All Questions
Tagged with nt.number-theory diophantine-equations
787 questions
8
votes
1
answer
1k
views
Fundamental units of imaginary quartic fields
Let $F/{\mathbb Q}$ be an imaginary quartic extension (i.e. the degree $[K:{\mathbb Q}]=4$ and no embedding of $K$ in ${\mathbb C}$ has its image inside the real numbers). Then the unit group of the ...
- 3,399
8
votes
1
answer
206
views
Integral complete 4-partite graphs
For given block sizes $a<b<c<d$, consider the complete 4-partite graph $K_{a,b,c,d }$.
Can such a graph be integral, i.e. have only integer eigenvalues?
It is easy to see that the ...
- 13.4k
8
votes
4
answers
870
views
A cubic equation, and integers of the form $a^2+32b^2$
I am trying to determine whether there are any integers $x,y,z$ such that
$$
1+2 x+x^2 y+4 y^2+2 z^2 = 0. \quad\quad\quad (1)
$$
It is clear that $x$ is odd. We can consider this equation as quadratic ...
- 7,182
8
votes
1
answer
627
views
Hilbert 10th problem for cubic equations
Hilbert 10th problem, asking for algorithm for determining whether a polynomial Diopantine equation has an integer solution, is undecidable in general, but decidable or open in some restricted ...
- 7,182
8
votes
1
answer
382
views
Solutions of a Diophantine equation with large common divisors
There is a curious Diphantine equation showing up in my research:
$$ \frac{1}{a^2-1}+\frac{1}{b^2-1}=\frac{1}{c^2-1}+\frac{1}{d^2-1}. $$
I am trying to find its integer solutions where $a$, $b$, $c$ ...
- 5,169
8
votes
1
answer
868
views
Is it true that $\{x^3+2y^3+3z^3:\ x,y,z\in\mathbb Z\}=\mathbb Z$?
It is easy to see that no integer congruent to $4$ or $-4$ modulo $9$ can be written as the sum of three integer cubes. In view of this and Question 331163, I proposed the following conjecture in ...
- 15.6k
8
votes
0
answers
245
views
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
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
8
votes
0
answers
325
views
Integer solutions of $x^2=4+8y^2+13z^2$
I have been looking for integer solutions of certain Diophantine equations, one of the simplest examples being
$x^2=4+8y^2+13z^2$.
The ideal answer would be a way to parametrize all the integer ...
- 81
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
7
votes
1
answer
4k
views
Integer solutions of x^n + y^n = z^{n-1}
This is related to another question
I am interested in the non-trivial integer solutions of
$$ x^n + y^n = z^{n-1} $$
for $n \ge 4$. A solution is trivial if $xyz=0$ or $x = \pm y$.
There are ...
- 25.4k
7
votes
2
answers
3k
views
4900, a particularly square number
I read in "Letters to a young mathematician" that 4900 is the only square integer that is the sum of consecutive squares (I believe he meant by that "starting from 1", but maybe that's not even ...
- 1,801
7
votes
2
answers
1k
views
Are there consecutive integers of the form $a^2b^3$ where $a$, $b$ > 1?
Let $S$ = { $a^2b^3$ : $a, b \in \mathbb{Z}_{>1}$ }.
Does there exist $n$ such that $n$, $n+1 \in S$?
Motivation: I was thinking about Question on consecutive integers with similar prime ...
- 870
7
votes
5
answers
941
views
Perfect powers in the solutions of a certain Pell equation
The fundamental solution of the Pell equation $$x^{2}-3y^{2}=1$$ is $2+\sqrt{3}$.
It seems that if $x_{n}+y_{n}\sqrt{3}$, $x_{n}, y_{n} \in \mathbb{N}$, is a solution of the said Pell equation and $x_{...
- 87
7
votes
3
answers
511
views
On the equation $a^n + b^n = c^2$
I am interested in the possible natural solutions of the equation $a^n + b^n = c^2$ where $n \geq 4$ is fixed. I am not sure if it is well-known or not, so any suggestion would be helpful.
- 81
7
votes
3
answers
611
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 ...
- 8,842
7
votes
2
answers
604
views
Density version of the Erdős-Graham conjecture
In 2003 E. S. Croot [Ann. of Math. 157(2)(2003), 545-556] proved the Erdős-Graham Conjecture which states that if $\{2,3,\ldots\}$ is partitioned into finitely many subsets then one of the subsets ...
- 15.6k
7
votes
4
answers
884
views
Extending rational Diophantine triples to sextuples
(This is a follow-up to a previous post.) A rational Diophantine $m$-tuple is a set of rationals {$a_1,a_2,\dots a_m$} such that (with $i\neq j$), all $a_i a_j+1$ is a square. Problem: Find a class of ...
- 12.6k
7
votes
2
answers
641
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 ...
- 7,182
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
7
votes
2
answers
1k
views
Is it true that $\{x^3-2x+y^3-2y+z^3-2z: x,y,z\in\mathbb Z\}=\mathbb Z$?
