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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
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
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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
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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
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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\...
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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
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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
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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 $N$. ...
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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
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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
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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
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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
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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 ...
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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
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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
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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
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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
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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
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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 ...
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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
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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
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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
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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
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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 $x$ ...
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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
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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
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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 ...
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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 ...
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1answer
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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
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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
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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 ...
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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
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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)=...
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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 ...
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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
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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
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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 ...
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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_{{...
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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.