Questions tagged [diophantine-equations]
Use for questions about finding integer or rational solutions to polynomial equations.
5,583 questions
2
votes
2
answers
237
views
Generating a recursive formula for solutions to this equation
Given an equation:
$y^2 = -x^3 + ((6n+3) \cdot x - (36n^3 + 54n^2 + 27n - 4))^2 $ where $n,x,y$ are non-zero integers,
for which value(s) of $n$ and $x$ is the term $\dfrac{(2(36n^3 + 54n^2 + 27n - 4))...
- 147
1
vote
2
answers
359
views
Find the least integral value of $x$ such that $x^2 = y^2 + 2000$.
I successfully solved this question on my own by using a different method, but when I read the suggested solutions, I don't really understand the following part.
$x$ is the least when $y$ is the ...
- 117
7
votes
2
answers
685
views
Find $n\in\mathbb{Z}^+$ such that there exists $x\in\mathbb{Z}^+$ where $4x^n+(x+2)^2$ is a perfect square
Find $n\in\mathbb{Z}^+$ such that there exists $x\in\mathbb{Z}^+$ where $4x^n+(x+2)^2$ is a perfect square.
I know and can prove that $n=1$ is not a solution, for $n=2$ there's $x=4,30,\dots$, for ...
- 9,890
4
votes
1
answer
163
views
Form of elliptic curve where torsion of order 4 is apparent
While studying elliptic curves, I came upon this form for an elliptic curve with a point of order $3$: $x^3+y^3+z^3=3xyzp$. Unlike other forms of elliptic curves, the order $3$ torsion is apparent: ...
- 509
0
votes
2
answers
82
views
Don't understand just this one part of a solution for pell's equation
$x^2-dy^2=1$
$(x-y\sqrt{d})(x+y\sqrt{d})=1$
So $(x_1,y_1)$ is the smallest positive integer value for which this equation holds, which are also the fundamental solutions.
$(x_1-y_1\sqrt{d})(x_1+y_1\...
- 89
1
vote
1
answer
101
views
Integer Solutions to $m^3+m^2+m=l^2$ [closed]
I tried looking at the Problem modulo 4. Trivially one gets, that $m$ hast to be an even integer. After this step I do not see any further conclusion. Thank you in advance, it is a problem resulting ...
1
vote
2
answers
130
views
Diophantine equation: finding when a degree $2$ expression in $x$ and $y$ is a perfect square.
For what values of $x,y \in \mathbb{Z}$ the expression $(2x-y)² + 4x + 1 - 6y$ is square of an integer?
I figured out that the expression equals $1$ for $$(x,y) = (n(n+3)/2 , n(n+1)) \quad \text{and} ...
- 1,520
3
votes
2
answers
251
views
$a^2 = 3^b + 37$
Find all solutions in integers to $a^2 = 3^b + 37$.
I came up with this problem myself. (I chose $37$ so there would be a relatively large solution.)
But is there a nice way to approach this problem? ...
- 524
1
vote
1
answer
133
views
Integer points of the expression $\frac{4q}{2q-pq+2p}$ where $p,q\in\mathbb Z$.
This expression is encoded with a geometric meaning as follows. Consider a polyhedron with congruent polygon faces, each with $p$ sides. If there are $F$ number of polygons, that is, faces of the ...
- 2,603
18
votes
4
answers
965
views
Criteria for a number being a square-pyramidal number
The $n$-th square-pyramidal number is the sum of the first $n$ squares:
$$
P_n = \sum_{k=1}^n k^2 = \frac{n(n+1)(2n+1)}6.
$$
Suppose we have a number $K \in \mathbb{N}$. How can we test if $K$ is a ...
- 435
0
votes
0
answers
84
views
How to Solve a modified elliptic curve problem
In studying Diophantine equations, the following was thrown up (as an example):
$my^2 = 28x^3 - 5x^2 + 2805x + 2323$
I am trying to solve for $x$ where $x$, $m$ and $y$ are all unknown.
Another ...
- 160
6
votes
3
answers
286
views
Find all pairs of primes $p,q$ such that $16p^2+13q^2+5p^2q^2$ is a perfect square.
Find all pairs of primes $p,q$ such that $16p^2+13q^2+5p^2q^2$ is a perfect square.
Hello, I am Alek and I have this as a part of my homework. I know the problem must be easy-solvable, because it's ...
user1495349
0
votes
0
answers
98
views
How is Kroneker's Theorem different than the Lonely Runner Conjecture?
While thinking about adding a pairwise coupling between the speeds of the lonely runner conjecture runners it seems we can rediscover a version of the Kuramoto model and if there is no coupling we ...
