All Questions
Tagged with diophantine-equations reference-request
62 questions
38
votes
5
answers
10k
views
Are nontrivial integer solutions known for $x^3+y^3+z^3=3$?
The Diophantine equation
$$x^3+y^3+z^3=3$$
has four easy integer solutions: $(1,1,1)$ and the three permutations of $(4,4,-5)$. Elsenhans and Jahnel wrote in 2007 that these were all the solutions ...
- 2,350
31
votes
5
answers
8k
views
Fermat's proof for $x^3-y^2=2$
Fermat proved that $x^3-y^2=2$ has only one solution $(x,y)=(3,5)$.
After some search, I only found proofs using factorization over the ring $Z[\sqrt{-2}]$.
My question is:
Is this Fermat's original ...
- 3,866
30
votes
9
answers
10k
views
Diophantine equation with no integer solutions, but with solutions modulo every integer
It's probably common knowledge that there are Diophantine equations which do not admit any solutions in the integers, but which admit solutions modulo $n$ for every $n$. This fact is stated, for ...
- 10.3k
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
20
votes
2
answers
2k
views
On a result attributed to W. Ljunggren and T. Nagell
I've read in a number of places that, building on previous work of T. Nagell, W. Ljunggren proved in 1 that the Diophantine equation
$$\frac{x^{n}-1}{x-1} = y^{2}$$
doesn't admit solutions in ...
- 8,842
15
votes
4
answers
575
views
Are all partial consecutive harmonic subsums distinct?
Let $b \gt a \geq 0$ be integers, and as elsewhere let $H_n$ be $\sum^n_{i=1} 1/i$. A partial consecutive harmonic subsum is a number $H(a,b)$ of the form $H_b - H_a$ (with $ H_0=0$). If $c=a$ and $...
- 13k
13
votes
1
answer
760
views
Infinitely many integer solutions to $X^4+Y^4-18Z^4= -16$
We found infinitely many integer solutions to
$$X^4+Y^4-18Z^4= -16 \qquad (1)$$.
The interesting part in this diophantine equation is the sum of
the reciprocals of the degrees is $3/4 < 1$, which ...
- 25.4k
12
votes
3
answers
411
views
(Non-)Existence of curves of low degree on affine and projective varieties
I am interested in papers that investigates the existence or non-existence of curves of low degree (relative to the degree of the ambient variety). The starting example is that of surfaces and ...
- 26.9k
12
votes
1
answer
598
views
Fermat last theorem : proof of a criterion by Cauchy
In 13 Lectures on Fermat's Last Theorem, Ribenboim states the following theorem (on page 7) attributed to Cauchy:
If the first case of Fermat's theorem fails for the exponent $p$, then the sum:
$$ 1^{...
- 121
10
votes
5
answers
771
views
Reference request: Diophantine equations
I am looking for a textbook, or preferably lectures, on the subject of Diophantine equations. I am familiar with the basic principles of modular arithmetic, conics and the Hasse Principle, and the ...
- 2,811
10
votes
4
answers
1k
views
Number of solutions of linear homogenous Diophantine equation inside a box
Let $a_1, ..., a_d$ be positive reals and consider the linear Diophantine equation
$$
\sum_i a_in_i = 0.
$$
I am interested in estimating the number of integer solutions of this equation inside a ...
- 960
9
votes
1
answer
682
views
On the exact reference of a cute Diophantine problem
The problem asks to prove that the Diophantine equation $x^{3}+y^{3} = (x+y)^{2}+(xy)^{2}$ does not have any solutions in natural numbers $x, y$.
I believe that this problem appeared in the section ...
