Questions tagged [diophantine-equations]
Use for questions about finding integer or rational solutions to polynomial equations.
3,549 questions
3
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1answer
70 views
Do solutions over $\mathbb{F}_{p^n}$ tell me anything about solutions over $\mathbb{Q}_p$?
Yo, this is probably some nonsense stupid question, but I will ask it anyway.
Let $X$ be some variety over $\mathbb{Z}$. Suppose that I know $X (\mathbb{F}_{p^n})$ for every $n$. Does it tell me ...
1
vote
1answer
34 views
General solution for a Diophantine equation with more than two variables
Consider the Diophantine equation $$k_0a+k_1b+k_2c+k_3d+\cdots=1$$ where $a,b,c,d,\cdots$ are variables and suppose that a solution obtained through the Euclidean Algorithm is $a_0,b_0,c_0,d_0,\cdots$....
1
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2answers
42 views
diophantine equation $|4m^2-n^{n+1}| \le3$
please can someone give me a hint in this equation?
$|4m^2-n^{n+1}|\le3$ for non zero integers, find for which numbers this equation holds
I found roots as $m,n=0; m,n=1; m=0$ and $n=1$
I tried ...
1
vote
1answer
23 views
Is it possible for $\sum_{i=1}^n x_i^3 > \sum_{i=1}^n y_i^3 $ when $\sum_{i=1}^n x_i < \sum_{i=1}^n y_i $ where $x_i,y_i \in \mathbb{N}$
I.e. is it possible for a sum of $n$ positive integers to be strictly less than another sum of $n$ positive integers but be strictly greater when it's the sum of their cubes?
$x_1 + x_2 + ... + x_n &...
1
vote
0answers
38 views
Is it true that $(2k+1)(2^{4k+1}+(2k-1)^{4k+1})=a^2+(4k+1)b^2$ has no positive integer solution?
Is it true that the equation $$(2k+1)(2^{4k+1}+(2k-1)^{4k+1})=a^2+(4k+1)b^2$$ has no positive integer solution ?
Let $A=(2^{4k+1}+(2k-1)^{4k+1})/(2k+1)$,
if $q$ is a prime factor of $A$ such that $x^...
3
votes
2answers
113 views
+100
Finding all combinations that sum up to a specific number with given constrains
This is a continuation of this problem. Find all combinations that sum up to a specific number
Think of following values.
...
3
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0answers
63 views
Modularity and the most general $(p, q, r)$ case in the BeChDaYa paper.
In the Bennet, Chen, Dahmen, Yazdani paper, Generalized Fermat equations: A miscellany, on page 24 in section 6 entitled "Future Work", they say:
"A limitation of the modular method at present is ...
0
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0answers
15 views
Sharing a Factor as a Diophantine Equation [closed]
It's well known that if $A,B$ share a factor in common called $k$ then that means there is a Diophantine equation
$$ A = u_1 k, B = u_2 k$$
With solution $A,B,k, u_1, u_2 $
Involving 5 variables. ...
2
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0answers
63 views
Find all positive integers $a, b, c$ such that $(a^{3}+b)(b^{3}+a)=2^{c}$. [duplicate]
Find all positive integers $a, b, c$ such that $(a^{3}+b)(b^{3}+a)=2^{c}$.
I tried considering the prime factorization of both sides and using the exponent of 2 in it: $$v_{2}(a^{3}+b)+v_{2}(b^{3}+a)=...
2
votes
0answers
80 views
Any process for solving more complex nonlinear Diophantine equations such as $(8+3n)m = 11$?
Is there any known process for solving nonlinear Diophantine equations such as the ones below?
$(8+3n)m = 11\;\;|\;\;n \in \{0,1\},\;m\in \Bbb Z^+$
$(5+(7+3x+2y)a+3z)b = 30\;\;|\;\; x,y,z \in \{0,1\},...
0
votes
3answers
57 views
Positive Integer Solutions to $\frac{1}{x}+\frac{1}{y}+\frac{1}{z}=\frac{4}{5}$
I'd like to ask how to generate all positive integer solutions to the equality $\frac{1}{x}+\frac{1}{y}+\frac{1}{z}=\frac{4}{5}$. How about all integer solutions?
Some solutions include $x=2, y=5, z=...
3
votes
3answers
89 views
Triangle numbers that are squares of triangle numbers.
