diophantine equations

Let $n\in\mathbb{N}$ be a positive integer. Denote by $f(n)$ the number of integral solutions of the following system $$ \left\lbrace\begin{array}{l} \sum_{i=1}^nx_i = 3n; \\ \sum_{i=1}^n (2i-1)x_i = ...
Puzzled's user avatar
  • 8,832
-1 votes
1 answer
94 views

Diophantine equation $546\cdot p+546\cdot q=1001\cdot r$ [closed]

$546\cdot p+546\cdot q=1001\cdot r$ $p,q$ odd primes, r positive integer. are there infinitely many solutions? And what if r is a Catalan number?
Enzo Creti's user avatar
1 vote
2 answers
228 views

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 ...
Puzzled's user avatar
  • 8,832
1 vote
0 answers
146 views

Exponential diophantine equation that I’m curious about

For which $x,y \in \mathbb{N} $ does the following hold? $\forall k \in \mathbb{N} \exists a,b,c,d \in \mathbb{N} \cup \{0\} : x^{a} + x^{b} = y^{c} + k y^{d} $. What sort of restrictions do we need ...
John Eaton's user avatar
1 vote
1 answer
162 views

Solutions to some cubic Diophantine equations

In searching for integral points on elliptic curves, I am encountering Diophantine equations of the following forms: $3m^3 - n^3 = {2^a}{3^b}$, $4m^3 - n^3 = {2^a}{3^b}{5^c}$, $5m^3 - n^3 = {2^a}{3^b}{...
student's user avatar
  • 121
1 vote
1 answer
246 views

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. ...
user avatar
5 votes
1 answer
299 views

Parity of number of solutions to Diophantine equations

By $MRDP$ resolution of Hilbert's tenth, we infer, counting number of solutions to Diophantine equations is undecidable. Is parity of number of solutions to Diophantine equations undecidable?
Turbo's user avatar
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1 vote
0 answers
120 views

When does a system of homogeneous quadratic equations have integer solutions?

I learned that in general, solving systems of quadratic Diophantine equations is a difficult problem. But I wonder if there are special (and non-trivial) types of systems that are easier to handle. ...
Louie's user avatar
  • 11
1 vote
0 answers
61 views

Set from a diophantine equation with similar statistics to primes

While doing some computational calculations with some diophantine equations, I came across with some sequences from solutions of quartic and quintic equations with slowly decreasing frequency, similar ...
rgvalenciaalbornoz's user avatar
7 votes
2 answers
723 views

Integer solutions of an algebraic equation

I'm trying to find integer solutions $(a,b,c)$ of the following algebraic equation with additional conditions $b>a>0$, $c>0$. $(-a^2+b^2+c^2)(a^2-b^2+c^2)(a^2+b^2-c^2) + 2 a b (-a^2+b^2+c^2)(...
Fedor Nilov's user avatar
4 votes
2 answers
740 views

On the equation $a^4+b^4+c^4=2d^4$ in coprime positive integers $a\lt b\lt c$ such that $a+b\ne c$

Background: The equation $$a^4+b^4+c^4=2d^4$$ has infinitely many positive integral solutions if we take $c=a+b$ and $a^2+ab+b^2=d^2$ further assuming that $GCD(a,b,c)=1$. Main problem: Find some ...
user avatar
0 votes
0 answers
172 views

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 ...
Bogdan Grechuk's user avatar

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