Questions tagged [diophantine-equations]
Use for questions about finding integer or rational solutions to polynomial equations.
3
votes
1answer
46 views
Solve the diophantine equation 71x +29y = 101
Solve the diophantine equation 71x +29y = 101
1.Euclidean algorithm
71 = 29*2 + 13
29 = 29*2 + 3
13 = 3*4 + 1
3 = 3*1 + 0
GCD(71,29) = 1
2. Write as linear equation (Euclidean algorithm ...
-1
votes
2answers
59 views
All natural number solutions of the equation
Can you find all natural number solutions of this equation?
I tried puting it in wolfram alpha and some other math problem solvers but they just solve it for one solution $$x = 2$$ and $$y = 1$$
$$y^{...
0
votes
0answers
12 views
Solving a non linear diophantine equation $px+y^{p-1}=2017$ [duplicate]
I had been tasked in an exam to solve this equation:
$px+y^{p-1}=2017$ where $p$ is a positive prime number, and $x$ and $y$ are natural numbers.
I was able to prove that if $p≥5$ then $p=7$, so now I ...
3
votes
2answers
44 views
Do there exist (non-trivial) prime solutions to the equations $p^2 = 1$ mod $q$, $q = 1$ mod $p$?
Question:
Do there exist odd primes $p$ and $q$ such that
$$p^2 = 1 + qt,\quad q = 1 + ps$$
for some positive integers $s,t$? I've written some code which has verified that no solutions exist for $p,q ...
1
vote
0answers
36 views
Is there better method than this one to solve quadratic diophantine equations?
Say we have the quadratic Diophantine equation :
$4x^2+9y^2=100$
I'm not really sure what the usual method to find integer solutions to these equations are , but one way I thought of was the ...
1
vote
2answers
72 views
Solve $a^3+b^3+3ab=1$ with $(a,b)\in \Bbb{Z}^2$
Solve the following equation for $(a,b)\in \Bbb{Z}^2$:
$$a^3+b^3+3ab=1$$
I tried all of the standard techniques I know. I tried modular arithmetic:
$$a^3+b^3+3ab\equiv 1 \pmod{3} $$
$$a^3+b^3\...
0
votes
0answers
44 views
How do you solve $xy=8$ and $x^y=y^x$? [duplicate]
I know answers are $4$ and $2$ but can't solve this.
2
votes
4answers
89 views
Infinitely many integer solutions of a cubic equation
Is it true that there are infinitely many pairs of integers $(m,n)$ such that
$m^3 + 5n^3 + m^2n = 1$? Or maybe $m^3 + 5n^3 + m^2n = -1$? The point is that I am trying to find a description of an ...
0
votes
1answer
35 views
Are the binomial coefficients unique?
Let $a,x,b,y$ be integers.
Can we find rationals $u,v,w,t$ such that $$(ax+by)^3=ux^3+vx^2y+wxy^2+ty^3\neq 0$$
where $$(u,v,w,t)\neq ( 1, 3a^2b, 3ab^2, 1)$$
The answer looks trivial but can one prove ...
1
vote
1answer
136 views
Can the solution to $n^2=pq+y^2$ help with the Golbach conjecture?
This question was inspired by the following question.
https://mathoverflow.net/questions/132532/goldbachs-conjecture-and-eulers-idoneal-numbers
Here, we are not looking to factor an integer $N$. ...
-1
votes
0answers
31 views
How can I solve these linear diophantine equations?
Just starting to learn (in 11th grade) linear diophantine equations, and I'm faced with these problems. Everybody said to me that they are quite easy.
Solve:
$x! - 1 = y^2$ in natural ...
0
votes
1answer
23 views
Is there a way to find positive values for unknown of Diophantine Equations?
I have a equation as follow ax+by =c where the value of x and y are unknown. This is a 2 ...
1
vote
3answers
31 views
The system of Diophantine equations with same solution
There is a system of Diophantine equations:
\begin{equation*}
\begin{cases}
368=x^7 (mod 407)\\
389=x^{11}(mod 407)
\end{cases}
\end{equation*}
However, solving each of them by hand is quite ...