What are the triangle numbers the are squares of other triangle numbers? I have found $1^2=1$ and $6^2=36$, but other than these examples I can't find any other triangle numbers that are squares of ...
3
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0answers
61 views
How many integer solutions are there to $x^4+y^4-x^2y^2=n$. Is there a generating function for this?
It would be kind of cool to get a closed form for the number of integer solutions $$x^4+y^4-x^2y^2=n$$ which we will let $\phi_n$ denote.
It would be cool because we could exploit $\sum_{n=1}^N\...
0
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1answer
24 views
Counting Integer Solution to $\frac{a}{x-a} \in Z, \frac{b}{x-b} \in Z, x \geq 1$ in Sublinear Time
Given two positive integers $a$ and $b$, I'd like to ask how many integer solutions $x > 0$ satisfy that both $\frac{a}{x-a}$ and $\frac{b}{x-b}$ are positive integers. A sublinear solution is ...
1
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1answer
36 views
Find $n(R\cap Q)$ if $R=\{(x,y)$:$x^2$+$3y^2$=$28,x$,$y\in\mathbb{Z}\}$ and $Q$=$\{(x,y)$:$x$>$y,x$,$y\in\mathbb{Z}\}$
Th relations $R=\{(x,y):x^2+3y^2=28,x,y\in\mathbb{Z}\}$ and $Q=\{(x,y):x>y,x,y\in\mathbb{Z}\}$, then the number of elements in $R\cap Q$ is ?
how do I find the integer solutions of the equation $x^...
1
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0answers
27 views
Diophantine Equations over polynomial rings
I'm trying to figure out a set of solutions to an ideal membership problem. Namely, if $f(x,y),g(x,y)\in \mathbb{C}[x,y]$ I want to figure out what constraints I need such that they satisfy
$$g(x,y)-...
3
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1answer
60 views
What Are the Limitations in Diophantine Equations?
I read a bit about Diophantine equations, and I learned Hilbert tried to see if a solution exists for problems such as
$x^2y^3z^5-23x^5y^4+11x^2=176$
Apparently, rather than the notion our ...
2
votes
2answers
95 views
Given $11x+17y +19z =2561$ , find minimum and maximum of $x+y+z $
Given diophantine equation $11x+17y +19z =2561$ , which $x,y,z \geq 1$
Find minimum and maximum value of $x+y+z$
I'm start with reduces equation to $11x+17y +19z =2514$ , which help us ...
-1
votes
1answer
36 views
A Criteria of Diophantine Equations on one variable, Is it wrong?
The following is a Home Work question provided by our instructor:
let a,b,c be positive integers. Prove that if a+b>c then the equation
ax+by = c has no solutions in positive integers.
I think ...
1
vote
1answer
66 views
$f(x) = 2^\alpha x^2+2^\beta x - 2^\gamma=y^2$
A few questions regarding a polynomial of degree 2:
$f(x) = ax^2 + bx - c$
where $a,b,c$ are given positive powers of 2.
Or more explicitly:
$f(x) = 2^\alpha x^2+2^\beta x - 2^\gamma$
where $\...
0
votes
0answers
22 views
How to interpret Fueter's description of Diophantine equations with no rational points?
I'm an amateur mathematician so please forgive me if this is a basic question, but could you help me understand the meaning of the following snippet from a journal article (referring to work of Fueter)...
1
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1answer
30 views
Quadratic Residues form a sub-semigroup, that cannot be acted on by non-residues
I was reading: https://homepages.warwick.ac.uk/~maslan/docs/T-seminar.pdf
Where they go over a proof that $x^2 - 5y^2 = 3z^2$ doesn't have integer solutions $(x,y,z)$ such that $z \ne 0$. Their ...
10
votes
2answers
105 views
Solving integer equation $a^3+b^3=a^2+72ab+b^2$
Find all pairs of positive integers $(a;b)$ that satisfy
$$a^3+b^3=a^2+72ab+b^2.$$
I have already solved this by letting $S=a+b$, $P=ab$. Then I have $S^3-S^2=(3S+70)P$, which will result in $3S+70$...
3
votes
0answers
64 views
Is there an upper bound to the number of weight $2$ newforms for any level $N$?
There are no newforms of weight $2$ for levels $1$ thru $10,12,13,16,18,22,25,28,60$.