- 1,101
2
votes
0
answers
57
views
How to get the integer solutions of $x^2+1=2y^3$ [duplicate]
What are the integer solutions of $x^2+1=2y^3$? I believe that (1,1) is the only solution.
- 41
2
votes
2
answers
129
views
Finding all integral points of an equation $x^2+3y^2=784$
Given a curve $C$, defined by the equation $x^2 + 3y^2 = 784$, find all the integral points on that curve, i.e. points $(m,n$) such that $m,n \in \mathbb{Z}$.
How do you do this? I tried assuming $m,n$...
2
votes
1
answer
116
views
How to check if a n equation has rational solution?
For example, how do I know if $(x-y)(x+y)=\frac{7}{2}$ has a solution $(x,y) \in \mathbb{Q}$? I can compute that the solution must be $y=\pm \frac{\sqrt{-7 + 2 x^2}}{\sqrt{2}}$, but how do I know ...
- 133
1
vote
0
answers
293
views
Expressing $2^n$ as a Sum of Four Squares Using Quaternion Norms
The problem is to find the positive integer solutions to the Diophantine equation
$2^n = a^2+b^2+c^2+d^2$. I was able to solve it using some modular arithmetic and a few case checks. The solutions I ...
4
votes
1
answer
536
views
Solution for find all positive integers a,b,c that satisfies $a^3+b+1=(b^2-c^2)^2$
I found this in a manga I was reading and it already has the answers laid out:
$(6,8,7); (46,7,19)$
I was wondering how someone even starts with this problem, I got to
$a^3+b+1=(b^2-c^2)^2$
$(a+1)(a^2-...
2
votes
1
answer
149
views
Euler bricks with same smallest side.
An Euler brick is a triple of positive integers $(a, b, c)$ with $a\leq b\leq c$ such that $a^2+b^2, a^2+c^2, b^2+c^2$ are all squares. We call $a, b, c$ the side lengths of said Euler brick.
As a ...
- 3,697
10
votes
2
answers
1k
views
Are there three distinct Pythagorean triples on six integers?
A Pythagorean triple is a solution to the equation $x^2+y^2=z^2$ over the positive integers. We write a Pythagorean triple as a tuple $(a, b, c)$, where $a\leq b\leq c$. I am interested in the ...
- 3,697
5
votes
2
answers
173
views
symmetric diophantine equation with $6$ variables
I want to find every solutions $\{x,y,z\},\{u,v,w\}\subset\mathbb{N}$ with $\{x,y,z\}\neq \{u,v,w\}$, $|\{x,y,z\}|=3=|\{u,v,w\}|$, and they fulfill the following equation
$$(x+y+z)(uv+uw+vw)=(u+v+w)(...
- 91
0
votes
0
answers
43
views
Prove that there are no integer solutions for $x^3+2y^3=4z^3$ [duplicate]
Prove that there are no integer solutions for $x^3+2y^3=4z^3$
My attempt:
Assume by contradiction that the equation $x^3+2y^3=4z^3$ has an integer solution. We'll choose the $(x,y,z)$ solution with ...
- 93
-1
votes
1
answer
113
views
Integer solutions to a Diophantine equation $2(q^2+s^2)(p^2+r^2)=(p^2-r^2)^2+(q^2-s^2)^2=(2pr)^2+(2qs)^2$ [closed]
I am trying to solve this Diophantine equation to find integer solutions but not succeed I need help to find integer solutions for
$2(q^2+s^2)(p^2+r^2)=(p^2-r^2)^2+(q^2-s^2)^2=(2pr)^2+(2qs)^2$
- 27
2
votes
2
answers
171
views
How do I find all natural numbers pairs $(a,b)$ such that $\frac{a^3 - b}{a^3 + b} = \frac{b^2 - a^2}{b^2 + a^2}$, with $a \le 10^5$ and $b < 10^5$?
The title isn't long enough, there are also conditions where $a \le 10^5$ and $b < 10^5$.
I've tried to pair $a^3 - b = b^2 - a^2$ and also $a^3 + b = b^2 + a^2$.
And I ended up getting $a = b = 1$....
- 49
7
votes
1
answer
380
views
Diophantine equation on 2 unknowns : $x^2 + y^2 = 2023(x-y)$
The question states: Find positive integers $x$ , $y$ such that $x^2 + y^2 = 2023(x - y)$.
The source is "Gazeta Matematica," a magazine meant for preparation for the national olympiad.
Now ...
