All Questions
Tagged with nt.number-theory diophantine-equations
787 questions
20
votes
2
answers
2k
views
Rational points on the "quintic circle" $x^5 + y^5 = 7$
I suspect that the curve $x^5 + y^5=7$ has no $\mathbb Q$ points, and a brief computer search verifies this hypothesis for denominators up to $10^4$. What techniques can be used to show that there are ...
- 1,329
20
votes
4
answers
3k
views
Striking applications of Baker's theorem
I saw that there are many "applications" questions in Mathoverflow; so hopefully this is an appropriate question. I was rather surprised that there were only five questions at Mathoverflow so far with ...
- 7,442
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
20
votes
3
answers
962
views
Does the equation $(xy+1)(xy+x+2)=n^2$ have a positive integer solution?
Does there exist a positive integral solution $(x, y, n)$ to $(xy+1)(xy+x+2)=n^2$? If there doesn't, how does one prove that?
- 317
19
votes
5
answers
5k
views
Which Diophantine equations can be solved using continued fractions?
Pell equations can be solved using continued fractions. I have heard that some elliptic curves can be "solved" using continued fractions. Is this true?
Which Diophantine equations other than Pell ...
- 655
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
19
votes
0
answers
1k
views
Can a number be palindromic in more than 3 consecutive number bases?
$2017:$ Was initially asked on MSE - but wasn't solved or updated there since.
Update $2019$: I've returned to this problem, made some progress and updated the post here. (I've basically rewritten ...
- 611
18
votes
1
answer
1k
views
Is $x^2+x+1$ ever a perfect power?
Using completing the square and factoring method I could show that the Diophantine equation $x^2+x+1=y^n$, where $x,y$ are odd positive and $n$ is even positive integers, does not have solution, but ...
- 841
18
votes
1
answer
1k
views
When is $(q^k-1)/(q-1)$ a perfect square?
Let $q$ be a prime power and $k>1$ a positive integer. For what values of $k$ and $q$ is the number $(q^k-1)/(q-1)$ a perfect square, that is the square of another integer? Is the number of such ...
- 533
18
votes
2
answers
1k
views
Lower bounds on the easier Waring problem
The easier Waring problem asks for the least number $v=v(k)$ such that every every integer is a sum of $v$ $k$'th powers with signs, i.e. every $n\in \mathbb{N}$ is of the form $$n=x_1^k\pm x_2^k\pm\...
- 7,836
18
votes
3
answers
2k
views
More elliptic curves for $a^4+b^4+c^4+d^4 = (a+b+c+d)^4$?
(Note: See also the $a^4+b^4+c^4 = 1$ version in this old MSE post.)
The equation discussed in a paper by Jacobi and Madden,
$$a^4+b^4+c^4+d^4 = (a+b+c+d)^4 = z^4\tag1$$
or equivalently,
$$(p-2q + ...
- 12.6k
18
votes
2
answers
2k
views
What is the taxicab number for rational fourth powers?
The taxicab number is the smallest integer that can be expressed as a sum of two positive integer cubes in two different ways, and it is equal to $1729=12^3+1^3=10^3+9^3$. There are generalizations to ...
- 7,182
18
votes
3
answers
1k
views
Not-lonely runners
The lonely runner conjecture
has several formulations.
They all involve a number $n$ runners running on a circular track,
each with a different speeds, and the conjecture is that each runner is ...
- 151k
18
votes
1
answer
1k
views
Torsion points of abelian varieties in the perfect closure of a function field
The following is a problem, which was recently brought to my attention by H. Esnault and A. Langer.
Let $K$ be the function field of a smooth curve over the algebraic closure $k$ of the finite field ...
- 3,076
18
votes
0
answers
1k
views
Is the set of integers of the form $a/(b+c)+b/(a+c)+c/(a+b)$ computable?