Looking at the list given in Sloane's integer sequence A127788 it appears that the number of newforms increases ...
0
votes
1answer
94 views
How $u | a,b,c,d$ and thus $u=1$?
If $a=a_1t, b=b_1t, c=c_1t, d=d_1t$ and -
$a_1+b_1i=\frac{u}{v}m\bar p$,
$a_1-b_1i=\frac{u}{v}p\bar m$,
$c_1=\frac{u}{v}p\bar p, d_1=\frac{u}{v}m\bar m $
then how can we imply that $u | a,b,c,d$ ...
4
votes
2answers
37 views
Diophantine Equation - least values for n : absolute values
Find all solutions to the diophantine equation
$323x + 278y = 7$
Choose also a solution for which $|x| + |y|$ is as small as possible.
My approach to this is to do the usual euclidean algorithm
$...
0
votes
2answers
41 views
Odd Formed Diophatine Equation Help
I am relatively new at solving these kind of equation and was wondering if someone can help with a step by step for an odd formed Diophantine equation. The particular equation I am trying to solve is $...
4
votes
5answers
100 views
Competition Diophantine Equation
I am looking for the number of integer solutions to this system of equations
$$a^2+b^2+c^2+d^2=2500$$ and
$$(a+50)(b+50)=cd$$
I tried moving terms around in the first equation and using the ...
0
votes
2answers
59 views
Integer Solutions Explaination
I know how to solve linear diophantine equations, but I was wondering if someone can give me a step by step to solve something like $2x^2 + 2x - 5y = -1$? I cannot find a lot of resources on this ...
0
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2answers
122 views
Existence of solutions to $a^2x^2 + bx = y^2$
Is there a way to determine if there are or aren't any integer positive solutions $(x,y)$ to the equation $a^2x^2 + bx = y^2$ depending on the values of $a$ and $b$?
I tried to deal with it using ...
2
votes
1answer
93 views
The minimum value of perimeter of the triangle
Let there be $\triangle ABC$ having integral side lengths such that angle $$ \angle A=3 \angle B $$ Then find the minimum value of its perimeter.
I did this with trigonometry and got an equation as: $...
1
vote
1answer
41 views
Cubic Diophantine with two variables
The question is : prove that $y^2 = x^3+(x+4)^2$ have no solutions in positive integers $x,y$.
I tried to play with the equation and get to $ x^3 = (y+x+4)(y-x-4)$, if $p|x$ then $p^3 | x^3$ and $ p |...
2
votes
2answers
73 views
How do you construct the Frey curve for (2,3,p)?
In Darmon's paper on p.14 he lists a table of signatures $(p,q,r)$ and constructed Frey curves. How do you construct the Frey curve he gives for $(2,3,p)$?
The curve he gives for this signature is:
...
0
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2answers
48 views
Solution for three variable Simultaneous Equations
I have two Equations :
$$1)\; abc=1$$
$$2)\; a+b+c=1$$
And the constraint that $a,b,c$ are positive real numbers.
I have to prove that there exist no solution for the given constraint.
My attempt:
I ...
1
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0answers
78 views
Cremona 2.14.1 Why is $c_4$ and $c_6$ complex when they should be rational?
In Cremona's online book Chapter 2, in order to calculate the lattice invariant we have:
$\tau=\omega_1/\omega_2$
Set $q=e^{2\pi i\tau}$
(2.14.1) $c_4 =(2\pi/\omega_2)^4(1+240 \sum_{n=1}^{\infty} n^...
1
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0answers
111 views
How to solve $a^{b^2} = b^a$ in positive integers
I have this exercise:
$a,b$ are positive integers. Get all $a,b$ numbers that satisfies $a^{b^2} = b^a$
Is hard to me, but I have these advances:
$a,b$ must be both even or both odd, since $a, b$ ...
11
votes
1answer
171 views
Find $a,b,c \in \Bbb N$ such that $n^3+a^3=b^3+c^3$
Is it true that for every $n \in \Bbb N$, there exist $a,b,c \in \Bbb N$ satisfing
$$
n^3+a^3=b^3+c^3,
$$
where $\operatorname{gcd}(a,b,c)=1$ and $b,c\ne n$?
I checked all positive integers less ...
0
votes
0answers
92 views
Number of solutions of quadratic system of diophantine equations
Let $c_0,\dots,c_k$ be some known nonnegative integers.