A well known conjecture states that
$$\{x^3+y^3+z^3:\ x,y,z\in\mathbb Z\}=\{m\in\mathbb Z:\ m\not\equiv\pm4\pmod 9\}.$$
For $m=33,\, 42$ an integer solution to the equation $x^3+y^3+z^3=m$ was only ...
- 15.6k
7
votes
1
answer
315
views
Rational perfect power values of $y(y+1)$
This is hard, so I am looking for partial results and how hard it is.
Let $n>4$. Is it true that the hyperelliptic curve $x^n=y(y+1)$
doesn't have rational point with $x \ne 0$?
If necessarily ...
- 25.4k
7
votes
1
answer
691
views
What are the solutions of this Diophantine equation?
Besides $(x, y, z)=(0, 0, 0)$ and $(1, 1, -2)$ (and their permutations) are there any other integer solutions to the equation
$$3(x^{3}+y^{3}+z^{3})+3(x^{2}+y^{2}+z^{2})+(x+y+z)=0 $$ ?
- 71
7
votes
1
answer
6k
views
The Jones-Sato-Wada-Wiens polynomial for prime numbers and differential calculus?
After works of Davis, Matijasevic, Putman and Robinson between 1960 and 1970, we know that every recursively enumerable set of numbers can be represented by a polynomial.
In particular, it's the case ...
7
votes
2
answers
929
views
English translation of Voronoi's dissertation
I am looking for an English translation of Voronoi's doctoral dissertation, "On a generalization of the Algorithm of Continued Fractions." I can only find it in the original Russian.
- 173
7
votes
1
answer
386
views
Does this equation have more than one integer solution?
Consider the following diophantine equation
$$n = (3^x - 2^x)/(2^y - 3^x),$$ where $x$ and $y$ are positive integers and $2^y > 3^x$.
Does $n$ have any other integer solutions besides the case ...
- 79
7
votes
2
answers
957
views
Primitive integral solutions to $x^2+y^3=z^2$
The Diophantine equation
$$x^2+y^3=z^2$$
has solutions $(\pm 1,2,\pm 3)$ and $(\pm 13,3,\pm 14)$.
Brown [Int. Math. Res. Not. IMRN 2012, no. 2, 423–436; MR2876388] states that "there are ...
- 1,719
7
votes
3
answers
581
views
Uniform bounds on the number of integer points on a family of elliptic curves
Let $P(x,y)$ be a binary cubic polynomial with integer coefficient. Let $n$ be an integer. Suppose the (complex) curve $P(x,y)=n$ is nonsingular, so is an elliptic curve. Is there any bound on the ...
- 5,169
7
votes
3
answers
2k
views
Diophantine equation - $a^4+b^4=c^4+d^4$ ($a,b,c,d > 0$)
How can I find the general solution of $a^4+b^4=c^4+d^4$ ($a,b,c,d > 0$)?
And how did Euler find the solution $158^4+59^4=133^4+134^4$?
- 181
7
votes
1
answer
618
views
$(2x^2+1)(2y^2+1)=4z^2+1$ has no positive integer solutions?
Equation
$$(2x^2+1)(2y^2+1)=4z^2+1$$
has no solutions in the positive integers. Its true?
- 73
7
votes
1
answer
480
views
Imprimitive solutions to $x^2+y^3=z^7$
Poonen, Schaefer, & Stoll give the primitive solutions to $x^2+y^3=z^7$:
$$
(±1, −1, 0), (±1, 0, 1), ±(0, 1, 1), (±3, −2, 1), (±71, −17, 2),\\
(±2213459, 1414, 65), (±15312283, 9262, 113), (±...
- 9,114
7
votes
1
answer
376
views
$xyz = \frac{7}{16}\left(\frac{2x - y - z}{3}\right)^3$ in nonvanishing integers
From research completely unrelated to Number Theory I stumbled onto the following equation:
$$
xyz = \frac{7}{16}\left(\frac{2x - y - z}{3}\right)^3
$$
for $x, y, z$ integers, $x,y,z \neq 0$. Are ...
user22139
7
votes
2
answers
765
views
Integer solutions of an algebraic equation
I'm trying to find integer solutions $(a,b,c)$ of the following algebraic equation with additional conditions $b>a>0$, $c>0$.
$(-a^2+b^2+c^2)(a^2-b^2+c^2)(a^2+b^2-c^2) + 2 a b (-a^2+b^2+c^2)(...
- 691
7
votes
2
answers
397
views
An equation involving perfect numbers
Let $s,x_1,x_2,\cdots, x_s$ be natural numbers not neccesarily distinct.
I am interested in solving the equation $$(x_1+x_2+\cdots +x_s)^s=2^s(x_1\cdot x_2\cdots x_s)^2$$
Some Notes:
I have found ...
- 3,866
7
votes
2
answers
2k
views
How to prove that this equation has only one solution?
I can't find a way to prove that the following equation has only one solution :
$$
X = \frac{2^Q - 1}{2^{P+Q} - 3^P}
$$
with $X,P,Q$ integers $> 0$.
One trivial solution is $X = 1, P = 1, Q = 1$....
