All Questions
Tagged with nt.number-theory diophantine-equations
787 questions
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A system of nonlinear Diophantine equations whose positive solutions are not coprime
Consider the following system of Diophantine equations:
$$v_1k_1=k_1^3-k_2^3+k_3^3 \\
v_2k_2=k_1^3+k_2^3-k_3^3 \\
v_3k_3=-k_1^3+k_2^3+k_3^3 \tag{1}$$
where $v_1,v_2,v_3$ and $k_1,k_2,k_3$ are integer-...
- 303
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0
answers
80
views
Solution modulo $9$ of certain linear equation implies triviality modulo $3$
Question: Let $k \geq 2$ and $r \geq 4$ be two natural numbers. We are given eight integers $\nu_{ij} \geq 0$ for every $1 \leq i \leq k$ and $1 \leq j \leq r$ such that the following two conditions ...
- 119
5
votes
2
answers
363
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Are there integer solutions of $m^4+m^2n^2+n^4=k^2$?
The recent question about Sets of integers with same sum and same sum of reciprocals, with its wealth of solutions, raises naturally the question: Can we require all those numbers to be squares? Of ...
- 13.4k
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67
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System of linear diophantine equations with many small solutions?
Let $n$ be positive integer, $k$,$B$ fixed positive integers.
Let $f_i(x_1,x_2...x_n)$ be a system of $n-k$ linearly independent linear
equations over the integers.
Let $S(f_i,k,B)$ be the set of ...
- 25.4k
7
votes
0
answers
394
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What are the integer solutions to $2x^5+3y^5=5z^5$?
This equation has the obvious integer solution $(x,y,z)=(1,1,1)$. By Faltings's theorem, the equation has finitely many primitive integer solutions (those with $\gcd(x,y,z)=1$).
What is the complete ...
- 435
5
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0
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205
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On the equation $x^4+y^3+z^2+1=0$
Are there infinitely many triples of integers $(x,y,z)$ satisfying the equation
$$
x^4+y^3+z^2+1=0 \quad ?
$$
This is one of the simplest-looking equations (another one is famous $x^3+y^3+z^3=3$) for ...
- 7,182
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0
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241
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Conjecture about some recurrent primes
I want to know if there are conjectures similar to this one, I know there is the Bell primes conjecture or Gardner conjecture (mentioned in this page https://en.wikipedia.org/wiki/Bell_number), but ...
26
votes
1
answer
943
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What are the integer solutions to $x^4+2y^4=3z^4$?
This equation has the obvious integer solution $(x,y,z)=(\pm 1,\pm 1,\pm 1)$. By Faltings's theorem, the equation has finitely many primitive integer solutions (those with $\gcd(x,y,z)=1$).
What is ...
- 435
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0
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148
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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
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1
answer
235
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Methods of finding integer solutions beyond the reach of direct search
Consider a classical problem: given a polynomial Diophantine equation $P(x_1,\dots,x_n)=0$, determine whether it has an integer solution. While this problem is undecidable in general, we may still ...
- 7,182
1
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1
answer
275
views
How to prove that this equation has no other integer solutions?
To find the integer solutions of an indeterminate equation and prove that there are no other solutions, where all variables are positive integers and n can be regarded as a constant, let's first ...
- 11
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138
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Eisenstein triples (and triangles with rational sides and a rational-degree angle) in Pascal's triangle
This question leads to a follow-up: are there any Eisenstein triples (satisfying $a^2\pm ab+b^2=c^2$) in one row of Pascal's triangle apart from the following:
$\binom{23}{8}^2+\binom{23}{8}\binom{23}{...
- 2,370
11
votes
1
answer
598
views
How to prove this problem about ternary quadratic form?
Is this right? And how to prove it ?
For $n \equiv 1,2 \bmod 4$
$$ \Bigg|\ \mathbb Z^3\cap\Big\{(a_1,a_2,a_3)\ \Big|\
a_1^2+a_2^2+a_3^2=n \Big\}\Bigg| \\
= \frac12\Bigg|\mathbb Z^3\cap\Big\{(a_1,...
- 113
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vote
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162
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Infinite number of decompositions into sum of four cubes
The context is the sum-of-four-cubes problem (see here).
I ask myself the following question (I asked a similar question on MSE). Do we know if any integer $k=a^3+b^3+c^3+d^3$ can be decomposed in an ...
- 655
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1
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150
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On Mordell equation $y^2=x^3+k$ [closed]
Have the Mordell equation $y^2=x^3+k$ solved for all integer $k$ or not?
Please Could you tell me about a good review papers about such equation.
- 17
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votes
1
answer
316
views
Which integers can be expressed as $P(t)^2 + Q(t)^2 + R(t)^5$?
Inspired by this article and that one, I have two questions:
(1) Is the question of whether every integer can be expressed in the form $x^2 + y^2 + z^5$ ($x$, $y$, $z$ in $\mathbb{Z}$) an open problem?...
