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
2
votes
3
answers
286
views
Specific Diophantine Equation Appearing in Faa Di Bruno Formula
In a Faa Di Bruno Formula there is an equation:
$m_1$+2*$m_2$+3*$m_3$+...n*$m_n$=n
Is there any general solution for this equation.
For example for
$m_1$+$m_2$+$m_3$+...+$m_n$=n, there is a ...
- 185
2
votes
2
answers
324
views
Four-Square Theorem for Negative Coefficient
What integers are not in the range of $a^2+b^2+c^2-x^2$ (for all integer combinations of a, b, c, and x)? This form is similar to that of Lagrange's Four-Square Theorem, for which the answer would be ...
- 1,547
2
votes
1
answer
261
views
Small solutions of $x^2-a^3 y^2=\pm 1$
We are interested in small integer solutions to the Pell equation:
$$x^2-a^3 y^2=\pm 1 \qquad (1)$$
Where in $\pm 1$ you can chose either sign.
$(x^2,a^3 y^2)$ are consecutive powerful numbers.
$abc$ ...
- 25.4k
2
votes
1
answer
587
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 ...
- 1,019
2
votes
2
answers
239
views
Concise formulation of set of equation systems
I have the following set of equation systems, and I would like to find a short, formal way to write it down. My main difficulty is that I cannot find a good way to write the indices of the variables $\...
- 155
2
votes
2
answers
823
views
Enumerating solutions to an underdetermined non-homogenous linear system of Diophantine equations
I have a large, under-determined system (60 equations and 116 unknowns) of linear Diophantine equations. I am aware of the algorithms typically used to solve these systems, which is not my question.
...
- 21
2
votes
3
answers
945
views
Algorithm to count number of positive integer solutions of $x^2(8x-3)=y^2z$?
Given the Diophantine equation
$$ x^2(8x-3)=y^2z, $$
is there a way to efficiently count the number of solutions that satisfy $x+y+z\leq n$, where $n$ is a fixed given integer?
Also, for any fixed $x$...
- 416
2
votes
4
answers
686
views
solution of the equation $a^2+pb^2-2c^2-2kcd+(p+k^2)d^2=0$
i am wondering if there is a complete solution for the equation $a^2+pb^2-2c^2-2kcd+(p+k^2)d^2=0$ in which $a,b,c,d,k$ are integer(not all zero) and $p$ is odd prime.
- 29
2
votes
1
answer
269
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
$$
\left\lbrace\begin{array}{l}
\sum_{i=1}^nx_i = 3n; \\
\sum_{i=1}^n (2i-1)x_i = ...
- 8,998
2
votes
1
answer
278
views
Diophantine equations and ergodic theorems
In the paper by Akos Magyar, Diophantine Equations and Ergodic Theorems, one states in page 923 the following theorem:
Theorem 1: Let $Q(m)$ be a nondegenerate polynomial and $\Lambda$ is ...
2
votes
1
answer
196
views
Software for $S$-unit equation
Is there any implementation available of an algorithm which solves in full generality the $S$-unit equation $x+y=1$ in a number field? It seems that Magma solves $ax+by=c$ but only in the algebraic ...
- 509
2
votes
1
answer
231
views
Equations involving arithmetic functions of primorials
Let $\sigma(n)=\sum_{1\leq d\mid n}d$ the sum of divisors, $\varphi(n)$ the Euler's totient function and we denote the primorial $\prod_{k=1}^n p_k$ as $N_n$, where $p_k$ denotes the $k$-th prime ...
2
votes
1
answer
462
views
Counting algebraic points of bounded height
Let $K$ be a number field and $X\hookrightarrow\mathbb P^n_K$ be a projective variety of degree $\delta$ (with respect to universal bundle) and dimension $d$. We denote the set
$$S(X;D,B)=\{\xi\in X(\...
- 403
2
votes
1
answer
227
views
Bounds on near-zero integer linear combinations of numbers linearly independent over $\mathbb{Q}$
Let $\alpha_1, \alpha_2, \dots$ be an infinite sequence of real numbers such that any finite subset is linearly independent over $\mathbb{Q}$. Let $f(N)$ be the number of tuples $(m_1, \dots, m_N)$ ...
- 595
2
votes
1
answer
470
views
Non-negative integer solutions of x^2+y^3=n
I have the next equation: $x^2+y^3=n$. Where n is a positive integer constant.
I want to know the exact number of non-negative integer solutions.
Also I want to know what are those solutions. How ...
- 133
2
votes
5
answers
1k
views
Does the Diophantine equation $(x^2+ay^2)(u^2+bv^2) = p^2+cq^2$ admit a complete solution?
In this MSE question/thread, I have been discussing the equation
$$
(x^2+ay^2)(u^2+bv^2) = p^2+cq^2, \tag{$\star$}
$$
where $x,a,y,u,b,v,p,c,q$ are integers. I posed a conjecture which turned out to ...
- 1,055
2
votes
1
answer
258
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 ...