- 8,842
9
votes
2
answers
683
views
The Theory of Transfinite Diophantine Equations [closed]
The theory of Diophantine equations is one of the main stream research areas in number theory. There are many known results and unknown conjectures about the existence of non-trivial solutions for ...
user42090
9
votes
1
answer
419
views
Infinitely many solutions to $a^4+b^4+c^4=18$ over $\mathbb{Z}[i]$
We got infinitely many solutions to $a^4+b^4+c^4=18$ over
$\mathbb{Z}[i],i^2=-1$. Probably we can get infinitely many solutions
to $a^5+b^5+c^5=N$ over $\mathbb{Z}[\alpha]$ for algebraic $\alpha$.
...
- 25.4k
8
votes
2
answers
643
views
Around the diophantine equation $\frac{a}{2b+3c}+\frac{b}{2c+3a}+\frac{c}{2a+3b}=\text{odd integer}$, over positive integers
I am interested to know if a similar theorem that shows this answer of the post
Estimating the size of solutions of a diophantine equation (this MathOverflow, January 5th 2016) is feasible for a ...
8
votes
2
answers
1k
views
How are such sets of natural numbers called?
I heard about this problem an year ago, but I just can't remember the name.
The problem goes like this: study the sets
$\{a_1,a_2,\dotsc,a_m\}\subseteq\mathbb{N}$ such that if $1\leq i<j\leq m$,...
- 103
7
votes
5
answers
941
views
Perfect powers in the solutions of a certain Pell equation
The fundamental solution of the Pell equation $$x^{2}-3y^{2}=1$$ is $2+\sqrt{3}$.
It seems that if $x_{n}+y_{n}\sqrt{3}$, $x_{n}, y_{n} \in \mathbb{N}$, is a solution of the said Pell equation and $x_{...
- 87
7
votes
3
answers
611
views
Question on a crucial lemma in Euler's approach to Fermat's Last Theorem for $n=3$
As many of you may know, the illustrious L. Euler put forward a proof of the case $n=3$ of Fermat's Last Theorem via infinite descent. The thing is that, at a certain point, he resorted to the ...
- 8,842
7
votes
2
answers
929
views
English translation of Voronoi's dissertation
I am looking for an English translation of Voronoi's doctoral dissertation, "On a generalization of the Algorithm of Continued Fractions." I can only find it in the original Russian.
- 173
6
votes
2
answers
714
views
Origin and variations of problem on $4xy-x-y$ being square
One of the forms in which the Diophantine equation in question can be found in the literature is this:
Solve the equation \begin{eqnarray}z^{2} = 4xy-x-y \qquad \qquad (\ast)\end{eqnarray} in ...
- 8,842
6
votes
0
answers
410
views
Efficient solutions to general Bézout’s identity $a_1 b_1 + \dots + a_n b_n = 1$
Suppose I have integers $a_1, \dots, a_n$ which are coprime, meaning that
$$a_1 b_1 + \dots + a_n b_n = 1$$
has a solution in integers $b_1, \dots, b_n$.
I would like to get a bound saying ...
- 4,164
5
votes
5
answers
751
views
The diophantine equation $ \sum_{n=1}^{N} \frac{1}{x_{n}} = \prod_{k=1}^{N} \left(1-\frac{1}{x_{k}} \right) $
Background
I wonder if there are any rational numbers such that their Egyptian fraction (sum) representations are equal to their Egyptian product analogue. In other words, I am curious1 about ...
- 4,829
5
votes
1
answer
355
views
Research work on $ax^n-by^m=1$
I am looking for results on the equation $$ax^n-by^m=1 \tag 1 $$ where $\gcd(m,n)=1$ and $a,b,n,m$ are constants.
I found literature for $ax^n-by^n=1$ (R. A. Mollin, D. T. Walker) but couldn't ...
5
votes
1
answer
196
views
Relative size of Egyptian fraction denominators
Suppose we have a finite Egyptian fraction decomposition of a rational:
$$\frac{n}{m} = \sum_{i=1}^k \frac{1}{x_i}$$
such that
(i) $x_i>0$,
(ii) $x_i \neq x_j$ for $i \neq j$, and
(iii) $\gcd(m, ...
