Questions tagged [diophantine-equations]
Use for questions about finding integer or rational solutions to polynomial equations.
5,583 questions
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Prove $\frac{x^{2^n}-1}{x^{2^m}-1}$ is not a perfect square when $n, m$ have different parity
I am trying to prove the following number theory problem:
Problem:
Let $n, m$ be positive integers with different parity (one is even, the other is odd) and $n > m$. Prove that there is no integer $...
0
votes
0
answers
70
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From sum of three cubes to Desboves/Selmer elliptic curve
Let given $n \in \mathbb{Z}^+$ and equation $x^3 + y^3 + z^3 = n$ over $\mathbb{Q}$.
Let $x = -\dfrac{4 a^3-4 nb^3+1}{3 b},y = \dfrac{a^3-nb^3+1}{3 b},z = \dfrac{a}{b}$, where unknowns $a,b\in \mathbb{...
- 1,830
2
votes
1
answer
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Finding inverse of a $3 \times 3$ matrix with Cayley-Hamilton, and Diophantine equation
I am trying to use this formula to find the inverse of a $3 \times 3$ matrix.
$$ \mathbf A^{-1} = \frac{1}{\det(\mathbf A)} \sum_{s=0}^{n-1}\mathbf A^{s} \sum_{k_{1}, k_{2},\dots,k_{n-1}} \prod_{l=1}^{...
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answer
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Quintic residues and $p=x^2+125y^2$
Euler conjectured (and Gauss later proved) that:
If $p\equiv 1\pmod 3$, $2$ is a cubic residue mod $p$ iff $p=x^2+27y^2$ for some integer $x$ and $y$.
If $p\equiv 1\pmod 4$, $2$ is a quartic residue ...
- 509
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0
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166
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Elementary way of proving that $p^2-9=y^3$ has no solution
For prime $p>6$, I am trying to show that $p^2-9=y^3$ has no solution with elementary methods. Factoring $p^2-9$ and quadratic residues doesn't seem to work. It would be nice if I could factor $y^3+...
- 424
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votes
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Rational solutions for $x+y+\frac 1x+\frac 1y=2025$ without positivity assumption
A previous question asked for rational solutions of $x+y+\frac 1x+\frac 1y=2025$ without any positivity assumption. It has been closed since the absence of positive solutions is already known, but the ...
- 12.7k
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Rational points and sections on a family of genus-3 hyperelliptic curves
I am interested in the proof or disproof of some conjectures about rational points and sections over $\mathbb{Q}$ for the following family of genus-3 hyperelliptic curves:
$$
C_t: f(x,a)\, g(x,a)\, h(...
1
vote
1
answer
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An equivalent formulation for the twin prime conjecture
Consider Oeis $A065387$: $a(n)=\sigma(n)+\phi(n)$, where $\sigma(n)$ represents the sum of all divisors of $n$ and $\phi(n)$ is Euler's totient function.
I ask to prove that the assertion
the ...
- 3,064
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Find the number of ordered pairs of positive integers $(x, y)$ that satisfy the equation $\frac{1}{x} + \frac{1}{y} = \frac{1}{2004}$. [duplicate]
Find the number of ordered pairs of positive integers $(x, y)$ that satisfy the equation $\frac{1}{x} + \frac{1}{y} = \frac{1}{2004}$.
The answer is $45$, but I don't know how it is $45$. How to do ...
- 29
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votes
2
answers
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Rational points on a hyperelliptic curve defined by a product of two quartics
I am interested in finding all rational points on the hyperelliptic curve
$$
C: f(x)\, g(x) = y^2,
$$
where
$$
f(x) = \left(625x^4 + 3100x^3 - 11344x^2 + 6200x + 2500\right), \quad
g(x) = \left(961x^4 ...
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votes
2
answers
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views
There are no integer solution to $a^p+ p b^p+ (10p+1) c^p=0$ for any $p$ prime
I am trying to prove that $(0,0,0)$ is the only integer solution of
$$a^p+ p b^p+ (10p+1) c^p=0$$
I found this question on an Italian forum for the particular case in which $p=11$. The question was ...
- 3,900
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vote
0
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Parametrization $p x^2 + q y^2 = z^3 + r$
Let given $p,q \in \mathbb{Z}^+,r \in \mathbb{Z}$ and equation $p x^2 + q y^2 = z^3 + r$.
If exist solutions of Pell equation $qb^2-3pa^2=r\pm1$, then
$x = a(pa^2 - 3qb^2 + 3r)\\ y = b\\z = pa^2 + qb^...
- 1,830
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votes
3
answers
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Criterion that $ax^2+bx+c=n^2$ has integer solutions
Given $f(x)=ax^2+bx+c$, where $a,b,c\in\mathbb{Z}$, $a>0$ is square-free. If the coefficients involve parameters, for example
\begin{equation*}
f(x)=5p^2x^2+2qx+1,
\end{equation*}
where $p,q$ are ...