1
vote
1answer
30 views
Quadratic residue
Show that there are infinitely many pairwise coprime integers $d$ for which there is at least one integer $c$ so that $a^2 + b^2 \equiv c \pmod d$ has no integer solutions.
I see that c must be from ...
2
votes
4answers
77 views
Find all pairs $(m, n)$ of positive integers such that $m$ divides $8n+1$ and $n$ divides $8m+1$
I've found the pairs $(1,3),(1,9),(3,25)$ and $(13,21)$ up to order. But I have no idea how to prove that there are not other solutions. Any hints...? I've been trying for a few days but all I came up ...
-1
votes
1answer
65 views
How do solve an Interesting Diophantine Equation
I would like to know, for what integer values of $x$ makes $f(x)$ an integer for this equation, which I have derived from several other equations: $$f\left(x\right)=0.25x-0.5(33)+\left(\frac{1}{x}\...
1
vote
1answer
72 views
How to solve Quartic Diophantine Equations
I have this diophantine equation:$$x^4 - 4 x^2 y - 66 x^2 + 1089 = 0$$Is there a way to find all integer $x$ values that make $y$ also an integer without factoring or checking $x$ values within a ...
0
votes
0answers
44 views
solving a Diophantine equation through respective congruences?
Question:
1-Let $a,b$ be integers having the property that for any prime
power $t$ there exists an integer $v_t$ such that
$$
b\equiv a^{v_t}\pmod t.
$$
Then there exists an integer $v$ such that
$$
...
1
vote
2answers
59 views
Diophantine Equations of degree 3 and 4
I have these two Diophantine equations: $$0 = x^2 y - 36 x^2 - 12 x y + 36 x + 36 y - 9$$ and $$0 = x^2y^2 - 36 x^2 - 12 x y^2 + 36 x + 36 y^2 - 9$$ Each of them is slightly different from each other, ...
2
votes
4answers
109 views
How to prove that $2^n-1$ is not a perfect square for $n\ge 2$? [duplicate]
I have to prove that $2^n-1$ statement is not a perfect square if $n$ is greater than two. I tried some techniques but I didn’t manage to solve the problem.
0
votes
1answer
59 views
How many marbles?
One dozen of big marbles and small marbles is 132 gram. If one big marbles is 3 gram heavier than one small marbles, then specify the possibilities of how many are the big marbles and the small ...
0
votes
0answers
17 views
Is it possible to find $x$, knowing the following: $((z_1s_2 - z_2s_1) (v_1 s_1 - v_2 s_2))^{p-2} (\text{mod } p)$
This puzzled me over the last weekend, before everything let me say it's quite possible that the equation doesn't have a "solution" but it is a special case that follows from a solution. In any case, ...
0
votes
1answer
32 views
Pirate and Bags of Coin
A pirate captain has 63 bags of coin with the same amount of coin inside each of the bags. If he wants to divided the coins to his 23 henchman evenly, he has to add 7 more coins. How many coins inside ...
0
votes
2answers
32 views
Solution to Diophantine equation $19991112x + 2803y = 33$
I already found that $gcd(19991112,2803)=1$ so it does have solution. But I don't know how to find the solution.
Equation: Diophantine equation $19991112x + 2803y = 33$
1
vote
1answer
83 views
Integers in the form $\pm 2^a \pm 3^b \pm 5^c$
It follows by the theory of linear form in logarithms that "few" integers can be written as $2^a-3^b$, with integers $a,b\ge 0$ (see here). What about the case of more than two variables?
Question. ...
0
votes
1answer
37 views
Trouble understanding generalised form of Pell’s equation
A Pell’s equation is a diophantine equation in $x$ and $y$ of the form
$$x^2-d\cdot y^2=1$$
with $d$ a square-free integer.
The fundamental solution of a Pell’s equation is the smallest (with ...