- 655
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0
answers
146
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Interesting solutions of equation x^y = y^x [closed]
There is simple equation $x^y=y^x$. By taking logarithm we can see that it is equivalent to $\frac{\ln x}{x}=\frac{\ln y}{y}$. When we plot and inspect the function $f(x)=\frac{\ln x}{x}$, we can see ...
- 501
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2
answers
617
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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
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2
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332
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Diophantine equation of sixth degree [closed]
Show that the following equation admits infinitely many solutions:
$$x^3 + 2 y^6 - 2 z^6 = 1,\qquad \gcd(x,y)=\gcd(x,z)=\gcd(y,z)=1.$$
For example, $(79,5,8)$ is a solution .
- 47
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2
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215
views
Papers related to a diophantine equations about Magic square of squares for $n=3$
The open problem of magic squares of squares explained here. Consider the following magic square of squares:
$$
\begin{aligned}
&a^2&b^2&&c^2\\\\
&d^2&e^2&&f^2\\\\
&...
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
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votes
1
answer
827
views
Integer solutions for a simple cubic
What are the integer solutions to $x^3 - 7xy + y + 1 = 0$? A computation only finds $(0, -1), (-1, 0), (-6, 5), (-49, 342)$. This is surprisingly few.
Are these all of them? Is there an algebraic ...
- 83
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votes
1
answer
313
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On the Diophantine equation $2^a3^b + 4^c5^d = 2^b3^a + 4^d5^c$ [closed]
In this year's team selection test to the international mathematical Olympiad in Japan, a interesting diophantine equation appeared as the final problem of the second day.
Find all quadruples $(a,b,c,...
3
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1
answer
222
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Large integral points on the quadratic twist $ D y^2=x^3+A x +B$
For integers $A,B,D$ and $D$ squarefree let $E : y^2=x^3+A x + B$
and $E_D$ be the quadratic twist of the elliptic curve $E$:
$$ E_D : D y^2=x^3+Ax +B$$
$E_D$ is isomorphic to $ E'_D : y^2=x^3+D^2 A ...
- 25.4k
2
votes
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answers
52
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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
1
vote
1
answer
160
views
The existence of solutions of a Diophantine exponential equation
Given a prime number $p$ and a positive integer $n$, I am interested in the (non)existence of positive integer solutions $x,x_0,\dots,x_{p^n}$ of the following Diophantine equation
$$p^x+p^n=\sum_{i=0}...
- 41.8k
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votes
1
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126
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Proving that there are no integral points on a union of hyperbolas
I have a curve C: (x^2±x-y^2+1)(x^2∓x-y^2) where x,y ∈ Z+ that I want to prove has no non-trivial integral points other than (0,0),(1,0),(0,±1). I am having a hard time coming up with a solution.
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Can you "slice" a triangular number into three equal slices?
Problem statement:
Does there exist positive integers $a<b<c$ such that
$$1 + 2 + \dots + (a-1) = (a+1) + \dots + (b-1) = (b+1) + \dots + c?$$
(Note that $a$ and $b$ are not in the sums.)
...
- 257
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0
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197
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On the integer solutions of the equation $y^2 = x^3 + n$
Let $n$ be a nonzero integer. I am interested in the integer solutions $(x, y)$ to the equation $y^2 = x^3 + n$.
Let $S$ be the set of all integer solutions $(x, y)$ to this equation.
I am wondering ...
- 95
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votes
1
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763
views
Is 36 a sum of 4 rational fourth powers?
Hasse principle is known to hold for homogeneous quadratic equations, but fail for some 3- and 4-variable cubics, such as $5x^3+4y^3+3z^3=0$ or $15x^3+10y^3+4z^3+3t^3=0$. These counterexamples are ...
- 7,182
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votes
3
answers
830
views
About the units in $\mathbb{Z}[\frac{1+\sqrt{d}}{2}]$
Let's consider $K=\mathbb{Q}[\sqrt{d}]$ where $d$ is positive and square free. It is well known that the ring of integers is
$$
{O}_{K}=\mathbb{Z}\left[\frac{1+\sqrt{d}}{2}\right]
$$ if $d=1 \mod 4$ ...
2
votes
2
answers
270
views
Finding rational points on intersection of quadrics in affine 3-space
Consider the subvariety of Spec $\mathbb{Q}[x,y,z]$ cut out by the equations
\begin{eqnarray*} f_1: a_1x^2 - y^2 - b_1^2 & = & 0 \\
f_2 : a_2x^2 - z^2 - b_2^2 & = & 0
\end{eqnarray*}
...
- 7,527
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1
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261
views
Small solutions of $x^2-a^3 y^2=\pm 1$
We are interested in small integer solutions to the Pell equation:
$$x^2-a^3 y^2=\pm 1 \qquad (1)$$
Where in $\pm 1$ you can chose either sign.