The starting point of this question is the observation that the smallest positive integers $a,b,c$ satisfying
$$\frac{a}{b+c} + \frac{b}{a+c} + \frac{c}{a+b} = 4$$
are absurdly high, namely $$(...
- 48.9k
18
votes
0
answers
667
views
The lonely molecule
Suppose $n$ air molecules (infinitesimal points) are bouncing around in
a unit $d$-dimensional cube, with perfectly elastic wall collisions.
Let $k=n^{\frac{1}{d}}$.
For example, in 3D, $d=3$, with $n=...
- 151k
17
votes
2
answers
3k
views
Does the equation $241+2^{2s+1}=m^2$ have a solution?
Let $p$ be a prime congruent to $1$ mod. 8.
If $p= 17$ one has : $p+ 8 = 5 ^2$.
If $p= 41$ one has : $p+ 8 = 7 ^2$.
If $p= 73$ one has : $p+ 8 = 9 ^2$.
If $p= 89$ one has : $p+ 32 = 11 ^2$.
If $...
- 1,980
17
votes
3
answers
2k
views
Finding integer points on elliptic curves via divisibility conditions like $(a+b)^2 \mid (2b^3+6ab^2-1)$
Is the following conjecture correct?
Conjecture. The divisibility condition $(\alpha+\beta)^2 \mid (2\beta^3+6\alpha\beta^2-1)$ has no solutions in positive integers $1 \le \beta < \alpha < 2\...
- 1,055
17
votes
2
answers
2k
views
What is the smallest positive integer for which the congruent number problem is unsolved?
The congruent number problem is the problem of figuring out whether a given positive integer $N$ is the area of a right-angled triangle with all side lengths rational. According to Dickson's "History ...
- 41.4k
16
votes
6
answers
3k
views
Advances and difficulties in effective version of Thue-Roth-Siegel Theorem
A fundamental result in Diophantine approximation, which was largely responsible for Klaus Roth being awarded the Fields Medal in 1958, is the following simple-to-state result:
If $\alpha$ is a real ...
- 26.9k
16
votes
6
answers
9k
views
Methods for solving Pell's equation?
It is known that the minimum solution of Pell's equation $x^2-dy^2=\pm1$ can be found from the continued fraction expansion of $\sqrt d$. Are there other methods for finding the minimum (or any other) ...
- 161
16
votes
2
answers
857
views
Are there infinitely many positive integer solutions to $(3+3k+l)^2=m\,(k\,l-k^3-1)$?
I usually work in the field of differential geometry, but I have encountered the following problem in my research: Are there infinitely many positive integers $k,l,m\in\mathbb N^{>0}$ such that $$(...
- 6,825
16
votes
4
answers
1k
views
Random Diophantine polynomials: Percent solvable?
Suppose one generates a random polynomial
of degree $d$ with integer coefficients
uniformly distributed within $[-c_\max,c_\max]$.
For example, for
$d=8$, $|c_\max|=100$, here is one random polynomial:...
- 151k
16
votes
2
answers
1k
views
Representing $x^3-2$ as a sum of two squares
Prove that there exist infinitely many integers $x$ such that integer $P(x)=x^3-2$ is a sum of two squares of integers.
Ideally, I am looking for a proof method that also applies for other $P(x)$, ...
- 7,182
16
votes
2
answers
982
views
Prove $\frac{\text{Area}_1}{c_1^2}+\frac{\text{Area}_2}{c_2^2}\neq \frac{\text{Area}_3}{c_3^2}$ for all primitive Pythagorean triples
A while ago I asked this question on MSE here. After placing a bounty it got quite a bit of attention but unfortunately it has yet to be resolved. After getting some advice from MO Meta I have decided ...
- 211
16
votes
3
answers
1k
views
Number of solutions to polynomial congruences
Suppose I have $R$ homogeneous polynomials $F_1, ..., F_R$ with integer coefficients. Let $V$ be the affine variety defined by these polynomials over $\mathbb{C}$. I was wondering if some bound that ...