- 1,433
2
votes
1
answer
164
views
a b c triples with bounded prime factors
(i) For any fixed $B>0$, are there only finitely many triples $a,b,c$ of coprime positive integers, such that $a+b=c$ and all prime factors of $a,b,c$ are at most $B$?
(ii) For which $B$ all such ...
- 743
2
votes
1
answer
354
views
The Fermat-Catalan conjecture with signature $(2,n,4)$, $n\ge4$
The Fermat-Catalan conjecture is that for coprime $x,y,z$ and positive integers $a,b,c$ with $1/a+1/b+1/c<1$, the generalized Fermat equation $x^a + y^b = z^c$ has only finitely many solutions. I'm ...
- 141
2
votes
1
answer
139
views
Solutions to the Diophantine equation $a^xy+x=c$
Fix positive integers $a,c$ with $a>2$. Is it possible that the Diophantine equation
$$a^xy+x=c$$
has infinitely many solutions (in positive integers $x$ and $y$)?
- 21
2
votes
1
answer
418
views
On weaker forms of the abc conjecture from the theory of Hölder and logarithmic means
In this post (the content of this post is now cross-posted from Mathematics Stack Exchange see below) we denote the radical of an integer $n>1$ as the product of disctinct primes dividing it $$\...
2
votes
1
answer
248
views
Sum from combinatorics on nonnegative integer numbers
Let $n_1,n_2,\ldots,n_k\in\{0,1,2,\ldots\}$. Can you calculate the sum
$$
\sum_{n_1,n_2,\ldots,n_k\geqslant0}\mathbb{1}_\left\{n_1+\frac{n_2}{2}+\ldots+\frac{n_k}{k}<1\right\}?
$$
If it's helpful, ...
2
votes
1
answer
231
views
All the solutions of linear Diophantine equation [closed]
Given the linear Diophantine equation:
$ a_1 x_1 + a_2 x_2 + ... + a_n x_n = b$
and its particular solution $(x_1^*, x_2^*, ..., x_n^*)$.
How to write down all the solutions of this equation?
- 131
2
votes
1
answer
3k
views
Diophantine: $a^n + b^n + c^n = d^n$ and $a^n + b^n = c^n + d^n$
Let us consider the equation $a^n+b^n=c^n$ for positive integers $a,b,c$ and $n\ge 2$. The $n=2$ case has a well-known and beautiful parametrization known as Pythagorean triples. Fermat's Last Theorem ...
- 2,075
2
votes
1
answer
197
views
Mod n, are all higher powers also lower powers?
Since there are only finitely many residues mod $n$, there is some function $H(n) \le n$ such that for all integers $n>1$, $r$, and $e>H(n)$, if $r$ is an $e$-th power mod $n$ then there is some ...
- 9,114
2
votes
1
answer
180
views
Only trivial solution to a pair of constrained linear diophantine equations
Given positive integer $n$, we are looking for a set
of $n$ positive integers $a_i$.
The following linear integer program must have only
the trivial integer solution of all ones.
$0 \le x_i \le \frac{...
- 25.4k
2
votes
1
answer
532
views
Diophantine equation: $n^2=c(4ab-a-b)-b$?
I asked the following question here, but I did not get a full answer, so I put it here that may be some help.
Let $n$ be a positive integer. The Diophantine equation
$$
n^2=c(4ab-a-b)-b,\qquad (a,b,...
- 841
2
votes
1
answer
202
views
Feasibility of constrained multivariable diophantine equations
Let $d$ be day, $m$ be month and $y$ be year fields of a date. I want to find few dates of format
$$(d^2\, mod\,\, 2 + (my + d^3) \,mod \,4) = 2$$
Is there a method to solve this type of equation or ...
- 131
2
votes
1
answer
366
views
System of congruences
I have a system of $n$ congruences.
the generic $m$_th congruence of the system ($m = 1,\dots,n$) is in the form:
$(p_n\#)^2(\displaystyle\sum_{\substack{i=1\\i\neq m}}^n{\frac{x_iy_m}{p_ip_m}}+\...
- 21
2
votes
1
answer
211
views
Finite sequences which happen to be the sequence of orders of elements of a simple group
Let $n\in \Bbb N$. Any finite group $G$ with $|G|=n$ has a solution for the Diophantine equation
$$\sum_{d|n}x_d\phi(d)=n~~~~~~~~~~~~~~~~~~~~~~~(1)$$
where $\phi$ is the Euler's totient function, $d$ ...
- 1,145
2
votes
1
answer
208
views
Integral values of rational map
This question is related to this post on Math.MO.
A theorem of B.Segre tells us that if there is one rational point on a non-singular cubic surface $X$ over $\mathbb{Q}$, then the surface is ...
- 3,337
2
votes
1
answer
727
views
Infinite solutions of a diophantine equation [closed]
Given the Diophantine equation$$ax^2+bxy+cy^2+dx+ey+f=0$$
if the coefficients $(a,b,c,d,e,f)$ are chosen among all the prime numbers, we have infinite equations. Is it possible to prove that the ...