- 861
3
votes
0
answers
106
views
Extending the twin‑prime polynomial Diophantine pair $(n^2+p,n^2+p+2)$ to a triple
While playing with the identity
$$
p(p+2)+1=(p+1)^2,
$$
I noticed that for any integer $n$ and any integer $p$ (twin prime not actually needed here),
$$
(n^2+p)(n^2+p+2)+1=(n^2+p+1)^2.
$$
So the pair
$...
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139
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On the Diophantine equation $x^{3}+y^{3}=(x+y)^{m}-(xy)^{n}$ — finiteness and partial results
I've been exploring some symmetric Diophantine forms that blend additive and multiplicative structures, and one equation I found keeps bugging me:
$$
x^3 + y^3 = (x + y)^m - (xy)^n,
$$
with integer $x,...
1
vote
1
answer
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On the number of integer solutions of the diophantine equation $x^n+a_nx^{n-1}+\dots+a_1x+a_0=y^n$ for $n\gt1$
I am interested in finding the answer to the following problem:
Given a monic polynomial with integer coefficients $P_n(x)=x^n+a_nx^{n-1}+\dots+a_1x+a_0$ of degree $n\ge2$, is it possible for the ...
- 225
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Simple/Elegant proof of a weakened form of the Cannonball Problem?
I am wondering if anyone has a simple and elegant proof of the following fact:
The diophantine equation $\frac{n(n+1)(2n+1)}{6}=m^2$ has finitely many integral solutions.
Of course, this is a very ...
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votes
1
answer
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views
Find all prime $p$, $q$ such that $p - q$ and $pq - q$ are both perfect cubes
Problem 4 of USAMO 2022 was the following number theory problem.
Find all pairs of primes $(p, q)$ for which $p-q$ and $pq-q$ are both perfect squares.
In the MOP homework of the same year, a ...
- 399
1
vote
2
answers
133
views
How to solve system of 2 arbitrary bivariate quadratic equations over finite field?
I'm in the process of needing a solver for bivariate quadratic system of 2 equations over finite field - this is to estimate the time complexity of breaking an algorithm that I'm designing.
Most ...
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votes
3
answers
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views
Determinants of two matrices and their sum, entries are consecutive primes
In general determinants are not additive (related question). Nevertheless, matrices $A$ and $B$ can be chosen so that $\det A+\det B=\det\left(A+B\right)$ (related answer).
For which positive integer ...
- 1,575
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244
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Efficient algorithm for solving Diophantine equation $x ^ 2+y ^ 3+z ^ 5=w ^ 7$ with $\gcd (x, y, z)=1$.
My Mathematica Codes:
...
- 365
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For which smallest $n$ does there exist natural numbers $a_1,\dots,a_n$ making $\dfrac{(a_1+\cdots+a_n)^2-1}{a_1^2+\cdots+a_n^2}$ an integer?
Recently, at an olympiad I came across a problem that left me stuck.
Let $n\ge 2$ be a natural number. For which smallest $n$ do there exist natural numbers $a_1,a_2,\dots,a_n$ such that
$$
\frac{\...
- 487
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votes
1
answer
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No solutions in perfect squares for $ (a^{2}-b^{2})^{2}=c^{2}(a^{2}+b^{2}) $ [duplicate]
About a week ago, I discovered an interesting property of a certain class of triangles, but I have not been able to prove it.
Claim:
If a triangle $ \triangle ABC $ satisfies
$$ \angle A - \angle B =...
- 4,461
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votes
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views
An optimization problem over the positive integers
Let $x,y,z \in \mathbb{N}$ be three distinct positive integers. Find the minimum of $$ \frac{(x+y)(y+z)(z+x)}{xyz} $$
If the numbers are not distinct, then indeed the answer would have been 8, via AM-...
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1
answer
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Asymptotics for $n = a^2 + b^2 + c^2 $ in exactly one way
Consider integers $n$ such that
$$ n = a^2 + b^2 + c^2 $$
for integers $a,b,c$ such that $0 <a \leq b \leq c$ in exactly one way.
How fast do these numbers grow ?
The sequence starts like
$$3, 6, 9,...
- 18.4k
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Rational pre-images of a quadratic polynomial
Suppose I'm given rational numbers $\{y_1,\dots,y_m\} \subset \mathbb{Q}$. It is easy to construct a quadratic polynomial $f(x) = ax^2+bx+c$ such that all of the values $y_i$ are the $y$-coordinate of ...
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views
Prove that there are no other solutions to $12a^4+6a^2+1=n^2$ other than $a=0$.
I am trying to solve the Diophantine equation
$$
12a^4+6a^2+1 = n^2.
$$
Rearranging, we can write
$$
(2n)^2 - 3(4a^2+1)^2 = 1.
$$
Let $Y = 4a^2+1$. Then
$$
3Y^2 = (2n-1)(2n+1).
$$
Since $n$ is odd, we ...
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votes
0
answers
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Find the largest positive integer n for the equation a + b + c + d = gcd(a, b) + lcm(c, d). [closed]
I'm currently working on a number theory problem and could use some insights or alternative approaches. The problem is as follows:
Find the largest positive integer $ n $ such that for every positive ...