- 2,235
4
votes
2
answers
343
views
Algorithm for computing rational points if the rank of Jacobian is 0
Is there a general algorithm that can compute in finite time all rational points on any curve of genus $g\geq 2$ whose Jacobian has rank $0$?
If not, for what special cases such algorithm is known? ...
- 7,182
4
votes
1
answer
319
views
What about $n^{\frac{1}{x}+\frac{1}{y}}+n^{\frac{1}{y}+\frac{1}{z}}=n^{\frac{1}{z}+\frac{1}{x}}$ over positive integers?
I've considered the following equation for positive integers $x,y,z\geq 1$, and for positive integers $n\geq 2$
$$n^{\frac{1}{x}+\frac{1}{y}}+n^{\frac{1}{y}+\frac{1}{z}}=n^{\frac{1}{z}+\frac{1}{x}},\...
4
votes
1
answer
217
views
Diophantine equation for generating computably enumerable set
By Matiyasevich's theorem, each member of computably enumerable set can be obtain from a diophantine equation system. For prime numbers, this system of diophantine equation is found. My question is:
...
- 4,784
4
votes
1
answer
386
views
Is this problem of Schinzel and Tijdeman misquoted? It appears easy with Pell equations
In Diophantine equations over the twentieth century: a (very) brief overview
, p. 5
Problem Let $f(x) \in \mathbf{Z}[x]$ be an irreducible polynomial of degree at least 2. Do the Diophantine ...
- 25.4k
4
votes
1
answer
729
views
Is there a error/typo in the proof related to Goormaghtigh equation in Yann Bugeaud's paper?
I found the following theorem in a paper by Yann Bugeaud (page 12) ,
the theorem was not written in detail, to be specific,following two lines on page 13 were not understandable-
I think this ...
- 267
4
votes
0
answers
506
views
Collatz conjecture and a diophantine equation
Let $M \ge 2$. Inspired by the Collatz iteration / algorithm ($M=2$), I tried the following function:
$$C_M(n) = n/M, \text{ if } n \equiv 0 \mod M, \text{ otherwise } (M+1)n+\{(M-n) \mod M \}$$
We ...
- 3,074
4
votes
0
answers
332
views
Diophantine equations over cyclotomic fields
Let $\mathbb{Q}^{\text{ab}}$ be the compositum of all finite abelian extensions of $\mathbb{Q}$. Explicitly, $\mathbb{Q}^{\text{ab}}$ is the field obtained from $\mathbb{Q}$ by adjoining all roots of ...
- 11.3k
3
votes
1
answer
247
views
4-th order diophantine equation
I met in many places the equation
$(a^4-b^4)(c^4-d^4)=\square$
It is well known that this was investigated by Euler.
But I was unable to find the general solution of this equation. Could you please ...
- 185
3
votes
0
answers
148
views
Solutions of a quadratic Diophantine equation over algebraic integers
The problem is not very exactly Diophantine in the classical way. I am trying to find some algebraic integer solutions in a number ring to a quadratic equation over the same ring.
Precisely, let $\...
- 287
2
votes
1
answer
627
views
Particular case of Beal's Conjecture
Is it known that there exist no coprime positive integers $A$, $B$ and $C$ such that $A^3+B^4=C^3$? This is a particular case of Beal's Conjecture.
- 1,547
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
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
2
votes
0
answers
188
views
Solution of the Diophantine equation $x^4+y^4+z^4=2t^4$ are well-known? [closed]
Are solutions of the Diophantine equation $x^4+y^4+z^4=2t^4$ well-known?
I give a solution:
$x=m^2-n^2, y=m^2-2mn, z=n^2-2mn, t=m^2+n^2-mn$
- 477
2
votes
0
answers
207
views
n-ary quadratic forms with $S$-integer values
Let $Q(x_1,\ldots,x_n):=x_1^2+\cdots+x_n^2$ be an $n$-ary quadratic form.