$(x^2,a^3 y^2)$ are consecutive powerful numbers.
$abc$ ...
- 25.4k
5
votes
1
answer
393
views
On the equation $7x^3 + 2y^3 = 3z^2 + 1$
The question is whether there exist integers $x,y,z$ such that
$$
7x^3+2y^3=3z^2+1.
$$
After a similar equation On the equation $9x^3+y^3=z^2+3$ has been solved, this is one of the nicest cubic ...
- 7,182
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4
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Six consecutive positive integers with certain shape
Are there 6 consecutive positive integers, where each of them is a square or the product of a prime and a square ?
If they exist, one of those six integers A will be the product of 2 and a square of ...
- 321
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votes
1
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206
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Triangular repdigits
I would like to know whether $55$, $66$ and $666$ are the only triangular numbers that are repdigits, i.e., numbers at least $10$ whose digits w.r.t. base 10 all agree.
In more sophisticated terms, I ...
- 11.9k
5
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505
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Nice diophantine equations with large smallest solutions
Given a polynomial $P$ with integer coefficients in finitely many variables,
we denote by $v(P)$ the product of the absolute values of the non-zero coefficients
and the non-zero total degrees of the ...
2
votes
2
answers
1k
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Sum of three square is a square and sum of their product taken two at a time is also a square
Let $a^2 + b^2 + c^2 = X^2$ and
$$(ab)^2 + (ac)^2 + (bc)^2 = Y^2$$
Such that $a,b,c,x,y$ are all non zero Integers.
How to find All solutions ?
Is there any parametrization which gives Infinitely ...
- 147
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votes
1
answer
1k
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The new shortest open cubic equations
Section 8.3.2 of recent book [1] studies the following problem. Define the length of a Polynomial Diophantine equation as the sum of degrees of monomials plus sum of base 2 logarithms of the ...
- 7,182
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671
views
Do there exist positive integers $m$, $n$, $p$, $q$ such that $m>1$, $p\neq q$, $p$ and $q$ divide $mn^2 - 1$, and $mn$ divides $p - q$?
Do there exist positive integers $m$, $n$, $p$, $q$ such that $m>1$, $p\neq q$, $p$ and $q$ divide $mn^2 - 1$, and $mn$ divides $p - q$?
It seems numerically up to $n \leq 10^6$ that for $m=3$ or $...
- 191
31
votes
1
answer
2k
views
Can $9xy$ divide $1+x^2+x^3+y^2$?
Can $9xy$ divide $1+x^2+x^3+y^2$ for integers $x,y$? Equivalently, do there exist integers $x,y,z$ such that
$$
1 + x^2 + x^3 + y^2 + 9 x y z = 0 \quad ?
$$
This equation arises in the search for the ...
- 7,182
0
votes
0
answers
122
views
On the (hyper?)elliptic curve $y^2=x^2-x^3z^2+z-1$
The question here is if there exists $x,y,z\in\mathbb Z$ such that$$y^2=x^2-x^3z^2+z-1\label{1}\tag{1}$$other than the trivial solution$$x=0,y^2+1=z\text{ for all }y\in\mathbb Z\label2\tag2$$I know ...
- 145
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votes
2
answers
482
views
Difficult elliptic curves for $a^4+b^4+c^4+d^4 = (a+b+c+d)^4$?
Similar to the case $x^4+y^4+z^4 = 1$ discussed in this MO post, define the system,
$$x^4+y^4+z^4+1 = (x+y+z+1)^4\tag1$$
$$\frac{x^2+x+1}{(x+y+1)(x+z+1)}=u\tag2$$
$$\frac{y^2+y+1}{(y+z+1)(y+x+1)}=v\...
- 12.6k
13
votes
1
answer
666
views
On the equation $9x^3+y^3=z^2+3$
The question is whether there exist integers $x,y,z$ such that
$$
9x^3+y^3=z^2+3.
$$
This is one of the nicest (if not the nicest one!) cubic equations for which I do not know whether integer ...
- 7,182
5
votes
4
answers
476
views
A cubic equation, and integers of the form $a^2+192b^2$
This question resembles my previous question A cubic equation, and integers of the form $a^2+32b^2$ , but seems to be more difficult.
We are trying to determine whether there are any integers $x,y,z$ ...
- 7,182
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
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
1
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0
answers
121
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Situations where the number of solutions to a linear Diophantine equation is always even
I have a number theory situation that I hope someone will recognize as a known situation and can direct me to some relevant papers in the literature.
This came out of some numerical experiments run by ...
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
1
vote
0
answers
136
views
On the equation $q(\mathbf{x}) = 1$ for $q$ a quadratic form
Let $q(\mathbf{x}) = q(x_1, \cdots, x_n)$ be a quadratic form with integer coefficients. For $n \geq 3$, is there a reasonable theory for the set of integer solutions to the equation
$$\displaystyle q(...
- 26.9k