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 ...
935 questions
10
votes
4
answers
1k
views
The smallest solution to $2^{2k}-1=\text{powerful}$
Integer is powerful if all the exponents in its factorization are at least $2$.
Every powerful integer can be written in the form $a^2 b^3$.
For odd $k$, define $F(k)=2^{2k}-1=(2^k-1)(2^k+1)$.
This ...
- 25.4k
22
votes
2
answers
1k
views
Why is 1331 the only cube of the form $x^2 + x − 1$?
The Wikipedia (https://en.wikipedia.org/wiki/1000_(number)#1300_to_1399) mentions that 1331 is the only cube of the form $x^2 + x − 1$, for $x = 36$. What is the proof?
- 459
0
votes
0
answers
112
views
The number of solutions of $2^xpx+k=y^2$
Let's consider the family of diophantine equations
$$2^xpx+k=y^2$$
being $p\gt2$ a prime and $k$ a positive integer.
An example is given by the equation
$$2^x\cdot3x+97=y^2$$
that presents, at least, ...
- 1,008
1
vote
2
answers
435
views
On the equation $x^3 + y^3 =cz^3$
What are the characteristics of the values of $c$ for which the equation $x^3 + y^3 = cz^3$ has pairwise coprime non-zero integral solutions where $x \neq \pm y$ ? For instance, it is known that $c$ ...
- 11
7
votes
2
answers
911
views
Triangular numbers of the form $x^4+y^4$
Recall that triangular numbers are those $T(n)=n(n+1)/2$ with $n\in\mathbb N=\{0,1,2,\ldots\}$. Fermat ever proved that the equation $x^4+y^4=z^2$ has no positive integer solution. So I think it's ...
- 15.6k
8
votes
1
answer
639
views
On Markoff-type diophantine equation
Do there exist integers $x,y,z$ such that
$$
x^2+y^2-z^2 = xyz -2 \quad ?
$$
Why this is interesting? First, this equation arose in an answer to the previous Mathoverflow question What is the smallest ...
- 781
8
votes
1
answer
806
views
Can $x^4+y^4+1$ be a perfect power?
Recall that a perfect power has the form $x^m$ with $m,x\in\{2,3,\ldots\}$. Motivated by Fermat's result that the equation $x^4+y^4=z^2$ has no positive integer solution, here I ask the following ...
- 15.6k
3
votes
0
answers
135
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, ...
1
vote
0
answers
161
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 ...
- 15.6k
3
votes
1
answer
233
views
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
$$\left(\frac ap\right)=\left(\frac bp\right)=\...
- 15.6k
6
votes
1
answer
332
views
Is $xz+1 $ a proper divisor of $a_3z^3+a_2z^2+a_1z+1$ finitely often?
Given a polynomial $P=a_3z^3+a_2z^2+a_1z+1, z >0$ with non-negative integer coefficients $a_1, a_2, a_3\ne 0$, it appears if $P$ is not factorizable then there are finitely many positive integers $...
- 319
10
votes
0
answers
217
views
Are the nonnegative rationals diophantine with only two quantifiers?
Definition: A subset $D\subseteq \mathbb{Q}$ is diophantine if it is the projection of the zero set of a polynomial, i.e. there exists a polynomial $f\in\mathbb{Q}[X,Y_1,\dots,Y_n]$ for some $n$ such ...
- 2,051
3
votes
0
answers
198
views
Positive integers $m$ such that $2m^2-1=x^4+y^4$ for some $x,y\in\{0,1,\ldots\}$ with $|x-y|>1$
I note that
$$2(n^2+n+1)^2 -1= n^4+(n+1)^4.$$
This leads me to pose the following question.
Question 1. Are there infinitely many positive integers $m$ such that $2m^2-1=x^4+y^4$ for some $x,y\in\...
- 15.6k
4
votes
0
answers
238
views
Diophantine consequences of the Buzzard–Diamond–Jarvis conjecture
Serre's modularity conjecture famously implies Fermat Last Theorem. More generally, Serre's conjecture implies that certain generalized Fermat equations have no non-trivial solutions (see Section 4.1 ...
- 595
5
votes
0
answers
240
views
Existence of $\{0,1\}$-solution to a system of linear equations with coefficients in $\{0,1\}$
Crossposted at Theoretical Computer Science SE
A problem I study reduces to a system of linear equations $A\mathbf{x}=\mathbf{1}$ where $A$ is an $m\times n$ matrix with each entry $a_{ij}\in\{0,1\}$....
- 123
4
votes
1
answer
571
views
Relation between stacky curves and "M-curves"
A tame stacky curve over a field $k$ is a geometrically connected proper smooth DM stack of dimension 1 which has a dense open substack which is a scheme, and whose automorphism group of each ...
- 1,364
7
votes
0
answers
274
views
Is decidability reducible to unique decidability (perhaps in multilinear polynomial situations)?
Given a Diophantine equation it is not decidable if it has integer solution.