Given a finite set of (rational) primes $S$ is there an algorithm or theorem that describes all solutions to $Q(x_1,\ldots,...
- 1,639
2
votes
0
answers
243
views
Hurwitz integers and $F_4$
The Hurwitz integers are
$$
\mathcal H=
\{a+bi+cj+dk:a,b,c,d\in\mathbb Z\;\text{ or } \;a,b,c,d\in \tfrac12+\mathbb Z\}.
$$
I want to know if there is a formula, for $m\in\mathbb Z$, for the number ...
- 2,446
1
vote
2
answers
751
views
Computational complexity of solution of Pell equation and more
What is computational complexity for computing integral solution of Pell equation .It seems to be in P ,and could any one give an algorithm and reference for proof of it's complexity?
And more,could ...
- 3,094
1
vote
2
answers
127
views
Number of integers $x \leq B$ such that $f(x)\mid g(x)$ for coprime polynomials $f,g$
Let $f, g \in \mathbb{Z}[x]$ be coprime polynomials.
I am interested in an upper bound for
$$
N(B) = \# \{ x \in [-B, B] \cap \mathbb{Z}: f(x)\mid g(x) \}.
$$
I assume there must be something known ...
- 3,625
1
vote
2
answers
349
views
Rational solutions to $P(x,y)=0$ for $P$ reducible over ${\mathbb C}$
There are facts in Mathematics that are so "obvious" and "well-known" that no-one includes a proper proof. An example is:
Theorem: If polynomial $P(x,y)$ with rational coefficients ...
- 7,182
1
vote
1
answer
250
views
Link between integral points on varieties and solutions to Diophantine equations
Let $k$ be a number field, $S$ a finite set of places of $k$ including the infinite ones and $F(X_1,\dots,X_n)$ a polynomial in $k[X_1,\dots,X_n]$.
I am looking for notes, books or surveys detailing ...
1
vote
1
answer
560
views
A good introduction to S unit equations
I was looking up some stuff when I stumbled across S unit equations. It seems to me that they are quite helpful in number theory, as given in this paper.
http://faculty.nps.edu/pstanica/research/...
- 230
1
vote
2
answers
752
views
basis of the lattice generated by the integer points inside a subspace of R^L
Consider $K$ linearly independent vectors $\mathbf{a}_1, \mathbf{a}_2, ..., \mathbf{a}_K \in \mathbb{Z}^L$, where $1 \leq K<L $. Hence, the span of $\lbrace\mathbf{a}_1, \mathbf{a}_2, ..., \mathbf{...
- 51
1
vote
1
answer
364
views
Good references to study Baker's theory
I am studying diophantine equations and I need the theory of Bakers, Can you advise me about good books, or lectures on Baker's theory?
- 17
1
vote
1
answer
330
views
A paper by W. Ljunggren
I am looking for the following paper by Ljunggren, Wilhelm: "Zur Theorie der Gleichung $x^2 + 1 = Dy^4$", Avh. Norske Vid. Akad. Oslo. I., 1942 (5): 27
The main result of this paper which I am ...
- 2,404
1
vote
0
answers
106
views
Mahler's proof of $S$-unit equation
Many modern proofs of the (ineffective) finiteness of solutions of the $S$-unit equation $x+y=1$ use Roth's theorem. In particular it is used Lang's version of Roth's theorem which takes in account ...
- 321
1
vote
0
answers
84
views
Beyond pure rational and integral solutions to cubic equations
I started reading Silverman and Tate’s introductory book on elliptic curves. In the introductory chapter they mention that for the Bachet equation $x^2 - y^3 = c$, there are infinitely many rational ...
- 4,466
1
vote
0
answers
87
views
Diophantine equation $z=(ax+by+c)/(dxy)$, references? [closed]
I am looking for some sources (books or papers) which discuss the Diophantine equation
$$
z=\frac{ax+by+c}{dxy}
$$
where $a,b,c,d$ are given positive integers. Could anyone give some references?
...
- 841