- 399
2
votes
0
answers
132
views
Solving a system of exponential Diophantine equations
I am trying to find solutions to the system $p^x+q=q^y+r=r^z+p$ where $p,q,r$ are primes and $x,y,z$ are integers greater than $2$.
So far, I have only found that at most one among $x$, $y$, and $z$ ...
- 121
2
votes
0
answers
52
views
Infinitely many coprime solutions of $F(x,y)= k(a_1 x + a_2 y)^2 z^2$?
This might be related to an open problem.
Let $F(x,y)$ be homogeneous degree 4 squarefree polynomial
with integer coefficients and
$h(x,y)=a_1 x + a_2 y$ and $\gcd(F,h)=1$ and $k$ be integer.
Consider ...
- 25.4k
2
votes
0
answers
214
views
Question on digital sum of the square of $n$
If we set $f(n)=$ the digital sum of $n$，for example, $f(2024)= 2+0+2+4= 8$.
Are there any $n>375501$ in solutions to the equation $f(n^2)=9,$ except $n=10k$, $n=10^a+10^b+1$, $n=5 \cdot 10^a+1$ or ...
- 321
2
votes
0
answers
119
views
The connection of Faltings height and Tate module
Suppose $K$ is a number field, $S$ a finite set of places of $K$, and $A$ is an abelian scheme over $\mathcal O_{K,S}$. I want to ask is there some connections between the Faltings' height $h_F(A)$ ...
- 775
2
votes
0
answers
184
views
Will Coppersmith's method work for this bivariate modular polynomial shape?
I have a bivariate modular polynomial of shape
$$f(x,y)=x^2y-g(x)\equiv 0\bmod q$$
where
$q=(2p-1)(2p+1)$ is a product of two primes $2p-1$ and $2p+1$,
$g(x)\in\mathbb Z[x]$ is of degree four and
$f(...
- 13.9k
2
votes
0
answers
78
views
Is the continued fraction of a constructible number special in some way?
Rationals have finite CF and quadratic have periodic CF. CF in turn can be represented in terms of the modular group SL2(Z), e.g. using the standard generators S(z)=-1/z and T(z)=z+1. On the other ...
2
votes
0
answers
101
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 ...
- 121
2
votes
0
answers
87
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 $f$, ...
- 25.4k
2
votes
0
answers
96
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,...
- 25.4k
2
votes
0
answers
93
views
Integers solutions of products of truncated Riemann zeta functions
Let $n \in \mathbb{N}$ be a positive integer.
It is possible to prove that the equation $F_1(n)=m$ where $m \in \mathbb{Z}$ and
$$
F_1(n)=(1+2+\ldots+n)\cdot\left(\frac{1}{1}+\frac{1}{2}+\dots+\frac{1}...
- 1,343
2
votes
0
answers
356
views
Classifying solutions of a certain Diophantine Equation
The following question arose from a problem I am working on.
Let $N, k$ be positive integers. Consider the Diophantine equation in $a, b, c$:
$$
\frac{1}{a} + \frac{N - 1}{b} = \frac{N^k}{c}
$$
with ...
- 791
2
votes
0
answers
201
views
On GCD and lattice reduction
$LLL$ algorithm is vectorized version of Euclidean algorithm for $GCD$.
Even the $m=2$ case known to Lagrange and Gauss does not have an $NC$ algorithm for shortest vector.
If $GCD$ is in $NC$ and in ...
- 13.9k
2
votes
0
answers
40
views
Reference for bound $x_{0} \le n^{2}$ in solutions to the Diophantine equation $\left(\sum_{i=1}^{n} x_{i}\right)^{2} = x_{0}\prod_{i=1}^{n}x_{i}$?
Using Vieta jumping, one can prove that, if $x_{0}, x_{1}, \dotsc,
x_{n} \in \mathbb{Z}_{\ge 1}$ are such that
\begin{equation*}
\left(\sum_{i=1}^{n} x_{i}\right)^{2} =
x_{0}\prod_{i=1}^{n}x_{i},...
2
votes
0
answers
242
views
Solving $x^k+y^k+z^k=w^k$ non-trivially in strictly positive integers
Consider the equation $x^k+y^k+z^k=w^k$ in $x$, $y$, $z$ and $w$ with $k\in\mathbb{N}_{\geq2}$.
If we look for solutions that are strictly positive and non-trivial i.e. $x\neq-y$, $x\neq w$ etc... ...
- 4,862
2
votes
0
answers
115
views
Reference request: "A result of Siegel" related to Ramanujan-Nagell type equations
Wikipedia refers to the Diophantine equation
$ x^2 + D = AB^n $
as an "equation of Ramanujan–Nagell type". It also says that "A result of Siegel implies that the number of solutions in ...
2
votes
0
answers
266
views
On the equations $x^yy^z=z^x$ and $w^x+x^y+y^z=z^w$
Recently, I considered the equation
$$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
2
votes
0
answers
252
views
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 ...
- 2,143
2
votes
0
answers
125
views
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 ...
- 163