Questions tagged [diophantine-equations]
Use for questions about finding integer or rational solutions to polynomial equations.
5,583 questions
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Inferring corollary from a theorem in Shorey-Tijdeman book on Exponential Diophantine Equations
I am studying the book 'Exponential Diophantine Equations by T.N. Shorey and R. Tijdeman.' I have a doubt in chapter 1.
Let $P \geqslant 3$. Let $p_1, \ldots, p_s$ be given (rational) prime numbers ...
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Weil height of algebraic number
We know the basic fact that if $n \equiv 0 \pmod p$ then $|n| \geq p$ (provided $n \neq 0$). Let $\alpha$ be a non-zero algebraic number and suppose that there is a prime ideal $\mathcal{P}$ in $\...
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Solutions to $a(x^n-x^m)=(ax^m-4)y^2$ [duplicate]
This problem comes from the 2002 Art of Problem Solving China Team Selection Test. It is problem 3 on quiz 4.
Find all groups of positive integers $(a,x,y,n,m)$ that satisfy $a(x^n-x^m)=(ax^m-4)y^2$ ...
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Show that $x^2+4y^2=z^4$ has no integer solutions if $x, y, z >0$ [closed]
I want to show that $x^2+4y^2=z^4$ has no integer solutions if $x, y, z >0$.
If $x$ and $z$ are even, regardless of $y$'s parity, then I can show by infinite descent, but if $x$ and $z$ are odd I'm ...
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What are the known integer solutions to $a^3+b^3=n^5 $? Found:$3^3+6^3=3^5$ to be smallest positive integeral solution.
I came across an interesting and seemingly rare integer identity:
$$3^3 + 6^3 = 3^5.$$
That is:
$$27 + 216 = 243.$$
This satisfies the equation:
$$a^3 + b^3 = n^5.$$
This seems unusual to me since:
...
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Divisor classes of Cubic surface
I have been trying to understand cubic surfaces using the thesis On the Parametrization over $\mathbb{Q}$ of Cubic Surfaces by René Pannekoek. (Backup link.) On page 9 it talks about the Picard group ...
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Finding all integer points on the elliptic curve $y^2 = (x^3-x) / 6 + 1$
I am trying to find all integer solutions $(x,y)$ to the following equation:
$$
\frac{x^3-x}{6} + 1 = y^2 \quad \cdots\ (\ast)
$$
(Note: By setting $n=x-1$, this problem is equivalent to finding all ...
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Diophantine $(a^3 - a - 1)b = c^3 - c - 1$
How to solve the diophantine equation
$$(a^3 - a - 1)b = c^3 - c - 1$$
for integers $a,b,c>1$ ?
Is the expected number of solutions ,denoted $f(c)$, for $a_n$ such that $(a_n^3 - a_n - 1)b_n = c_n^...
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Diophantine equation $1176x=y^{4}$
My Question: Solve the following indefinite equation.
$$1176x=y^{4}$$
My attempt: $1176=2^3\cdot3\cdot7^2$, so we have $2^3\cdot3\cdot7^2\cdot x=y^{4}$. Further simplification yields:
\begin{equation}
...
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Solutions for Diophantine equation $a^2+b^3+c^4=d^5+e^6+f^7$ [closed]
I have infinitely many integer solutions for such an equation at a lower level, having the form
$$
a'^2 + b'^3 = c'^4 + d'^5
$$
For example,
$$
255^2 + 8^3 = 16^4 + 1^5
$$
and
$$
14332523^2 + 7776^3 = ...
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$p$-adic solution to $x^2 + x^4 + x^8 = y^2$
So I found a rather introductory video of $p$-adic numbers of Veritasium that you could find here. The video included an outline of a solution to the Diophantine equation $x^2 + x^4 + x^8 = y^2$.
...
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How to solve the diophantine $94 \cdot 10^{16} + 1 = a^2 + b^2$? [duplicate]
Let $a,b$ be positive integers. How to efficiently solve the diophantine equation
$$94 \cdot 10^{16} + 1 = a^2 + b^2$$
?
The solutions $a,b$ are unique (up to switching $a$ and $b$) because $94 \cdot ...
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Proving there are no integer solutions to the following equation
The context is the use of $2$-isogeny descent to calculate the rank of the elliptic curve $E:y^2 = x(x^2 +x -7)$. One of the steps involves finding all the integer solutions $(r,m,l,n)$ of:
$$r^2 l^4 +...