I. Is there a Diophantine set $\mathcal D_{unique}$ satisfying the properties
every member in $\mathcal D_{unique}$ is a ...
- 13.9k
-2
votes
1
answer
494
views
Why wolfram alpha gives integers solutions for some equations of the form $ x^3 +(k\times10^n)^3 + z^3=0 $?
I have tried to get representations of some integers as sum of three cubic of the form $x^3+(k*10^n)^3+z^3$ with $k$ is integer and $n$ is a postive integer, I took this example : $(48807585839879)^3-(...
3
votes
0
answers
308
views
Is there an integer-valued quadratic polynomial $P(x,y)$ such that $\{P(x,y)+2^k:\ x,y\in\mathbb Z\ \text{and}\ k\in\mathbb N\}=\mathbb Z^+$?
I seek for very sparse representations of positive integers. Let
$$\mathbb N=\{0,1,2,\ldots\}\ \ \ \text{and}\ \ \ \mathbb Z^+=\{1,2,3,\ldots\}.$$
Recall that a polynomial $P(x,y)$ is integer-valued ...
- 15.6k
4
votes
0
answers
176
views
Natural numbers in the form $\lfloor\frac{a^3+b^3}2+\frac{c^3+d^3}6\rfloor$
Let $\mathbb N=\{0,1,2,\ldots\}$. Several years ago I proved that
$$\{aw^3+bx^3+cy^3+dz^3:\ w,x,y,z\in\mathbb N\}\not=\mathbb N$$
for any positive integers $a,b,c,d$ (cf. http://maths.nju.edu.cn/~...
- 15.6k
0
votes
1
answer
228
views
On the equation $y^2 = x^3 - z^3$ [closed]
What is the parametric form of the rational solutions of the equation $y^2 = x^3 - z^3 ?$
4
votes
1
answer
273
views
Is $\prod_{k=1}^nk^{\sigma(k)}$ a square or a cube for some $\sigma\in S_n$?
Note that for any permutation $\sigma\in S_5$ the product $\prod_{k=1}^5k^{\sigma(k)}$ is neither a square nor a cube.
Question. Let $n>5$ be an integer. Is the product $\prod_{k=1}^nk^{\sigma(k)}$ ...
- 15.6k
0
votes
0
answers
266
views
Is there a permutation $\tau\in S_n$ with $\tau(1)^{\tau(2)}+\cdots+\tau(n-1)^{\tau(n)}+\tau(n)^{\tau(1)}$ a square?
Let $n>1$ be an integer, and let $S_n$ be the symmetric group of all the permutatins of $\{1,\ldots,n\}$.
I'm curious whether there is a permutation $\tau\in S_n$ such that
$$\tau(1)^{\tau(2)}+\...
- 15.6k
3
votes
0
answers
116
views
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 ...
- 31
2
votes
0
answers
266
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 $$(x,y,z)=(n^n,n^{n-1},n^n),\ (n^{2n^2},n^...
- 15.6k
8
votes
4
answers
2k
views
Status of $x^3+y^3+z^3=6xyz$
In
Erik Dofs, Solutions of $x^3 + y^3 + z^3 = nxyz$, Acta Arithmetica 73 (1995) pp. 201–213, doi:10.4064/aa-73-3-201-213, EuDML
the author has studied the Diophantine equation
\begin{equation}
x^3+y^...
- 371
5
votes
1
answer
345
views
Quadratic Diophantine equations with all values prime
Given a quadratic Diophantine equation over the integers in two variables, can we say much about when it has only finitely many solutions with the additional assumption that both variables are prime?
...
- 6,969
1
vote
0
answers
102
views
Finding number fields over which Diophantine equations are solvable
Given a Diophantine equation $f(x_1, \dots, x_n) \in \mathbb{Z}[x_1, \dots, x_n]$ and a family of number fields $K$ (say, the number fields of a specified degree and signature), are there techniques ...
- 479
6
votes
3
answers
876
views
Is there a simple proof that $Ax^3+By^3=C$ has only finitely many integer solutions
One can use Thue's 1909 result to show that the Diophantine equation $Ax^3 + By^3 = C$ ($A,B$ not perfect cubes, $C\neq 0$) has finitely many integer solutions $(x,y)$.
But does there exist a simple ...
- 4,280
26
votes
1
answer
2k
views
The "stubborn" solutions to sums of three cubes
It is conjectured (see [1]) that for any integer $k\not\equiv \pm 4\pmod 9$ there are infinitely many integer solutions to
$$
a^3+b^3+c^3=k.
$$
Numerical investigations of this conjecture show that ...
- 7,193
4
votes
1
answer
271
views
$n$ variables Diophantine
Let $n \ge 2$ be a positive integer. Do there exist $n$ non-zero distinct integers such that the sum of their square is a perfect square and their product is a nth power?
For $n=2$ the answer is no, ...
- 3,153
-2
votes
2
answers
148
views
Mordell like equation [closed]
This looks like a mordell like equation
X²=Y³-25056
How to solve it?
