All Questions
Tagged with nt.number-theory diophantine-equations
787 questions
8
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0
answers
245
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Hilbert 10th problem for genus 2 equations
Hilbert 10th problem, while undecidable in general, remains open for 2-variable equations: we do not know if there is an algorithm that, for polynomial $P(x,y)$ with integer coefficients, decides ...
- 7,182
11
votes
1
answer
540
views
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 ...
- 1,109
2
votes
1
answer
360
views
Positive divisors of $P(x,n)=1+x+x^2+ \cdots + x^n$ that are congruent to $1$ modulo $x$
This is a follow-up question to Positive integer solutions to the diophantine equation $(xz+1)(yz+1)=z^4+z^3 +z^2 +z+1$
Let \begin{equation}
P(x,n)= 1+x+x^2+ \cdots + x^n, \end{equation}
\begin{...
- 319
5
votes
0
answers
215
views
Integer points of rational function in 2 variables
Is there an algorithm that, given polynomials $P(x)$ and $Q(y)$ with integer coefficients, decides whether there exists integers $x$ and $y$ such that $\frac{P(x)}{Q(y)}$ is an integer?
This is a ...
- 7,182
0
votes
1
answer
87
views
Diophantine equations that involve Lehmer means with all digits equal to $1$ in their $x-$adic expansions
In this post I present my variations of the problem involving Nagell-Ljunggren equation, that is explained in pages 10 and 11 of Highlights in the Research Work of T. N. Shorey by R. Tijdeman, from ...
6
votes
2
answers
743
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Positive integer solutions to the diophantine equation $(xz+1)(yz+1)=z^4+z^3 +z^2 +z+1$
Let $P(z) = z^4 +z^3 +z^2 +z+1$ where $z$ is a positive integer.
While working with the diophantine equation $(xz+1)(yz+1)=P(z)$, I was able to construct a seemingly infinite and complete solution set ...
- 319
0
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0
answers
138
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A diophantine equation involving partial sums of exponentials similar than the equation in Fermat's Last Theorem
I'm curious about the following diophantine equation from my invention: I don't know if this is in the literature, I wrote it using creativity in an attempt to write a variant of the equation in ...
6
votes
1
answer
357
views
Can $2^n\pm n$ with $n>2$ be a triangular number?
Recall that triangular numbers are those
$$T(n)=\frac{n(n+1)}2\ \ (n=0,1,2,\ldots\}.$$
Clearly, $$2^1-1=1=T(1),\ \ 2^1+1=3=T(2),\ \ 2^2+2=6=T(3).$$
Question. Is there an integer $n>2$ with $2^n-n$ ...
- 15.6k
9
votes
1
answer
702
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Software for detecting Brauer-Manin obstructions?
In the context of another MO question, the following question arose: Does there exist any software for detecting Brauer–Manin obstructions to the existence of integer solutions to a single polynomial ...
- 82.7k
6
votes
2
answers
896
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How can I find all integer solutions of $3^n - x^2 = 11$
I know that $n$ can't be even because of the following argument:
Let $n = 2p$. Then we can use the difference of two squares and it becomes like this :
$(3^p + x)(3^p - x) = 11; 3^p + x = 11 , 3^p - x ...
- 169
21
votes
1
answer
1k
views
Is "almost-solvability" of Diophantine equations decidable?
Say that a Diophantine equation is almost-satisfiable iff for each $n\in\mathbb{N}$ it has a solution mod $n$. Trivially genuine satisfiability over $\mathbb{N}$ implies almost-satisfiability, but the ...
- 21.2k
7
votes
1
answer
880
views
A family of Diophantine equations with no integer solutions but solutions modulo every integer
Selmer's curve is the equation $3x^3 +4y^3 +5z^3=0$. This equation is famous for having non-trivial solutions in every completion of $\mathbb{Q}$ but only having the trivial solution in the rationals. ...
- 6,969
72
votes
3
answers
8k
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Can you solve the listed smallest open Diophantine equations?
In 2018, Zidane asked What is the smallest unsolved Diophantine equation? The suggested way to measure size is substitute 2 instead of all variables, absolute values instead of all coefficients, and ...