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Prove that the product of four consecutive positive integers is not a power of any positive integer with exponent greater than or equal to 2.
For the case of the product of two or three consecutive integers, this is easy to prove (using coprimality and expressing 1 in terms of the factors quickly leads to a contradiction). However, for the ...
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What is the maximum number of distinct prices whose product equals their sum?
A price is a number $ \frac{x}{100}$ with a positive integer $x$. How many distinct prices can there be such that their product equals their sum, i.e., what is the largest $n$ such that positive ...
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Solvability of the negative Pell's equation $x^2-dy^2=-1$ where $d=a^2\pm1$ or $d=a^2\pm2$
Tinkering around with the Euler-Muir theorem, I noticed a pattern in the palindromes and Euler-Muir polynomials corresponding to $d=a^2\pm1$ and $d=a^2\pm2$ and was able to prove that
$x^2-dy^2=-1$ ...
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Rational points on a plane quartic curve
I'm trying to find the rational points on the plane quartic curve
$$(y^2-x^2)^2 - 2(y^2+x^2) + \frac{409}{9} = 0$$
Equivalently, the homogenized curve is
$$(y^2-x^2)^2 - 2z^2(y^2+x^2) + \frac{409}{9}z^...
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$4n^2$ is not a sum of four distinct odd perfect squares
So I was seeing an answer to question where the op ask if $4^n$ can be written as a sum of four distinct squares. I solved that problem but after I change the problem and tried to solve whether $4n^2$ ...
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Three natural numbers whose sum is equal to their product. [duplicate]
A few days ago my professor told us that
$ \tan^{-1}x + \tan^{-1}y + \tan^{-1}z = \pi $ then xyz = x+y+z (we proved it).
He asked us to find all the possible natural numbers that satisfy the relation
$...
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Are there infinitely many $n\in\mathbb{N}$ which equals the sum of square of its smallest $k$ divisors?
For $n\in\mathbb{N}$ , let $d_1<d_2<…<d_m$ be all divisors of $n$, where $d_1=1$, $d_m=n$, and $m=\tau(n)$ is the number of (positive) divisors of $n$ .
Questions: There are two questions ...
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Find all positive integer solutions to the Diophantine equation $3x^{2}- 5y^{2}= 7$
Problem: Find all positive integer solutions to the Diophantine equation:
$$3x^{2}- 5y^{2}= 7$$
Attempted approach:
Rewriting the equation:
$$x^{2}= \frac{5y^{2}+ 7}{3}$$
We want the right-hand side ...
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Suppose $a$ and $b$ are integers such that both $a + 3b$ and $3a − b$ are the squares of positive integers. What is the smallest possible value?
Suppose $a$ and $b$ are integers such that both $a + 3b$ and $3a − b$ are the squares of
positive integers. What is the smallest possible value of these squares?
Here is my approach:
Assume, $a+3b=x^...
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Solve $1+ x_{1}+ 2 x_{1} x_2+ \cdots+ \left ( n- 1 \right ) x_{1} x_{2}\cdots x_{n- 1}= x_{1} x_{2}\cdots x_{n}$
Let $n\in\mathbb{N}^{\ast}$ be arbitrary and fixed. Solve the equation $$1+ x_{1}+ 2x_{1}x_2+ \cdots+ \left ( n- 1 \right )x_{1}x_{2}....x_{n- 1}= x_{1}x_{2}....x_{n}$$
in distinct natural numbers $x_{...
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Existence of bounded solutions to a diophantine linear equation mod $p$
Given a large prime $p$ and integers $a,b\in \mathbb{Z}_p$, consider the equation,
$$
ax+b \equiv y \bmod p,\quad (1)
$$
where we impose the restriction that the solution must satisfy $$0<x,y<p^{...
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Solving $((ax+b)\bmod p)<\sqrt p$ for integer $x\in[0,\sqrt p)$
Given prime $p$ and integers $a,b\in[1,p)$, what's a method to solve $((ax+b)\bmod p)<\sqrt p$ for integer $x\in[0,\sqrt p)$ ?
I'm interested in huge $p$, thus trying all $x$ from $0$ to $\sqrt p$ ...