1
vote
1
answer
240
views
Help me solve this Diophantine equation $xy(x^2-y^2)=pq(p^2-q^2)=rs(r^2-s^2)$
I am solving this Diophantine equation
$$xy(x^2-y^2)=pq(p^2-q^2)=rs(r^2-s^2)$$
where $(x^2+y^2)^2+(p^2+q^2)^2=2(r^2+s^2)^2$,
but I am not able to proceed.
I use Euler method $pq(p^2-q^2) = rs(r^2-s^2)$...
- 27
1
vote
4
answers
137
views
$\frac{(a + b + c)^2}{a^2 + b^2 + c^2}$ integral with $1 \le a,b,c \le 30$
My friend recently sent me the following problem:
For how many ordered triples of positive integers $(a,b,c)$ with $a,b,c \le 30$ is $\frac{(a + b + c)^2}{a^2 + b^2 + c^2}$ an integer?
We’ve made a ...
- 469
1
vote
0
answers
74
views
Two questions on two-term Machin-like formulae
It is known that there are exactly four two-term Machin-like formulae that allow for the estimation of $\pi$:
$$
\begin{aligned}
{\tfrac {\pi }{4}}&=\arctan {\tfrac {1}{2}}+\arctan {\tfrac {1}{3}}\...
- 161
1
vote
1
answer
125
views
Generating Function Calculation
Suzanne went to the bank and withdrew dollar $800$. The teller gave her this
amount using dollar $20$ bills, dollar $50$ bills, and dollar $100$ bills, with at least
one of each denomination. How many ...
- 2,674
2
votes
2
answers
279
views
How can we find general solution of $8x^2-7=y^2$ in integers.
$8x^2-7=y^2$
I tried this:
$8(x^2-1)=y^2-1$
$8(x^2-1)=(y-1)(y+1)$
Let $x^2-1 = a*b$
I made cases:
$y-1 = 8a , y+1 = 1b$
$y-1 = 4a , y+1 = 2b$
$y-1 = 2a , y+1 = 4b$
$y-1 = 1a , y+1 = 8b$
but wasn't ...
- 73
-1
votes
2
answers
117
views
Squares in arithmetic progression form [duplicate]
I was reading a pdf (Progressions of Squares by Tom C. Brown1, Allen R. Freedman2, and Peter Jau-Shyong Shiue2) (link: https://www.sfu.ca/~vjungic/tbrown/tom-9.pdf) and there the author on Page no. 1, ...
- 145
-2
votes
1
answer
118
views
Positive integral solutions to ternary form $Ax^ky-Bxy^k=Cz^{k+1}$ [closed]
Given any choice of integers $A,B,C,k\in\Bbb{Z}$ with $C,k>0$ and $A>B$, does the ternary form
$$Ax^ky-Bxy^k=Cz^{k+1},$$
have a solution in the positive integers?
Original question:
Given the ...
- 31
0
votes
1
answer
102
views
Ternary Nonhomogeneous Diophantine Equation
The known ternary Diophantine equation is as follows
\begin{align*}
(1 - 2X) (X^2 - X + Y^2) + 2 Z^3=0.
\end{align*}
This inhomogeneous ternary Diophantine equation has two singular points, and these ...
- 1,259
2
votes
0
answers
126
views
What is the relationship between rational points and integer points on algebraic varieties?
Let $C$ be a plane conic defined over the integers by some equation $f(x,y)=0$. If $C$ has a rational point $P$, then in fact $C$ has infinitely many and all of them can be found by drawing lines with ...
- 234
-1
votes
1
answer
170
views
My question Was to know how to solve such questions: find all integer solutions for $x^{6} + x^{3}y = y^{3} + 2y^{2}$ [closed]
how to solve such questions: find all integer solutions for $$x^{6} + x^{3}y = y^{3} + 2y^{2}$$
I couldn't figure out a way to solve it. My idea was to take $x^{3}=k$
and then solve the quadratic and ...
5
votes
4
answers
260
views
To construct a pythagorean triple out two linear equations
My question may appear a bit odd, maybe it does not even make sense.
The following is my problem: Given two linear equations $f(t)$ and $g(t)$, with $f(t)^2-g(t)^2 =Y^2$. Find $t$, so that $Y$ is an ...
- 69
2
votes
1
answer
94
views
Algebraic expression for the geometric group law on the unit circle
A group law can be defined on the unit circle by the following geometric definition
where $N$ is an arbitrary, but fixed point on the circle (which plays the role of the identity element).
Assume ...
- 423
7
votes
1
answer
407
views
If $a>2b$ then $(a^2)!+b^2≠c^2$ and $4$ is the only square solution to the Brocard's problem
Let $f$ be the arithmetic function $f(x)=\left|a-\frac{x!}{a}\right|$ where $a$ is the divisor of $x!$ which is the closest to $\sqrt{x!}$.