- 3,625
16
votes
4
answers
930
views
Integer matrices whose determinant equals their norm
Let $M$ be an $2 \times 2$ matrix, with all entries in $\mathbb{N}$:
$$
M=
\begin{bmatrix}
a & b \\
c & d
\end{bmatrix} \;.
$$
So
$$
\mathrm{det}(M) = a d - b c \; .
$$
The
Euclidean norm
(...
- 151k
16
votes
3
answers
1k
views
Is Multilinear Hilbert's tenth problem version undecidable?
A multilinear polynomial $f\in\mathbb Z[x_1,\dots,x_t]$ has terms only of form $$b\prod_{i=1}^tx_i^{a_i}$$ where $a_i\in\{0,1\}$ and $b\in\mathbb Z$.
Is there no general purpose algorithm for ...
- 13.9k
16
votes
1
answer
1k
views
Solve this Diophantine equation $(2^x-1)(3^y-1)=2z^2$
Find the positive integers $(2^x-1)(3^y-1)=2z^2$ have three solutions
$$(1,1,1),(1,2,2),(1,5,11)$$I already know $(2^x-1)(3^y-1)=z^2$ has no solution. See: P.G.Walsh December 2006 [On Diophantine ...
- 4,280
16
votes
2
answers
410
views
$3$-ranks of elliptic curves and representations $p=ax^3+by^3$
Let $p$ be a prime with $p\equiv2\pmod3$ and $E_p$ the elliptic curve $y^2=x^3+9p^2$
which has a rational $3$-torsion point. Let $\alpha$ from $E_p(\mathbb Q)$ to $\mathbb Q^*/{\mathbb Q^*}^3$ be the $...
- 13.1k
15
votes
2
answers
2k
views
sum of three cubes and parametric solutions
The first paragraph in the following link asserts that the equation $x^3+y^3+z^3=2$ has finite many parametric solutions over $\mathbb{Q}$, i.e., there are finite many polynomial triples $(x(t),y(t),z(...
- 3,337
15
votes
1
answer
612
views
Dividing squares by sums
This question is out of curiosity and came to me thinking about another MO question which is linked below.
Question: Do there exist positive integers $a,b,c$ such that $\gcd(a,b,c) =1 $ and each of ...
- 7,875
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
15
votes
4
answers
1k
views
Number of $\mathbb F_p$ points constant mod $p$?
I have some affine varieties $X$ defined over $\mathbb Z$, and associated integers $c(X)$, with the property that $\# X_{\mathbb Z/p} \equiv c(X) \bmod p$ for all $p$. (In particular $c(X)$ is usually ...
- 27.8k
15
votes
1
answer
2k
views
Fermat's Bachet-Mordell Equation
Fermat once claimed that the only integral solutions to $y^2 = x^3 - 2$ are $(3, \pm 5)$.
Fermat knew Bachet's duplication formulas (more precisely, Bachet had a formula for computing what we call $-...
- 32.6k
15
votes
0
answers
631
views
Does every integer $n>1$ have the form $a^2+b^2+3^c+5^d$ with $a,b,c,d$ nonnegative integers?
Lagrange's four-square theorem states that every nonnegative integer is the sum of four squares. I have tried to replace two of the four squares by two powers. This leads to my following question: ...
- 15.6k
14
votes
7
answers
1k
views
Diophantine equation $3^n-1=2x^2$
How to solve a Diophantine equation like $$3^n-1=2x^2$$. One can easily see that the parity of $n$ and $x$ will be same and equation further can be seen taking if $$n\equiv0\pmod3\quad \text{then }x \...
14
votes
4
answers
1k
views
Find all solution $a,b,c$ with $(1-a^2)(1-b^2)(1-c^2)=8abc$
Two years ago, I made a conjecture on stackexchange:
Today, I tried to find all solutions in integers $a,b,c$ to
$$(1-a^2)(1-b^2)(1-c^2)=8abc,\quad a,b,c\in \mathbb{Q}^{+}.$$
I have found some ...
