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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3
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0answers
95 views

Will an integer combination of some number of copies of the set of powers of 2 and the set of powers of 3 always have natural density 0?

Consider a Diophantine equation of the form $$(c_1 2^{x_1} + \dots + c_n 2^{x_n}) + (c_{n+1} 3^{x_{n+1}} + \dots + c_m 3^{x_m}) = y$$ where $x_1, \dots, x_m, y$ are our variables (here $x_1, \dots, ...
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Looking for a better way to solve quadratic integer equation with 2 variables [closed]

I have a mathematical problem (a+n)^2-c-(b+m)^2=0 a, b, c, m and n are positive integer or zero a, b, c we know, m and n we must calculate. In all cases there is solution. We are looking for solution ...
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92 views

On the equations $(x^n+1)(y^n+1)=z^2+1$ and $(x^n-1)(y^n-1)=z^2+1$

Note that $$(1^2+1)(2^2+1)=10=3^2+1\ \ \mbox{and}\ \ (1^4+2^4)(5^4+6^4)=8^4+13^4.$$ Today I tried to find positive integers $x,y,z$ satisfying $(x^4+1)(y^4+1)=z^4+1$ but failed. In view of this ...
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Pythagorean triples and quadratic residues modulo primes

QUESTION. Are my following conjectures true? How to prove them? Conjecture 1. For each prime $p>100$, there are $a,b,c\in\{1,\ldots,p-1\}$ such that $$\left(\frac ap\right)=\left(\frac bp\right)=\...
5
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1answer
225 views

Is $xz+1 $ a proper divisor of $a_3z^3+a_2z^2+a_1z+1$ finitely often?

Given a polynomial $P=a_3z^3+a_2z^2+a_1z+1, z >0$ with non-negative integer coefficients $a_1, a_2, a_3\ne 0$, it appears if $P$ is not factorizable then there are finitely many positive integers $...
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Are the nonnegative rationals diophantine with only two quantifiers?

Definition: A subset $D\subseteq \mathbb{Q}$ is diophantine if it is the projection of the zero set of a polynomial, i.e. there exists a polynomial $f\in\mathbb{Q}[X,Y_1,\dots,Y_n]$ for some $n$ such ...
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104 views

Positive integers $m$ such that $2m^2-1=x^4+y^4$ for some $x,y\in\{0,1,\ldots\}$ with $|x-y|>1$

I note that $$2(n^2+n+1)^2 -1= n^4+(n+1)^4.$$ This leads me to pose the following question. Question 1. Are there infinitely many positive integers $m$ such that $2m^2-1=x^4+y^4$ for some $x,y\in\...
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161 views

Diophantine consequences of the Buzzard–Diamond–Jarvis conjecture

Serre's modularity conjecture famously implies Fermat Last Theorem. More generally, Serre's conjecture implies that certain generalized Fermat equations have no non-trivial solutions (see Section 4.1 ...
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Positive integers written as $x^4+T(y)^2+T(z)+2^w$ with $x,y,z,w\in\mathbb N$ and $T(n)=n(n+1)/2$

Triangular numbers are those $T(n)=n(n+1)/2$ with $n\in\mathbb N=\{0,1,2,\ldots\}$. Clearly $$|\{(x,y,z,w)\in\mathbb N^4:\ x^4+T(y)^2+T(z)+2^w\le N\}|=O(N\log N)\ \ \text{for}\ N\ge2,$$ since $1/4+1/4+...
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Existence of $\{0,1\}$-solution to a system of linear equations with coefficients in $\{0,1\}$

Crossposted at Theoretical Computer Science SE A problem I study reduces to a system of linear equations $A\mathbf{x}=\mathbf{1}$ where $A$ is an $m\times n$ matrix with each entry $a_{ij}\in\{0,1\}$....
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Relation between stacky curves and “M-curves”

A tame stacky curve over a field $k$ is a geometrically connected proper smooth DM stack of dimension 1 which has a dense open substack which is a scheme, and whose automorphism group of each ...
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Is decidability reducible to unique decidability (perhaps in multilinear polynomial situations)?

Given a Diophantine equation it is not decidable if it has integer solution. I. Is there a Diophantine set $\mathcal D_{unique}$ satisfying the properties every member in $\mathcal D_{unique}$ is a ...
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1answer
205 views

Why wolfram alpha gives integers solutions for some equations of the form $ x^3 +(k\times10^n)^3 + z^3=0 $?

