Stack Exchange Network

Stack Exchange network consists of 175 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.

Visit Stack Exchange

Questions tagged [diophantine-equations]

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

0
votes
3answers
45 views

Solution to a Third degree diophantine equation

I have two diophantine equations of the third degree viz.$$2b_1^3l_1+3b_1^2l_1^2+b_1l_1^3=k$$ and $$2b_2^3l_2+3b_2^2l_2^2+b_2l_2^3=k$$ The aim is to find distinct values of $(l_i,b_i)$ which satisfy ...
1
vote
0answers
28 views

Pythagorean Quadruples and Stereographic Projection

I am trying to solve Diophantine equation $a^2+b^2+c^2=d^2$ by transforming this equation, assuming $d \neq 0$, into a sphere $ (\frac{a}{d})^2 + (\frac{b}{d})^2 +(\frac{c}{d})^2 = x^2 + y^2 +z^2 =1$ ...
3
votes
1answer
92 views

Show that the equation $\frac{x^n-1}{x-1}=4y^2$ has only two solutions

Prove or disprove Let $x,y,n$ be postive integers, show that the equation given below has only two solutions $$\frac{x^n-1}{x-1}=4y^2$$ I have found that $(x,n,y)=(3,2,1)$ and $(x,n,y)=(7,4,10)$ ...
0
votes
1answer
35 views

Find and integer solution to an equation where the x coordinate is greater than a certain value

So I have $9160x+4240y=1000$ Using the EA to find the gcd, I get that it is equal to 40. Solving for the RHS and working backwards, I have: $$40=680-160*4$$ $$160=4240-680$$ $$680 = 9160-4240*2$$...
0
votes
0answers
17 views

A discrepancy in the used general solution of linear diophantine equation in David M. Burton book.

In David Burton book "seventh edition", the general solution of linear diophantine equation is given below (in page 34): Then in page 76 after linking linear congruences with linear diophantine ...
-1
votes
1answer
24 views

How to find solutions to a linear Diophantine equation restricted to given intervals of x and y? [on hold]

For example: find all $x$, $y$ satisfying $10x + 20y = 300$ and $-10 < x < 10 , 0 < y < 20$. I wanted to write a program for this.
0
votes
3answers
31 views

Using Diophantine Equation to find the solutions of another equation

If $17x+51y=85$, find the value of $19x+57y$ I know I could use substitution and figure this out but i wanted to use Diophantine equation. I'm just a little confused because I know that $\gcd(17,51)=...
0
votes
1answer
28 views

Solutions to simple Diophantine Equation [closed]

Find all solutions to the equation $3^x - 3^y = 3$ where $x, y$ are positive integers (or show there are none). I know there are no solutions to this, so I know I can set up the proof for 2 cases: $x ...
2
votes
0answers
60 views

Question about “Primes of the form $x^2+ny^2$”: Proper ideals are invertible

I am reading through Cox's book Primes of the form $x^2+ny^2$ and I am stuck with some proofs in Chapter 7 (I have the 2nd edition). There, the author presents the following Lemma: Lemma 7.5: Let $...
0
votes
1answer
38 views

Area of a digital disk

Considering the number of integer solutions of $x^2+y^2\le n^2$, a digital disk, which is obviously asymptotic to $\pi n^2$, how can we find a tight upper bound of the form $$an^2+bn+c\ ?$$ The ...
0
votes
1answer
37 views

What property of rational solutions of a Diophantine equation makes this graph impossible?

Yesterday I read up on Diophantine equations and the property in which two rational solutions of such an equation make a line that gives a third rational solution to the equation. I was thinking about ...
2
votes
2answers
70 views

Need help to understand this solution of $2^mp^2 +1=q^5$

The Question- Find all triples $(m,p,q)$ where $ m $ is a positive integer and $ p , > q $ are primes. ****$2^mp^2 +1=q^5$**** After trying my best to solve this problem, and progressing a ...
1
vote
1answer
102 views

Solving a Diophantine equation $3a^3+3b^3=a^3+c^3$

A friend of mine gave me the following problem Find all integers solutions to $$3a^3+3b^3=a^3+c^3$$ Of course $(0,0,0)$ is a solution, and I think that there are no others, but I can’t prove it. I ...
1
vote
1answer
40 views