- 4,280
14
votes
1
answer
3k
views
A hard diophantine equation: $m!+27=n^3$
I would like prove that the following diophantine equation is unsolvable: $m!+27=n^3$.
Thanks in advance.
14
votes
4
answers
2k
views
Is (n,m)=(18,7) the only positive solution to n^2 + n + 1 = m^3 ?
It's hard to do a Google search on this problem.
If I was using Maple correctly, there are no other positive solutions with n at most 10000.
I know some of these Diophantine questions succumb to ...
- 377
14
votes
1
answer
763
views
Is 36 a sum of 4 rational fourth powers?
Hasse principle is known to hold for homogeneous quadratic equations, but fail for some 3- and 4-variable cubics, such as $5x^3+4y^3+3z^3=0$ or $15x^3+10y^3+4z^3+3t^3=0$. These counterexamples are ...
- 7,182
14
votes
3
answers
2k
views
Diophantine equation: Egyptian fraction representations of 1
According to the OEIS (A002966) there are 294314 solutions in positive integers to the equation
$$\sum_{i=1}^7\frac{1}{x_i}=1$$ assuming $x_1\leq x_2\leq\cdots\leq x_7$.
Similarly for 8 summands there ...
- 1,639
14
votes
1
answer
408
views
Can you "slice" a triangular number into three equal slices?
Problem statement:
Does there exist positive integers $a<b<c$ such that
$$1 + 2 + \dots + (a-1) = (a+1) + \dots + (b-1) = (b+1) + \dots + c?$$
(Note that $a$ and $b$ are not in the sums.)
...
- 257
14
votes
1
answer
612
views
What are the rational solutions to $y^4=x^3+x+1$?
What are the rational solutions to $y^4=x^3+x+1$?
This equation is interesting because it has substitution $y^2=z$ that reduces it to elliptic curve $z^2=x^3+x+1$. Sometimes, the existence of such ...
- 7,182
13
votes
7
answers
3k
views
Special arithmetic progressions involving perfect squares
Prove that there are infinitely many positive integers $a$, $b$, $c$ that are consecutive terms of an arithmetic progression and also satisfy the condition that $ab+1$, $bc+1$, $ca+1$ are all perfect ...
- 625
13
votes
2
answers
938
views
On Generalizations of Fermat's Conjecture
We know the following facts:
(1) For all $1\leq n\leq 2$ the equation $x_{1}^{n}+x_{2}^{n}=x_{3}^{n}$ has a solution in $\mathbb{N}$.
(2) For all $3\leq n$ the equation $x_{1}^{n}+x_{2}^{n}=x_{3}^{n}...
user36136
13
votes
1
answer
455
views
Universality of $y^4-x^3$ mod $p$
For pedagogical reasons, I got interested in the equation $y^4-x^3=a$ over $\mathbf F_p$.
To my surprise (maybe I'm naive), there is only one couple $(p,a)=(13,7)$ for which there is no solution, at ...
- 1,980
13
votes
3
answers
2k
views
A heuristic for the density of solutions to Diophantine equations
Let $f\in\mathbb{Z}[X_1,\ldots,X_n]$ be a Diophantine equation which, for the purposes of this question, I will assume is homogeneous and nonsingular on $\mathbb{R}^n\setminus\{0\}$ (so that $\nabla f\...
- 17.1k
13
votes
1
answer
760
views
Infinitely many integer solutions to $X^4+Y^4-18Z^4= -16$
We found infinitely many integer solutions to
$$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
13
votes
1
answer
4k
views
Effect of abc conjecture on Fermat's Last Theorem
A website ( http://www.math.unicaen.fr/~nitaj/abc.html#Consequences ) says that the $abc$ conjecture implies that there are only finitely many solutions to the equation $x^n+y^n=z^n$ with $\gcd(x,y,z)=...
- 2,075