The exact equation is
(36x)²=(6y)³-25056
Is there any website has records of the equation x²=y³+k
For k>25000
- 1
1
vote
0
answers
216
views
How to solve special Diophantine equation systems (which one can solve by hand) with the computer?
I have a quadratic Diophantine equation system which is possibly not homogeneous and has some mixed and some linear terms.
But I know that there are only finitely many solutions over the integers.
One ...
- 1,755
4
votes
3
answers
286
views
Finding Pythagorean quadruples on a given plane?
In 2D one cannot construct Pythagorean triples $x^2+y^2=m^2$ ($x,y,m\in\mathbb{Z}$) that lie on every line through the origin (e.g., a Pythagorean triple with $x=y$ would require $\sqrt{2}$ to be ...
- 41
7
votes
1
answer
625
views
Action of the symmetric group $S_3$ on an elliptic curve $E$ defined over $\mathbb{Z}$
I came up with the following question on a facebook group: find the positive integer solutions of the equation $$\frac{x}{y+z}+\frac{y}{x+z}+\frac{z}{x+y}=4$$
Now clearly this is very difficult, ...
- 1,343
3
votes
1
answer
217
views
Does the equation in positive integers $(n,y)\,\prod_{k=1}^n(p_k^2-1)=y^2\,$ only have the solution $(3,24)$?
Does the equation in positive integers $\,(n,\,y)$
$$\prod_{k=1}^n(p_k^2-1)=y^2$$
only have the solution $(3,\,24)\,$?
I asked a more general question here.
The computational complexity of the problem ...
- 1,008
10
votes
0
answers
205
views
Does the diophantine equation $\,\prod_{k=1}^n(p_k^{x_k}-1)=y^2\,$ have always at least a solution for $\,n\gt2\,$?
P.G.Walsh proved in this paper that the diophantine equation $\,(2^{x_1}-1)(3^{x_2}-1)=y^2\,$ has no solution in positive integers $\,x_1$, $\,x_2\,$ and $\,y$.
If we generalize the previous equation ...
- 1,008
9
votes
2
answers
644
views
Diophantine equation $3(a^4+a^2b^2+b^4)+(c^4+c^2d^2+d^4)=3(a^2+b^2)(c^2+d^2)$
I am looking for positive integer solutions to the Diophantine equation $3(a^4+a^2b^2+b^4)+(c^4+c^2d^2+d^4)=3(a^2+b^2)(c^2+d^2)$ for distinct values of $(a,b,c,d)$.
There are many solutions with $a=b$ ...
- 506
8
votes
0
answers
271
views
Restricted divisor summatory function
I have a problem that boils down to prove that the succession $\{a_n\}$ tends to infinity, where
$$a_n:=1+\sum_{0\leq j<n}D_{2j+1}(n-j)$$
and $D_{m}(n)$ is the number of divisors $d>1$ of $n$ ...
- 361
0
votes
0
answers
143
views
A conjectural limit involving primorial and factorial
It is well known that the abc conjecture implies that the there are only finitely many solutions to Brocard problem, as shown by Overholt in Overholt, Marius (1993), "The diophantine equation $n! ...
- 1,175
6
votes
1
answer
231
views
How can the number of rational points depend on the choice of height function?
Let $V/\mathbb{Q}$ be a subvariety of $\mathbb{P}^n$. There are many plausible choices of height function, some differing only by constant factors: $\max |x_i|$ (for $(x_0,x_1,\dotsc,x_n)$, $\gcd(x_1,\...
- 20.2k
1
vote
0
answers
154
views
Cubic surface in $\mathbb{P}^3$ singular along a line
Maybe it is a stupid question but I'm not able to find the answer anywhere else.
My goal is to prove in an "algebraic geometry fashion" that $\sqrt{n}$ is not a rational number for $n$ not a ...
- 1,343
2
votes
1
answer
196
views
Software for $S$-unit equation
Is there any implementation available of an algorithm which solves in full generality the $S$-unit equation $x+y=1$ in a number field? It seems that Magma solves $ax+by=c$ but only in the algebraic ...
- 509
5
votes
0
answers
256
views
Are the numbers $\varphi(n^2)\sigma(n^2)\ (n=1,2,3,\ldots)$ pairwise distinct?
Let $\varphi$ be Euler's totient function, and let $\sigma(n)=\sum_{d\mid n}d$ for $n=1,2,3,\ldots$. Both $\varphi$ and $\sigma$ are multiplicative functions. It is easy to see that the numbers
$$\...
- 15.6k
4
votes
1
answer
675
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 ...
- 4,280
19
votes
1
answer
679
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 ...
- 15.6k
0
votes
0
answers
122
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 ...
- 13.9k
0
votes
0
answers
183
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 $ ...
- 785
0
votes
0
answers
336
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 ...
- 101
0
votes
0
answers
115
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 ...
- 13.9k