Diophantine equations are polynomial equations $F=0$, or systems of polynomial equations $F_1=\ldots=F_k=0$, where $F,F_1,\ldots,F_k$ are polynomials in either $\mathbb{Z}[X_1,\ldots,X_n]$ of $\mathbb{Q}[X_1,\ldots,X_n]$ of which it is asked to find solutions over $\mathbb{Z}$ or $\mathbb{Q}$. Topics: Pell equations, quadratic forms, elliptic curves, abelian varieties, hyperelliptic curves, Thue equations, normic forms, K3 surfaces ...
Questions tagged [diophantine-equations]
902 questions
26
votes
1 answer
Is there an online encyclopedia of Diophantine equations (OEDE)?
Hello all!
I'm just wondering if there is an online encyclopedia of Diophantine equations (OEDE), analogous to the OEIS for sequences.
While trying to solve one Diophantine equation, I reduced the solution to that of a very similar Diophantine…
Kieren MacMillan
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- 1
- 10
- 21
8
votes
3 answers
Sum of two consecutive squares equals difference of two consecutive cubes
To celebrate my birthday, I like to find interesting number theoretic
properties of my new age. My upcoming 61st birthday was challenging, but then
I noticed that $61 = 5^2 + 6^2 = 5^3 - 4^3$, the sum of two consecutive squares
and the difference of…
Ernest Davis
- 449
6
votes
3 answers
Non-negative integer solutions of a single Linear Diophantine Equation
Consider the following linear Diophantine Equation::
ax + by + cz = d ------------ (1)
for all, a,b,c and d positive integers, and relatively prime, and assume a>b>c without loss of generality.
Can we find a lower bound on d which ensures at…
TGM
- 63
5
votes
4 answers
Diophantine problem
I have reduced a knotty research problem to the following reasonable looking form:
Given any two integers $a$ and $b$, show that there are natural numbers $x_1,x_2,x_3$ and a (probaby negative) integer $n$, where $-3n < x_1+x_2+x_3$,…
Adam
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5
votes
5 answers
Integer polynomials taking square values
Is there a way to determine a formula giving all integer values of $x$ for which the value of a polynomial $P(x)$ with integer coefficients is a square?
That is, is there a closed formula for:
$X = \{ x \in \mathbb{N} : \exists \ n \in \mathbb{N} :…
Mau
- 51
3
votes
1 answer
Solving elliptic equation in rational functions
Good afternoon,
I'm trying to solve an elliptic equation of the form
$$AY^2=4X^3+aX+b$$
where $A\in\mathbb{C}[z]$, $a,b\in\mathbb{C}$ and the unknowns $X,Y\in\mathbb{C}(z)$. In Mason ``Diophantine equation over function fields'', such equations are…
T. Combot
- 231
3
votes
3 answers
Integer points of an elliptic curve
I would like to find those integers $x,y$ that satisfies $y^2=x^3+1$. Is there some elementary way to find those?
Student
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2
votes
1 answer
Infinite solutions of a diophantine equation
Given the Diophantine equation$$ax^2+bxy+cy^2+dx+ey+f=0$$
if the coefficients $(a,b,c,d,e,f)$ are chosen among all the prime numbers, we have infinite equations. Is it possible to prove that the solutions of the infinite equations are infinite and…
Riccardo.Alestra
- 399
2
votes
2 answers
Parametrizing the solutions to a diophantine equation of degree four
Good evening,
Consider $x^4+y^4+z^4=2t^4$ where x,y,z,t integer.
Is it known how to find all parametrisation of this equation ?
If you have any parametrisation or reference of this equation, please post it
Thank you.
user81854
- 29
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2
votes
3 answers
Specific Diophantine Equation Appearing in Faa Di Bruno Formula
In a Faa Di Bruno Formula there is an equation:
$m_1$+2*$m_2$+3*$m_3$+...n*$m_n$=n
Is there any general solution for this equation.
For example for
$m_1$+$m_2$+$m_3$+...+$m_n$=n, there is a simple algorithm calculating this.
Thanks,
Gevorg.
veg_nw
- 185
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2
votes
2 answers
Help with this system of Diophantine equations
A couple hours ago, I'd posted a Diophantine equation question, but realized that I'd committed a rather preposterous blunder deriving it.
This is the actual question which I'm trying to solve:-
For a research problem that I'm working on, I need to…
Train Heartnet
- 285
1
vote
0 answers
Diophantine equation over Z[i]
I'm trying to generate the set of solutions of a specific diophantine equation over Z[i].
The equation is the following:
$$ z_1^2 + z_2^2 + z_1*z_2 + 39 = 0$$
with $ z_1$ (resp $z_2$) such that $\exists a,b \in \mathbb{Z} , z_1$ (resp $z_2$) $= a +…
Michel
- 21
1
vote
0 answers
Diophantine equation Oeis A159589
Considera the diophantine equation:
$y^2=x^2+(x+449)^2$.
Is there a method to solve this equation?
And why an Oeis sequence Is dedicated to this equation? Has this diophantine equation something special, has this diophantine equation some context?
Twiga
- 3
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1
vote
2 answers
Solutions to a system of Diophantine equations
In my research in a different field (representation theory), the following system of equations popped up:
$$
ax=by
$$
$$
xy+a+b-ax=p
$$
where $p\in\{0,1,2,3,4\}$ and $a,b,x,y$ are integers (I am also interested in the case where x and y are…
Irreducible
- 23
1
vote
1 answer
generalizations of the delone nagell equation
Are there any references that study integer solutions to cubic Diophantine equations of the form $x^3 + 2y^3 = 2^a 3^b$ for $\{a,b\}\subset \{0,1,2,3\}$? I am aware that Nagell solved $x^3 + 2y^3 = 1$ $(x,y)=(1,0)$ is the only solution. I read (in…
student
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