Solving the Diophantine equation $7x+4y=100$

I have attempted to solve this Diophantine equation: But the solution I got is different from the one given above, my general solution is: $x = -100 + 4t$ & $y = 200 - 7t$ , is the solution ...
4
votes
2answers
44 views

If $x, y, z$ are three distinct positive integers, where $x + y + z = 13$ and $xy, xz, yz$ form an arithmetic…

If $x, y, z$ are three distinct positive integers, where $x + y + z = 13$ and $xy, xz, yz$ form an increasing arithmetic sequence, what is the value of $(x + y)^z$ ? I've been trying to solve this by ...
0
votes
2answers
77 views

$1234x+567y=89$

I need to solve this with congruence. What I already done is : $$1234x+567y=89$$ $$1234x\equiv 89\bmod 567$$ $$1234\cdot17x\equiv 89\cdot17\bmod 567$$ $$20978x\equiv 1513\bmod 567$$ $$x\equiv -1513\...
2
votes
3answers
87 views

Number of possible integer values of $x$ for which $\frac{x^3+2x^2+9}{x^2+4x+5}$ is integer

How many integer numbers, $x$, verify that the following \begin{equation*} \frac{x^3+2x^2+9}{x^2+4x+5} \end{equation*} is an integer? I managed to do: \begin{equation*} \frac{x^3+2x^2+9}{x^2+4x+5} ...
0
votes
1answer
39 views

Diophantine equation $dx^2-y^2=d-1$ for non-square $d$

For $d=2$ or $3$ the continuous fractions for $\sqrt 2$ and $\sqrt 3$ help (at least partially), but for $d\ge 5$, this doesn't seem to work. Is there any known solution or technique to solve this ...
0
votes
1answer
21 views

exponential = polynomial diophantine equation

Let $a$ be a fixed constant like 2 or 3 (for example). Let $P(x)$ be a single variable polynomial in x with integer coefficients. What is known about the natural number solutions $(x,y)$ of $$P(x) = ...
1
vote
3answers
56 views

Solving the diophantine $y^3=2x^2$

I am trying to find the general solutions to the diophantine $y^3=2x^2$. Usually I can solve these using coprime/prime factorisation methods but I have tried and cannot seem to get the correct general ...
1
vote
1answer
23 views

Solvability of a Diophantine equation in mod 8 class

How does one solve an equation of the form $$ 3^a - b^3 \equiv 1 \ ({\rm mod}\ 8)? $$
0
votes
2answers
48 views

Pirates and Bag of Coins [duplicate]

A group of pirates with 17 members steal a bag of golden coins. When they share their loot evenly, it left with 3 coins. When they argue who should get the remaining of the coins, one of them is ...
1
vote
1answer
57 views

Find $1 \le a < b \le n$ such that $a\cdot b + a + b = n\cdot (n+1)/2$

Find $1 \le a < b \le n$ such that $$a\cdot b + a + b = \frac{n\cdot (n+1)}2$$ Is there a more efficient way than picking $a$ or $b$ and trying all values between $1$ and $n$ ?
-1
votes
3answers
111 views

Integer Solutions to a Two-Sheeted Hyperboloid

During some free time I had, I was wondering how to find the integer solutions $(x,y,z)$ to this generalized equation: $$z^2=axy+bx+cy+d$$ I am specifically looking for ways that do no involve ...
1
vote
1answer
35 views

Difference between diophantine equations $ax + by = c$ and $ax - by = c$

I had just solved this problem of Sierpinski Prove that the sequence $2^n - 3, \; n = 2,3,4, \dots$ contains infinitely many terms divisible by $5$ and $13$ but no terms divisible by $5\cdot 13$...
0
votes
1answer
38 views

How do you generate for a given solution for a linear diophantine equation more solutions

How can I generate for a given solution of a linear diophantine equation all solutions? For example let $21x+12y+9z=9$. I found one solution to be $(-3+3t,6-6t,t),t\in\mathbb Z$. How can I generate ...
0
votes
0answers
17 views