Consider the following system of equations in the unknown bits $a_i,b_i$, i.e. $a_i,b_i\in \{0,1\}$:
$$
\left\{\begin{array}{l}
a_0b_0& =&...
3
votes
0answers
57 views
When does an elliptic curve have accumulation points?
If the rank of an elliptic curve is greater than 1,then it has infinitely many rational points,I wonder how are these rational points distributed, especially, can we find an elliptic curve such that ...
1
vote
3answers
84 views
Given the equation $A(qr-p)=B(pr+q)$, when can $A=pr+q$ and $B=qr-p$ be true?
The following equations are known:
Given Equations
\begin{align}
A^2+B^2&=C^2\\
Ap+Bq&=Cr\\
p^2+q^2&=r^2+1\\
A&\neq B\\
Aq-Bp&=C
\end{align}
$A(qr-p)=B(pr+q)$ comes from the ...
2
votes
1answer
74 views
If $2^t = a^b \pm 1$, what are all the possible values of $t$?
Let $t$ be a positive integer such that $2^t = a^b \pm 1$ for some integers $a$ and $b$, each greater than $1$. What are all the possible values of $t$?
The question is taken from here. I know the ...
-1
votes
1answer
58 views
Solutions of $2^x-3^y=z$ with $z < 2^{x-y}$
I am looking for a source of the list of known solutions of:
$$2^x-3^y=z$$
with $x, y, z$ integer, $x, y, z > 0$ and $z$ "small". I would like to know especially if there are non-trivial (by ...
8
votes
2answers
233 views
Solve $a^2 - 2b^2 - 3 c^2 + 6 d^2 =1 $ over integers $a,b,c,d \in \mathbb{Z}$
Are we able to completely solve this variant of Pell equation?
$$ x_1^2 - 2x_2^2 - 3x_3^2 + 6x_4^2 = 1 $$
This has an interpretation as is related to the fundamental unit equation of $\mathbb{Q}(\sqrt{...
1
vote
1answer
20 views
Solution to rational Diophantine equations in fixed point
I'm trying to solve the following system of equations for $p$ and $q$, given fixed integers $x$, $y$ and $c$:
$$r = {{c x + p} \over {c y + q}} \ , \, \, \, r \in \mathbb{Z}$$
where
$$\{x, y, p, q, ...
0
votes
4answers
89 views
Proof that a diophantine equation does not have non-trivial solutions
Consider integer variables $x,y,z$ and the diophantine equation
$$z^2=6x^2+2y^2$$
I have the following proof that the above equation does not have a non-trivial integer solution i.e. the only solution ...
2
votes
2answers
55 views
Why we get inequality form an equation?
In the paper linear forms in the logarithms of real algebraic numbers close to 1, it is written on page 5 that-
$\varLambda \leq \frac{1}{by^n}$ (see equation 7 on page 5)
But we get it from an ...
4
votes
2answers
79 views
How we derive $y \geq (b+1)$ from $(b+1)b^n < (b+1)^n b$?
In the paper linear forms in the logarithms of real algebraic numbers close to 1, it is written on page 5 that-
$y \geq (b+1)$ since, $(b+1)b^n < (b+1)^n b$ and it was given that $(b+1)x^n - by^n ...
3
votes
0answers
105 views
Equation $\frac{1}{a_1}+\ldots +\frac{1}{a_{2018}} = 1$
Let $A_{n}=\{(a_1,a_2,\ldots,a_{n}): a_i\in\mathbb{Z_{>0}}|\ \ \frac{1}{a_1}+\ldots \frac{1}{a_{n}} = 1\}$.
My question. What is $|A_n|\operatorname{mod}2$, for $n=2018$? That is
what is the ...
2
votes
5answers
90 views
Pell's equation (or a special case of a second order diophantine equation)
Question Find integers $x,y$ such that $$x^2-119y^2=1.$$
So far I've tried computing the continued fraction of $\sqrt{119}$ to find the minimal solution, but either I messed up or I don't know where ...
2
votes
1answer
69 views
Large Cardinals and Diophantine Equations (Penelope Maddy)
Professor Penelope Maddy remarks without elaboration in her famous 'Believing the Axioms' essay that
'It should be mentioned that the Axiom of Inaccessibles also has a few extrinsic
merits. It ...