1
vote
1answer
33 views
Method to solve diophantine $Ax^2-By^3=C$ or $Ax-By^3=C$
Am doing a research where at a particular point i have to prove if $Ax^2-By^3=C$ or $Ax-By^3=C$ will give an integer solution and find some of the solutions. Letting $x^2$ in the first equation or $x$ ...
0
votes
2answers
52 views
Integers which can be written as sum of powers of $2,3$, and $5$
Is it true that every sufficiently large integer can be written in the form
$$
2^a3^b5^c+2^d3^e5^f
$$
for some integer $a,b,c,d,e,f \ge 0$?
0
votes
1answer
23 views
Diophantine Equations of Degree 2
During my studies, I have seen equations of this form $$xy+bx+cy+d=0$$ Is there a way to find solutions of equations of this form or the number of solutions of equations of this form without factoring ...
4
votes
1answer
93 views
Is the problem Find all $x,y \in \mathbf{N}$ such that $\binom{x}{2} = \binom{y}{5}$ solved?
I was recently browsing and came upon this document which gives some open problems involving Diophantine Equations.
Document: http://www.math.leidenuniv.nl/~evertse/07-workshop-problems.pdf
Upon ...
1
vote
0answers
48 views
Number of integer solutions of a nonlinear equations of the form $2^m-3^k=y$
Given a fixed odd integer $y\in\mathbb{Z}$ and a natural number $n\in\mathbb{N}$, can we compute the cardinality of the set $\{(m,k)\in\mathbb{N}^2:k\leq n, 2^m-3^k=y\}$ explicitly?
Thanks in advance ...
-4
votes
1answer
46 views
Counting integral solutions of a quadratic equation [closed]
What is the number of pairs $(x,y)$ of integers satisfying the following equation?
$$x^2+y^2+xy-x+y=2$$
2
votes
3answers
92 views
Does the equation $a^2+d^2+4=b^2+c^2$ have any solutions?
Does the equation have $a^2+d^2+4=b^2+c^2$ where $d<c<b<a$ have any integer solutions? This isn't a homework problem, but I need to know for a separate problem I'm doing. Wolfram Alpha isn't ...
1
vote
1answer
59 views
Solving Fractional Diophantine Equations
As my search to create an efficient factorization algorithm continues, I stumbled upon this equation for one of my test cases:$$\dfrac{3-n^2}{2n-12}=k$$ To continue, I need to know what integer values ...
1
vote
0answers
20 views
Bound of minimal solutions of a system of integer equalities
Hi I am obstructed by the following problem regarding the integer linear inequality in the following form:
$\mathbf{A}\cdot\mathbf{x}\geq\mathbf{b}$
Where $\mathbf A$ is $m\times n$ integer $(0, \...
6
votes
2answers
569 views
Solution of this Diophantine Equation
If $x$ and $y$ are prime numbers which satisfy $x^2-2y^2=1$, solve for $x$ and $y$.
My attempt:
$x^2-2y^2=1$
$\implies (x+\sqrt{2}y)(x-\sqrt{2}y)=1$
$\implies (x+\sqrt{2}y)=1$ and $(x-\sqrt{2}y)=...
0
votes
0answers
45 views
Constraint diophantine equations (open problem)
Given an odd number $N$ such that $N\equiv 3 \mod 4$, I am looking to find numerically or algebraically a positive natural number $k$ s.t, $1\leqslant k \leqslant \frac{N-3}{4}$, satisfying the ...
0
votes
1answer
39 views
Finding solutions in $\mathbb{Z}_{+}$ [closed]
Find all the triplets of positive integers $(a, b, c)$ such that: $a^2+b+3=(b^2-c^2)^2$
1
vote
1answer
74 views
Diophantine Inequalities
I am trying to research ways to come up with better ways of factoring, which is why in any answer, I would not like factoring as that would defeat the whole point. As I was working, I came across this ...