I have tried to get representations of some integers as sum of three cubic of the form $x^3+(k*10^n)^3+z^3$ with $k$ is integer and $n$ is a postive integer, I took this example : $(48807585839879)^3-(...
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Is there an integer-valued quadratic polynomial $P(x,y)$ such that $\{P(x,y)+2^k:\ x,y\in\mathbb Z\ \text{and}\ k\in\mathbb N\}=\mathbb Z^+$?

I seek for very sparse representations of positive integers. Let $$\mathbb N=\{0,1,2,\ldots\}\ \ \ \text{and}\ \ \ \mathbb Z^+=\{1,2,3,\ldots\}.$$ Recall that a polynomial $P(x,y)$ is integer-valued ...
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147 views

Natural numbers in the form $\lfloor\frac{a^3+b^3}2+\frac{c^3+d^3}6\rfloor$

Let $\mathbb N=\{0,1,2,\ldots\}$. Several years ago I proved that $$\{aw^3+bx^3+cy^3+dz^3:\ w,x,y,z\in\mathbb N\}\not=\mathbb N$$ for any positive integers $a,b,c,d$ (cf. http://maths.nju.edu.cn/~...
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On the equation $y^2 = x^3 - z^3$ [closed]

What is the parametric form of the rational solutions of the equation $y^2 = x^3 - z^3 ?$
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1answer
241 views

Is $\prod_{k=1}^nk^{\sigma(k)}$ a square or a cube for some $\sigma\in S_n$?

Note that for any permutation $\sigma\in S_5$ the product $\prod_{k=1}^5k^{\sigma(k)}$ is neither a square nor a cube. Question. Let $n>5$ be an integer. Is the product $\prod_{k=1}^nk^{\sigma(k)}$ ...
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243 views

Is there a permutation $\tau\in S_n$ with $\tau(1)^{\tau(2)}+\cdots+\tau(n-1)^{\tau(n)}+\tau(n)^{\tau(1)}$ a square?

Let $n>1$ be an integer, and let $S_n$ be the symmetric group of all the permutatins of $\{1,\ldots,n\}$. I'm curious whether there is a permutation $\tau\in S_n$ such that $$\tau(1)^{\tau(2)}+\...
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Variation in decidability of diophantine equations with field extension

Let $O_k$ be the ring of integers in a subfield $k$ of $\overline{\mathbb{Q}}$. Let's call an equation $p(x_1, \dots, x_n) = 0$ where $p$ is a polynomial in $n$-variables $x_1, \dots, x_n$ with ...
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191 views

On the equations $x^yy^z=z^x$ and $w^x+x^y+y^z=z^w$

Recently, I considered the equation $$x^yy^z=z^x\qquad(x,y,z\in\{2,3,\ldots\}).\tag{1}$$ The equation $(1)$ has infinitely many solutions with $x=z$ including $$(x,y,z)=(n^n,n^{n-1},n^n),\ (n^{2n^2},n^...
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376 views

Diophantine equation $x^3+y^9=z^6-1$

Consider the diophantine equation: $x^3+y^9=z^6-1$, for x, y, z positive. Has it the only solution: $x=6$ , $y=2$ and $z=3$?
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Status of $x^3+y^3+z^3=6xyz$

In Erik Dofs, Solutions of $x^3 + y^3 + z^3 = nxyz$, Acta Arithmetica 73 (1995) pp. 201–213, doi:10.4064/aa-73-3-201-213, EuDML the author has studied the Diophantine equation \begin{equation} x^3+y^...
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1answer
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Quadratic Diophantine equations with all values prime

Given a quadratic Diophantine equation over the integers in two variables, can we say much about when it has only finitely many solutions with the additional assumption that both variables are prime? ...
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87 views

Finding number fields over which Diophantine equations are solvable

Given a Diophantine equation $f(x_1, \dots, x_n) \in \mathbb{Z}[x_1, \dots, x_n]$ and a family of number fields $K$ (say, the number fields of a specified degree and signature), are there techniques ...
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3answers
530 views

Is there a simple proof that $Ax^3+By^3=C$ has only finitely many integer solutions