- 73
7
votes
2
answers
398
views
Given that $n > 3$ and $z$ is a Gaussian integer, when can $z^n \pm z$ be a rational integer?
I came across the following conjecture. If you have any thoughts on how to approach it, let me know.
Conjecture. For any integer $n > 3$ and any Gaussian integer $z$ that is not a unit, if $z^n - z$...
- 1,625
7
votes
1
answer
364
views
Rational Diophantine set for the non-squares
Related to Hilbert's Tenth problem.
Is there polynomial with integer coefficients $P(a,x_1,...,x_n)$
such that $P(A,X_i)=0$ has rational solutions $X_i$ iff $A$ is
not the square of integer (or as ...
- 25.4k
7
votes
1
answer
389
views
Why are some solutions of these diophantine equations off the usual patterns?
This is inspired by a recent question about complete multipartite integral graphs. I am wondering if more can be said about tripartite integral graphs with block sizes $a<b<c$. It is easy to see ...
- 13.4k
7
votes
2
answers
617
views
Genus 0 curves on surfaces and the abc conjecture
One of the most obvious methods to prove that a given Diophantine equation $P(x_1, \dots, x_n)=0$ has infinitely many integer solutions is to find polynomials $P_1, \dots, P_n$ in one variable $u$, ...
- 7,182
7
votes
3
answers
2k
views
Solution to a Diophantine equation
Find all the non-trivial integer solutions to the equation
$$\frac{a}{b+c}+\frac{b}{a+c}+\frac{c}{a+b}=4.$$
- 403
7
votes
3
answers
771
views
Something interesting about the quintic $x^5 + x^4 - 4 x^3 - 3 x^2 + 3 x + 1=0$ and its cousins
(Update):
Courtesy of Myerson's and Elkies' answers, we find a second cyclic quintic for $\cos\frac{2\pi}{p}$ with $p=\text{1 mod 10}$ as,
$$\frac{z^5}{\beta} = 10 z^3 - 20 n^2 z^2 + 5 (3 n^4 - 25 n^...
- 12.6k
7
votes
1
answer
454
views
Checking local solubility of varieties at "bad" primes
Let $X$ be a smooth variety defined over $\mathbb{Q}$. If we want to check that $X$ is locally soluble at a prime $p$, then it suffices to find a non-singular $\mathbb{F}_p$-point, which can be lifted ...
- 21.3k
7
votes
1
answer
315
views
Rational points on regular curves over global fields
Let $k$ be a global field and $C$ a smooth projective curve over $k$ which is not isotrivial. Then there is a well-known trichotomy:
If $g(C) = 0$ and $C(k) \neq \emptyset$, then $C \cong \mathbb{P}^...
- 21.3k
7
votes
2
answers
602
views
Bounds on Bézout coefficients
Let $0<a_1 \le a_2 \le \cdots \le a_n$ be positive integers such that $a_1 + \cdots + a_n = m$ and $\gcd(a_1,\ldots,a_n)=1$. Let $\mathbf a :=(a_1,\ldots,a_n)\in\mathbb Z^n$ and $\mathbf x:=(x_1,\...
- 277
7
votes
1
answer
463
views
A parametric elliptic curve for $x^4+y^4+z^4 = 1$?
Noam Elkies found that $x^4+y^4+z^4 = 1$ has infinitely many rational points $xyz \neq 0$ using an elliptic curve. We use a different approach that will produce pairs of solutions and a parametric ...
- 12.6k
7
votes
1
answer
751
views
On the shortest open cubic equation
The question is: are there any integers $x,y,z$ such that
$$
1+4 x^3+x y^2+2 y z^2 = 0 \quad\quad\quad\quad (1)
$$
The motivation is: Define the length of a polynomial $P$ consisting of $k$ monomials ...
- 7,182
7
votes
1
answer
327
views
Question about parametric representations of solutions to $x^3+y^3+z^3=n \in \mathbb N$
There are such representations for $n=1,2$. However, by the Wikipedia article, it seems that there are no known parametric (polynomial) representations $P,Q,R$ such that $(P(m))^3+(Q(m))^3+(R(m))^3=3$....
- 297
7
votes
1
answer
494
views
Gap principle for a diophantine inequality
Let $d$ be a positive, non-square integer, and let $B > 1$ be a real number. Consider the inequality
\begin{equation} |x^2 - dy^2| \leq B. \end{equation}
This inequality has infinitely many ...
- 26.9k
7
votes
2
answers
672
views
The equation $x^m-1=y^n+y^{n-1}+...+1$ in prime powers $x,y$
Does the equation $x^m-1=y^n+y^{n-1}+...+1$ have only finitely many solutions $(x,y,m,n)$ where $x,y$ are prime powers with $y>2$ and $m,n$ are integers with $m,n>1$?
This question arose in the ...
- 6,397
7
votes
2
answers
780
views
Some equalities involving prime powers
Let $p,a,b,x,y$ be positive integers where $p$ is an odd prime; $x$ and $y$ are odd; $p,x$ and $y$ are all coprime. I'm interested in finding examples of such numbers that satisfy this equation:
\...
- 11.2k