- 7,182
3
votes
1
answer
262
views
Determine if a 2-variable Diophantine equation has a finite or infinite number of solutions
Do there exist an algorithm, which, given a polynomial $P(x,y)$ with integer coefficients, determines whether Diophantine equation $P(x,y)=0$ has finite or infinite number of integer solutions?
Famous ...
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1
vote
1
answer
203
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On the Diophantine equation $m^2 - p^k = 2^r t$, where $r \geq 2$ and $\gcd(2,t)=1$
This question is an offshoot of this closely related MO question.
Here, we consider the Diophantine equation
$$m^2 - p^k = 2^r t,$$
where $r \geq 2$ and $\gcd(2,t)=1$.
Furthermore, we place the ...
6
votes
3
answers
606
views
Non-trivial solution to $\sum^{n}_{i=1}\sum^{n}_{j=1,j\ne i}(x_{i})^{(x_j)}=(\sum^{n}_{i=1}x_i)^{(\sum^{n}_{i=1}x_i)}$
This problem was first asked at Mathematics Stack Exchange, where it wasn't drawn much attention.
For ease of reading,
$$S=\sum_{i=1}^nx_i, S_p=\sum_{i=1,i\ne p}^nx_i, S^{[q]}=\sum_{i=1}^nx_i^q, S_p^{[...
9
votes
0
answers
563
views
Iterating Diophantine equations over Q to quickly get a large interval with just integer solutions
Hilbert's Tenth Problem was whether there is an algorithm which will answer whether any Diophantine equation has solutions (where we want integer solutions). Hilbert's Tenth has a negative solution by ...
- 6,969
10
votes
4
answers
1k
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The smallest solution to $2^{2k}-1=\text{powerful}$
Integer is powerful if all the exponents in its factorization are at least $2$.
Every powerful integer can be written in the form $a^2 b^3$.
For odd $k$, define $F(k)=2^{2k}-1=(2^k-1)(2^k+1)$.
This ...
- 25.4k
22
votes
2
answers
1k
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Why is 1331 the only cube of the form $x^2 + x − 1$?
The Wikipedia (https://en.wikipedia.org/wiki/1000_(number)#1300_to_1399) mentions that 1331 is the only cube of the form $x^2 + x − 1$, for $x = 36$. What is the proof?
- 459
0
votes
0
answers
112
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The number of solutions of $2^xpx+k=y^2$
Let's consider the family of diophantine equations
$$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$$
that presents, at least, ...
- 1,008
1
vote
2
answers
435
views
On the equation $x^3 + y^3 =cz^3$
What are the characteristics of the values of $c$ for which the equation $x^3 + y^3 = cz^3$ has pairwise coprime non-zero integral solutions where $x \neq \pm y$ ? For instance, it is known that $c$ ...
- 11
7
votes
2
answers
911
views
Triangular numbers of the form $x^4+y^4$
Recall that triangular numbers are those $T(n)=n(n+1)/2$ with $n\in\mathbb N=\{0,1,2,\ldots\}$. Fermat ever proved that the equation $x^4+y^4=z^2$ has no positive integer solution. So I think it's ...
- 15.6k
8
votes
1
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639
views
On Markoff-type diophantine equation
Do there exist integers $x,y,z$ such that
$$
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.$$
Today I tried to find positive integers $x,y,z$ satisfying $(x^4+1)(y^4+1)=z^4+1$ but failed. In view of this ...
- 15.6k
3
votes
1
answer
233
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Pythagorean triples and quadratic residues modulo primes
QUESTION. Are my following conjectures true? How to prove them?
Conjecture 1. For each prime $p>100$, there are $a,b,c\in\{1,\ldots,p-1\}$ such that
$$\left(\frac ap\right)=\left(\frac bp\right)=\...
- 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
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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
4
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1
answer
571
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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 ...
- 1,364
7
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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
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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
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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
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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 $y^2 = x^3 - z^3 ?$
4
votes
1
answer
273
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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
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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^...
- 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