For example, $f(5)=2$ since the divisors of $5!=120$ are: ${...
- 73
4
votes
1
answer
299
views
$x^2 + y^3 = 2024$ over the integers
How can you find all solutions to the equation $x^2 + y^3 = 2024$ over the integers?
This question came to me randomly, and checking with technology, there are a total of ten solutions:
$x = \pm 444, ...
- 524
1
vote
1
answer
103
views
Rational parametrization of $a^2+b^2-c^2-d^2=32$
Parametrization of $a,b\in\mathbb{Q}$ for $a^2-b^2=32$ is $a=p+\frac{8}{p}$, $b=p-\frac{8}{p}$ for $p\in\mathbb{Q-\{0\}}$.
Parametrization of the Pythagorean quadruple is $a=m^2+n^2-p^2-q^2$, $b=2(mq+...
- 157
1
vote
0
answers
43
views
Proving that solutions to equation in quadratic number fields are conjugates.
Say we have the equation $x^4 + y^4 = 8$ and we want to find the solutions to this in quadratic number fields.
By intuition, it would seem that solutions are conjugates. Namely, $x = a + b\sqrt{d}$ ...
0
votes
1
answer
102
views
How do you solve the equation, $\sqrt{20889 + 4 k}=n$, for integers solutions? [closed]
I was interested in knowing the integer solutions for the equation $\sqrt{20889 + 4 k}=n$. When I input that equation into Wolfram Alpha, it says the integer solution is the pair $(k,n)=(34,145)$. I ...
- 1
1
vote
4
answers
184
views
Solving the Diophantine equation $ab(a+b)=T$, where $a$ and $T$ are triangular numbers [closed]
How to solve the Diophantine equation, $ab(a+b)=T$, for positive integers, where $a$ is any given Triangular number & $T$ are also certain Triangular numbers?
As of, $(a,b,T)=(3,7,210), (3,82,...
0
votes
0
answers
112
views
Can $x^3 + 7$ Ever Be Square? [duplicate]
Can $x^3 + 7$ ever be a square integer, for integer $x$ ? If not, how can this be proved ?
I have tested with a quick Python script and have reached $x = $ approx. 14 billion, with no squares yet ...
-2
votes
1
answer
135
views
Prove that $6(6a^2+3b^2+c^2)=5d^2$ has no integral solutions [duplicate]
If $a,b,c,d$ are natural numbers, prove that $6(6a^2+3b^2+c^2)=5d^2$ has no solutions.
I did some simplifying and got that I need to prove $2a^2+b^2+3x^2=10k^2$ has no integral solutions where $a,b,x,...
-1
votes
1
answer
124
views
Find all non-negative integers $x^2+3y^2=z^2,\mathrm{gcd}(x,y)=1,x,y,z>0.$ [closed]
I use geometric methods to get the parametric form$\;x=p^2-3q^2,y=2pq,z=p^2+3q^2,\mathrm{gcd}(p,q)=1$, but it doesn't include the solution (1,1,2).
The process is as follows:
Elliptic equations $x^2+...
- 13
0
votes
0
answers
167
views
Find all positive integer triples $(x,y,z)$ such that $x^4+y^4=z^3+215z+1$
Find all positive integer triples $(x,y,z)$
such that $x^4+y^4=z^3+215z+1$
My Attempt
I began by analyzing the given equation:
$$x^4 + y^4 = z^3 + 215z + 1.$$
Here, $x$ and $y$ are positive integers, ...
-3
votes
1
answer
116
views
The diophantine $a(a+1)b(b+1) = c(c+1)d(d+1)$
Solve the diophantine equation
$$a(a+1)b(b+1) = c(c+1)d(d+1)$$
For nonzero integers $a,b,c,d$.
Solve the diophantine equation
$$a(a+1)b(b+1) = c(c+1)d(d+1)$$
For integers $a,b,c,d$ larger than zero....
- 18.4k
0
votes
0
answers
74
views
The Chinese Remainder Theorem and the yes/no question "for this general class of Diophantine Equations, is there an algorithm to find the solution"
I have recently been doing a side project and researching if there is a deep and beautiful mathematical structure in the process of recursion itself. My idea was: once we treat algorithms themselves ...
- 4,424
0
votes
1
answer
154
views
When are both $a^2 + b^2 + ab$ and $a^2 + b^2 - ab$ squares? [duplicate]
(Let an integer triangle be a triangle with only integer side lengths) I have been trying to find all pairs $m$ and $n$ so that both
The triangle with side lengths $m$ and $n$ with a $60$ degree ...
- 17