One can use Thue's 1909 result to show that the Diophantine equation $Ax^3 + By^3 = C$ ($A,B$ not perfect cubes, $C\neq 0$) has finitely many integer solutions $(x,y)$. But does there exist a simple ...
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1answer
989 views

The “stubborn” solutions to sums of three cubes

It is conjectured (see [1]) that for any integer $k\not\equiv \pm 4\pmod 9$ there are infinitely many integer solutions to $$ a^3+b^3+c^3=k. $$ Numerical investigations of this conjecture show that ...
4
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1answer
253 views

$n$ variables Diophantine

Let $n \ge 2$ be a positive integer. Do there exist $n$ non-zero distinct integers such that the sum of their square is a perfect square and their product is a nth power? For $n=2$ the answer is no, ...
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2answers
124 views

Mordell like equation [closed]

This looks like a mordell like equation X²=Y³-25056 How to solve it? The exact equation is (36x)²=(6y)³-25056 Is there any website has records of the equation x²=y³+k For k>25000
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180 views

How to solve special Diophantine equation systems (which one can solve by hand) with the computer?

I have a quadratic Diophantine equation system which is possibly not homogeneous and has some mixed and some linear terms. But I know that there are only finitely many solutions over the integers. One ...
4
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3answers
234 views

Finding Pythagorean quadruples on a given plane?

In 2D one cannot construct Pythagorean triples $x^2+y^2=m^2$ ($x,y,m\in\mathbb{Z}$) that lie on every line through the origin (e.g., a Pythagorean triple with $x=y$ would require $\sqrt{2}$ to be ...
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1answer
273 views

Action of the symmetric group $S_3$ on an elliptic curve $E$ defined over $\mathbb{Z}$

I came up with the following question on a facebook group: find the positive integer solutions of the equation $$\frac{x}{y+z}+\frac{y}{x+z}+\frac{z}{x+y}=4$$ Now clearly this is very difficult, ...
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1answer
193 views

Does the equation in positive integers $(n,y)\,\prod_{k=1}^n(p_k^2-1)=y^2\,$ only have the solution $(3,24)$?

Does the equation in positive integers $\,(n,\,y)$ $$\prod_{k=1}^n(p_k^2-1)=y^2$$ only have the solution $(3,\,24)\,$? I asked a more general question here. The computational complexity of the problem ...
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71 views

Relaxation of Waring's problem $n=\sum_{i=1}^{k+1} a_i x_i^k$

Waring's problem asks whether each natural number k has an associated positive integer s such that every natural number is the sum of at most s natural numbers to the power of k. We are investigating ...
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187 views

Does the diophantine equation $\,\prod_{k=1}^n(p_k^{x_k}-1)=y^2\,$ have always at least a solution for $\,n\gt2\,$?

P.G.Walsh proved in this paper that the diophantine equation $\,(2^{x_1}-1)(3^{x_2}-1)=y^2\,$ has no solution in positive integers $\,x_1$, $\,x_2\,$ and $\,y$. If we generalize the previous equation ...
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2answers
565 views

Diophantine equation $3(a^4+a^2b^2+b^4)+(c^4+c^2d^2+d^4)=3(a^2+b^2)(c^2+d^2)$

I am looking for positive integer solutions to the Diophantine equation $3(a^4+a^2b^2+b^4)+(c^4+c^2d^2+d^4)=3(a^2+b^2)(c^2+d^2)$ for distinct values of $(a,b,c,d)$. There are many solutions with $a=b$ ...
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185 views

Restricted divisor summatory function

I have a problem that boils down to prove that the succession $\{a_n\}$ tends to infinity, where $$a_n:=1+\sum_{0\leq j<n}D_{2j+1}(n-j)$$ and $D_{m}(n)$ is the number of divisors $d>1$ of $n$ ...
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64 views

When there are infinite many primitive solutions of Vinogradov system?

Forgive my ignorance if this is an obvious problem, and I can't solve it just because I didn't try to solve it with a pen or I am stupid. Consider Vinogradov's system, $$x_{1}^{j}+x_{2}^{j}+\cdots+x_{...
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79 views

A conjectural limit involving primorial and factorial

It is well known that the abc conjecture implies that the there are only finitely many solutions to Brocard problem, as shown by Overholt in Overholt, Marius (1993), "The diophantine equation $n! ...
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1answer
199 views

How can the number of rational points depend on the choice of height function?

