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Questions tagged [diophantine-equations]

Use for questions about finding integer or rational solutions to polynomial equations.

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
vote
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
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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
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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
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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
votes
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^...
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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
votes
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 ...
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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
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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
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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
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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
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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
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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
votes
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 ...