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Find all $x, y,z\in \Bbb{Z}^+$ such that $\dfrac{1}{x}+\dfrac{2}{y}+\dfrac{3}{z}=4$
Find all $(x, y,z), x,y,z\in \Bbb{Z}^+$ such that
$$\dfrac{1}{x}+\dfrac{2}{y}+\dfrac{3}{z}=4\quad\quad\quad\quad(1)$$
I tried $(1)\implies yz+2xz+3xy=4xyz \implies x(2z+3y) \equiv 0\pmod{zy}$.
I ...
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Transforming $\sum_{j= 1}^{i}jx_{j}\equiv n\pmod{b}$ into $\sum_{j= 1}^{i}x_{j}q^{j- 1}\equiv {n}'\pmod{a}$ for constrained coding
I am working on a constrained coding problem where I need to construct a sequence $\left ( x_{1}, x_{2}, \ldots, x_{n} \right )$ so that:
$$\sum_{j= 1}^{i}jx_{j}\equiv n\pmod{b}$$
where $x_{j}\in\left\...
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Using Rachinsky quintets and others to find $8\text{th}$ powers $x_1^8+x_2^8+x_3^8+x_4^8+x_5^8=y_1^8+y_2^8+y_3^8+y_4^8+y_5^8\,$?
I. Rachinsky quintets
In this previous post, it was shown that special Pythagorean quadruples can lead to $6$th powers. We go higher and use special Rachinsky quintets that lead to $8$th powers. In ...
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Solving $ab + bc + ca = xy + yz + zx = p$ with heuristics?
Euler’s idoneal numbers (also called suitable numbers or convenient numbers) are the positive integers D such that any integer expressible in only one way as $x^2 \pm Dy^2$ (where $x^2$ is relatively ...
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Diophantine equation $x^2+y^2=c$ [closed]
Question: Is it certain that the Diophantine equation $x^2+y^2=c$ has solutions? Where $c$ is a positive integer.
My thoughts:
If $c$ is a perfect square number, then this corresponds to the ...
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Controlling the $GCD$ of two numbers
I have a diophantine equation of the form
$$ax+by=c$$
We know that this equation has solutions if and only if $gcd(a,b)$ divides $c$. I want to control the equation in which it has no solutions by ...
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How do I check if I found all generators of the solution module of a linear homogeneous diophantine equation?
I have a linear diophtantine equation
$$28x + 30y + 31z=365$$
and want to find all its solutions. Using the Gauss-Euclid algorithm, I start with the matrix
$$\pmatrix{28 & 1 & 0 & 0 \\ 30 &...
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Integer solutions to $ab+1=x^2, ac+1=y^2, bc+1=z^2,$ and $\frac{x+z}y= \text{Integer}?$
(April 2025, three updates below.)
I. Problem
We are looking for positive integers $(a,b,c)$ and $(x,y,z)$ where $a<b<c\,$ solve,
$$ab+1 = x^2\\
ac+1 = y^2\\
bc+1= z^2$$
with the added ...
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Are there any positive integer solutions for $2^l = \frac{3^tk - 1}{2^tk-1}$ other than $l,t,k = 1$ [duplicate]
I'm currently working with the equation $$\frac{3^tk-1}{2^tk-1}$$, where $t, k \geq 1$ and both variables take integer values. I'm trying to show that the only case where this equation is equal to a ...
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On the quartic system $a^4+b^4+x_1^4=2y_1^4,\;b^4+c^4+x_2^4=2y_2^4,\;c^4+a^4+x_3^4=2y_3^4$
I. Quadruples
In the previous post about $x^4+y^4+z^4=3t^2$, quite a lot of solutions share common terms such as the "cyclic",
$$\color{blue}{11}^4+\color{blue}{23}^4+11^4\,=\,3u_1^2\\
\,\...
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Find all integer solutions for $x^2 - y^2 = z^n$ where $n > 2$.
I am not sure how to go about finding the solutions for this. I know Pell type equations are of the form $x^2 - Dy^2 = a$ along with the fact that the largest solutions for the case where $z = 1$ is ...
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Finding more integer solutions to $x^4+y^4+z^4=3t^2$ below a bound?
If $x^4+y^4+z^4=mt^2$ has one solution $xyzt\neq0$ then, using an elliptic curve, one can generally find infinitely many primitive solutions. We focus on $m=1,2,3$.
I. m = 1
Fauquembergue found that ...