General Solution of Linear Diophantine Equation with 2 and more than 2 variable

We know the theorem of linear diophantine equation from Bezout's identity that the solution is in ordered pair form: $\left(x + m \dfrac{b}{\text{gcd}(a,b)}\,,\,y-m \dfrac{a}{\text{gcd}(a,b)}\right)$ ...
1
vote
1answer
30 views

Diophantine with primes factorials

Let $p\in \Bbb{P},\alpha\in \Bbb{N}$. Find all of the solutions of the following equation: $$(p-1)!+1=p^{\alpha} \ \ \ ,p>6$$ My attempt We can rewrite the equation as follows: $$(p-1)!=p^{\...
1
vote
4answers
101 views

Diophantine equation of power 2

I want to solve the following equation: $x^2-17y^2=104$ where $x,y$ are integers. I don't know how to proceed here. I have tried with mod $13$ and mod $17$ but it doesn't work. Thanks in advance!
3
votes
2answers
89 views

Equation Involving Ratios

In some of my research, I found multiple equations of this form: $$\frac{ax+b}{cx+d}=k$$ where $a,b,c,d$ are all non-zero integers. Is there a way (that doesn't include factoring or checking within a ...
1
vote
2answers
60 views

Integer solutions to $x^2 - 15 y^2 = 15$

How would one show that $x^2 - 15 y^2 = 15$ has no integer solutions ? I got that $x = \pm \sqrt{15 (y^2 +1) }$ and then WLOG I assume that $x \in \mathbb{N}$ and $x = \sqrt{15 (y^2 +1) }$. From ...
1
vote
2answers
96 views

Prime numbers of the form $p=m^2+n^2$ such that $p \mid m^3+n^3-4$

Find all prime numbers $p$, for which there are positive integers $m$ and $n$ such that $p=m^2+n^2$ and $p \mid m^3+n^3-4$. I simplified this a little bit. $$m^2+n^2 \mid m^3+n^3-4 =(m+n)(m^2+n^2-mn)...
4
votes
1answer
95 views

A bizarre expression for cardinality involving summation of roots and floor function

Show that number of triples $(a,b,c)$ with $a,b,c\in [1,n]$ such that $ab=c$ is given by $$\bigl|\bigl\{(a,b,c)\in [1,n]^3:ab=c\bigr\}\bigr|=2\sum_{i=1}^{\left\lfloor\sqrt{n}\right\rfloor}\Big(\...
2
votes
2answers
117 views

Number of solutions to Markov Diophantine equation mod $p$

I am interested in the number of solutions $(x,y,z)\pmod{p}$ (with $p\ne2,3$) to $$x^2+y^2+z^2-3xyz\equiv0\pmod{p}$$ See Markov numbers. The solutions are ordered triples and do not include $(0,0,0)$. ...
-1
votes
2answers
32 views

find solutions to following equations?

This is an interview question. There are a total of 100 coins of values 5, 1, 0.2. I need to find a solution that satisfies the following constraints. 1) The total number of coins must be 100. 2) ...
0
votes
0answers
25 views

An equation defined by norm

Let $f$ be an Eisenstein polynomial of degree $n$ and the prime $p$. $\alpha$ is a root of $f$. Let $\mathbb{Q}(\alpha)=K$, Prove that for any $\gamma\in O_K$, there exist $a\in \mathbb{Z}$, such that ...
2
votes
1answer
75 views

Number of integral solutions to elliptic curve $\binom n2=\binom m3$.

I am wondering if there are infinitely many integral solutions to the equation: $$ {n \choose 2} = {m \choose 3}. $$ Also, do the solutions have a general form? From what I know, this is an elliptic ...
0
votes
0answers
21 views

Books on polynomial Pell equations

I am interested in polynomial Pell equations and their links with polynomial continued fractions, Padè approximants and the conditions under which they are solvable. Could you suggest me some books ...
1
vote
0answers
35 views

Solve the equation below for all m and n which are positive integers

Find all positive integers $m$ and $n$ such that $$m^2+2\cdot 3^n=m(2^{n+1}-1).$$
3
votes
1answer
73 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
60 views

All natural number solutions of the equation [closed]

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
47 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
52 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 ...
0
votes
2answers
77 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
45 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
108 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
151 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 $N$. ...
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 ...