Let $V/\mathbb{Q}$ be a subvariety of $\mathbb{P}^n$. There are many plausible choices of height function, some differing only by constant factors: $\max |x_i|$ (for $(x_0,x_1,\dotsc,x_n)$, $\gcd(x_1,\...
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0answers
124 views

Cubic surface in $\mathbb{P}^3$ singular along a line

Maybe it is a stupid question but I'm not able to find the answer anywhere else. My goal is to prove in an "algebraic geometry fashion" that $\sqrt{n}$ is not a rational number for $n$ not a ...
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1answer
73 views

Software for $S$-unit equation

Is there any implementation available of an algorithm which solves in full generality the $S$-unit equation $x+y=1$ in a number field? It seems that Magma solves $ax+by=c$ but only in the algebraic ...
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188 views

Are the numbers $\varphi(n^2)\sigma(n^2)\ (n=1,2,3,\ldots)$ pairwise distinct?

Let $\varphi$ be Euler's totient function, and let $\sigma(n)=\sum_{d\mid n}d$ for $n=1,2,3,\ldots$. Both $\varphi$ and $\sigma$ are multiplicative functions. It is easy to see that the numbers $$\...
5
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1answer
253 views

How to solve this equation $a^2+3b^2c^2=7^c$

Let $a,b,c$ be poistive integers,and such $\gcd(a,b)=\gcd(b,c)=\gcd(a,c)=1$,fine the all $a,b,c$ such $$a^2+3b^2c^2=7^c$$ I'm not sure that this question has been studied, but I've been trying for a ...
16
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1answer
460 views

Is it true that $\{x^4+y^2+z^2:\ x,y,z\in\mathbb Z[i]\}=\{a+2bi:\ a,b\in\mathbb Z\}$?

Recall that the ring of Gaussian integers is $$\mathbb Z[i]=\{a+bi:\ a,b\in\mathbb Z\}.$$ Clearly $$(a+bi)^2=a^2-b^2+2abi\ \ \mbox{and}\ \ (a+bi)^4=(a^2-b^2)^2-4a^2b^2+4ab(a^2-b^2)i.$$ Question. Is it ...
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0answers
109 views

Genus $0$ algebraic curves integral points decidable?

It is known there is an explicit algebraic variety in $\mathbb Z[x_1,\dots,x_t]$ a bounded $t>2$ whose integral zero-set is non-empty is undecidable. If the variety has genus $0$ is there anything ...
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0answers
164 views

A certain Pell Equation

Recently I came up with a positive solution $((x,y)\neq (\pm 1;0))$ to this diophantine equation $$ x^2-\left(w^2(2^{n-2}p)^2+2^n(2^{n-2}p)\right)y^2=1,\qquad n\geq 2, $$ where all variables are in $ ...
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0answers
99 views

Counting solutions of a equation involving prime powers

Let $p\geq 3$ be a prime number. Let $n\in\mathbb{Z}_{\geq 1}$, $q\in\mathbb{Q}$, $m\in\mathbb{Z}_{\geq 2}$. Set $$T_n(q,m)=\#\left\{(l_1,\cdots,l_{p^n})\in\mathbb{Z}_{\geq n+m}^{p^n}\middle\vert\sum_{...
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0answers
79 views

Efficiently find at least one lattice point on hyperbola of equation $axy+bx+cy+d=0$

I need an algorithm to efficiently find at least one lattice point on a hyperbola of equation $axy+bx+cy+d=0$. Lattice point means integer coordinates and equation with integer means diophantine ...
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0answers
83 views

Maximum number of integer solutions with some size constraints to bivariate polynomials?

Take a bivariate polynomial of total degree $d$ satisfying $d=d_x=d_y>1$ in $\mathbb Z[x,y]$ with coefficients bound in absolute value by $b$ ($d_x$ is $x$-degree and $d_y$ is $y$-degree). Given a ...
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0answers
73 views

Distribution of number of integer solutions in box to bivariate polynomials?

Take a bivariate polynomial of degree $d_x+d_y>\max(d_x,d_y)>1$ in $\mathbb Z[x,y]$ with coefficients bound in absolute value by $b$ ($d_x$ is $x$-degree and $d_y$ is $y$-degree). What is the ...

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