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Is the equation $m^3+4(mn)^3-8n^3=827, $ $\forall m,n\in\mathbb{N}$ sufficient to find the value of $m^2+n^2$?
If $m,n\in\mathbb{N}$ and:
$$m^3+4(mn)^3-8n^3=827$$
find the value of $m^2+n^2$.
I tried to modify LHS to
$$(m^2+n^2)(m-8n)+8m^2n-mn^2+4(mn)^3=827.$$
\begin{align}
&(m^2+n^2)(m-8n)+mn(8m-n)+4(mn)^...
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Finding sixth powers like $65^6 + 52^6 + 15^6 = 36^6 + 67^6 + 37^6$ using Pythagorean quadruples $65^2 = 52^2 + 15^2 + 36^2$?
I. Quadruples
In this post, we saw how infinitely many Pythagorean triples such as,
$$3^2+4^2-5^2=0\quad$$
can lead to 4th power equalities,
$$2^4+2^4+4^4+3^4+4^4=5^4\quad$$
It turns out that ...
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Finding equalities like $2^4+2^4+4^4+3^4+4^4 = 5^4$ using Pythagorean triples?
I. Ramanujan's parameterizations
Ramanujan gave just two quadratic parametrizations to,
$$a^4+b^4+c^4+d^4+e^4 = f^4$$
where $f$ is always integrally divisible by $5$, one of which is,
$$(2x^2+12xy-6y^...
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About the existential definition of binomail coffecients
The following result was proved in: Martin Davis, Hilbert's Tenth Problem is Unsolvable, The American Mathematical Monthly, Vol. 80, No. 3 (Mar., 1973), pp. 233-269. (https://www.math.umd.edu/~laskow/...
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Solving nonlinear diophantine equation
I have a nonlinear diophantine equation in four variables. The equation consists of the four variables $w, x, y, z$ and their combinations: $wx, yz, wz, xy$
The constants $A_n$ and $B_n$ are Bezout's ...
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3
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Polynomial with integer coefficients and Gaussian integer roots [closed]
Is there a name for a polynomial with (real) integer coefficients, and for which all of the roots are Gaussian integers (i.e. both real and imaginary parts are integers)?
what might be a test for ...
2
votes
4
answers
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Solving $\max(m^4,n^4)-56m^2n^2=2025$
I am trying to solve this problem, but unfortunately I cannot seem to be able to breakthrough? Without giving me the answer, could someone please give me some guidance on how to start solving? Thank ...
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Complementary Primitive Pythagorean Triples: A Systematic Construction Method
I have found a method to generate pairs of primitive Pythagorean triples (PPTs) that satisfy a complementary relationship. Specifically, given a PPT $a, b, c$ there exist two other PPTs $a_{c1}, b_{c1}...
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Do there exist any solutions to $n\sum\limits_{i=0}^{c-1}(4/3)^i=\sum\limits_{i=0}^{c-1}(4/3)^i2^{m_i}$
Context
This is a question that concerns the Collatz conjecture. Before you go after me for being a crank, let me just preface this by saying that this is a very direct corollary of a result proven by ...
2
votes
0
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Can this problem be solved with elliptic curves?
I’m working with a Diophantine equation where I express positive integers as $m^2 + n^3$ in three different ways(to show that there are infinite $c^2$ that can be expressed in this way). After finding ...
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what are all the rational points on the curve $ax^2-bxy +cy^2=d$ for integer values of $a,b,c,d$
I was playing around on desmos and found the equation $x^2-xy+y^2=1$ very interesting when you look at its rational points, which are $(0,1),(0,-1),(1,1),(-1,-1),(0,1),(0,-1)$. I then tried changing ...
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Show that 1 and -1 are the only units in the ring of integers of $\mathbb{Q}[\sqrt{m}]$ for $m$ squarefree, negative, and not equal to -1 or -3.
For reference this is exercise 13 in chapter 2 of Marcus's book Number Fields.
For the sake of simplicity I'll work in the case $m \not\equiv 1 \mod 4$ so that the ring of integers of $\mathbb{Q}[\...
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An exponential diophantine equation
I am interested in solving the following Diophantine equation: $3^x-2^y=121.$ I have considered the equation modulo 3 and 4, which allowed to deduce that $x$ is even and $y$ is odd. Hoawever, the ...