0
votes
0answers
36 views
number of answers in $a_1x_1 + a_2x_2 + … + a_nx_n = k$ equation
I was wondering to know how many possible answer does this equation have ?
$a_1x_1 + a_2x_2 + a_3x_3 + ... + a_nx_n = k$
where ai are coefficients and constant while xi are variable and a non-...
0
votes
0answers
31 views
Intersecting Integral points $(x,y,z)$ of $3x+5y+4z=45$ and $z^2+xy=15$?
I am trying to find all the intersecting integral points $(x,y,z)$ of the plane $$3x+5y+4z=45$$ and the one-sheeted paraboloid $$z^2+xz=15$$.
I noticed that
$$x=4t-(9-y)$$
$$z=-3t+2(9-y)$$
So I ...
0
votes
1answer
42 views
Solving algebra with multiple square root
I am currently solving an algebra and can't figure it out, could anyone help me on this?
$$2\sqrt{N + \sqrt{N^2+4c^2}} = \sqrt{N + \sqrt{N^2+3c^2}} + \sqrt{N + \sqrt{N^2+5c^2}}$$
Which I would like ...
-1
votes
1answer
70 views
Solving Diophantine system of degree three that contains 4 equations with 16 unknowns over $\mathbb Z_n$ [closed]
The following Diophantine system
$25333-123\,a_{{2}}a_{{1}}-a_{{2}}a_{{1}}a_{{3}}-478\,a_{{2}}b_{{1}}-a_
{{2}}b_{{1}}c_{{3}}-223\,c_{{2}}a_{{1}}-c_{{2}}a_{{1}}b_{{3}}-589\,c_{
{2}}b_{{1}}-c_{{2}}b_{{...
-2
votes
1answer
63 views
Integer arithmetic [closed]
Show that there are no positive integers n for which
$$n^4+2n^3+2n^2+2n+1$$
is a perfect square.
Are there any positive integers n for which $n^4 + n^3 + n^2 + n + 1$ is a perfect square? If ...
0
votes
1answer
29 views
System of three non linear equations with three unknowns with random coefficients
I have the system of equations:
$$
\begin{cases}
Ax + By + Cz &= D \\
Exy + Fxz + Gyz &= H \\
Ixyz &= J \\
\end{cases}
$$
Where $A,B,C,D,E,F,G,H,I,J$ are constant integers between 1 and 9.
...
1
vote
1answer
84 views
Solve the Diophantine equation $24x^4-5y^4=z^2$
I want to solve $24x^4-5y^4=z^2$ in integers not all zero, and to fix ideas, I want to find them such that x,y are coprime.
I've tried plugging in small values of $x$ and $y$ and they don't return a ...
0
votes
0answers
20 views
Algorithm for solving a diophantine equation
Is there an algorithm to obtain integral solutions for equation of the form
$\frac{N-x}{2x}=y$, with $N,x,y\in \mathbb{Z}^{+}$, $\mod(N,x) = 0$, $y\geq x$ and $x\leq\frac{\sqrt{8N+1}-1}{4}$. Or can ...
1
vote
1answer
46 views
Upper bound on the number of integer solutions to $y^p=x^2+2$, where $p$ is prime
I need to find an upper bound on the number of solutions of the Diophantine equation $y^p=x^2+2$, where $p$ is prime.
I have previously considered the equation $y^3=x^2+2$ and proved its solutions ...
0
votes
2answers
76 views
Triple Pythagorean with $a^2+b^2=c^4$
It is well known that there exist integer solutions to the equation $a^2+b^2=c^2$.
For example, an explicit formula for integer values of $a$ , $b$ , and $c$ is
\begin{align}a&=2mn \\ b&=m^2-...
2
votes
3answers
74 views
Solving nonlinear Diophantine equations with Euclid's Lemma
How do I use Euclid's Lemma to solve the Diophantine equation $x^2 \equiv 13$ mod $17$? From there, how do I solve the Diophantine equation $s^2 \equiv 13$ mod $289$?
Thanks in advance for any help.