- 307
2
votes
0
answers
139
views
When does the sum of reciprocal determine the sum uniquely? [closed]
Consider this problem:
if we know
$$\displaystyle\sum_{i=1}^n \frac{1}{x_i} = a,$$
that is $a$ is given, what can we then say about
$$\displaystyle\sum_{i=1}^n x_i$$
and the values the second sum can ...
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Number of solutions to the diophantine equation $1/a+1/b+1/c=1/p(k)$, where $p(k) = k$-th prime
Playing around with diophatine equations of the harmonic mean type I started with
$$\frac{1}{a}+\frac{1}{b}=\frac{1}{p}\tag{1}$$
where, for simplicity, $p$ is a prime number and asked for the number ...
- 13.5k
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$\frac{1}{a} + \frac{1}{b} + \frac{1}{c} + \frac{1}{abc} = \frac{m}{a + b + c}$ - Solve for natural solutions [closed]
The below problem is one of the shortlisted problems from the 2002 IMO Number Theory. Below that is my proposed proof. I would like to know the rigorousness of my attempted proof.
Problem:
Does the ...
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The no. of positive integral solution$ (a,b,c,d)$ satisfying $\frac1a+\frac1b+\frac1c+\frac1d=1$ with condition of $a<b<c<d$. [duplicate]
How can I find the number of positive integral solution$ (a,b,c,d)$ satisfying $\frac1a+\frac1b+\frac1c+\frac1d=1$ with condition of $a<b<c<d$.
I came about this question while practicing ...
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Which $n$ admit a square-product necklace on $\{1,\dots,n\}$?
For an integer $n\ge 2$, let $G_n$ be the simple graph on ${1,2,\dots,n}$ where distinct $a,b$ are adjacent iff $ab+1$ is a perfect square. A square-product chain is an ordering $(a_1,a_2,\dots,a_n)$ ...
user1675092
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answer
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views
If $a(b+1)(ab+1)$ is a perfect square, then does $(b+1) \mid a(ab+1)$ always hold?
Let $a, b$ be positive integers such that $a(b+1)(ab+1)$ is a perfect square. Then, is it necessarily the case that $(b+1) \mid a(ab+1)$?
Numerical evidence indicates that this conjecture holds for $a,...
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votes
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Diophantine equation $y^x-x=77$. Why must the factors in $(y^{x/2}+ x^{1/2})(y^{x/2}- x^{1/2})=77$ be integers? (Harvard entrance exam)
The solution of the Diophantine equation $y^x-x=77$ is obtained through factorization $$(y^{x/2}+ x^{1/2})(y^{x/2}- x^{1/2})=77$$ assuming that the two factors in brackets must be integers if $x$ and $...
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vote
1
answer
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views
Min $S$ satisfying $\frac{a}{S-a}\frac{b}{S-b}\frac{c}{S-c}=\frac{1}{60}$ and some constraints
Let $a$, $b$, $c$ be 3 natural numbers, each no less than 12. Let $S$ be a natural number such that:
$min(S-a,S-b,S-c)\ge12$
$\frac{a}{S-a}\frac{b}{S-b}\frac{c}{S-c}=\frac{1}{60}$
Find the minimum ...
3
votes
0
answers
93
views
Find all $x$ for which $x^3+2x+1$ is a power of 2 [duplicate]
I recently came across this problem:
Find all positive integers for which $x^3+2x+1$ is a power of $2$.
I tried splitting casework based on parity of $x$, firstly x must be odd, and writing $x=2y+1$, ...
- 57
6
votes
0
answers
181
views
Can a positive integer be the product of two or more consecutive integers in $4$ or more different ways?
These are the numbers that can be written in three different ways less than $700,000$:
$120=4\cdot5\cdot6=2\cdot3\cdot4\cdot5=1\cdot2\cdot3\cdot4\cdot5$
$720=8\cdot9\cdot10=2\cdot3\cdot4\cdot5\cdot6=1\...
- 1,047
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votes
0
answers
141
views
Find all primes $p$, such that $4p^3+2p-2$ is a perfect square.
I got this question from a reliable source in other forum.
This question is harder to me, because it can't be factored, at least in $Z$. We can put it as,
$p((2p)^2+2)=x^2+2$
Then $p$ has to be of the ...
3
votes
2
answers
136
views
Is there a computable sequence of Diophantine equations whose minimal height of integer solutions grows faster than any computable function?
This question is inspired by the busy beaver function and MRDP theorem. It is known that the busy beaver function grows faster than any computable function, that is,
$$
\lim_{n \to \infty} \frac{\...
- 650
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votes
1
answer
50
views
Is provability of existential arithmetic sentences/halting of Turing machines independent with theory
Let $D(x_1, x_2, …, x_n) = 0$ be a Diophantine equation of which coefficients are computable integers via Turing machines. Suppose we can prove in theory like ZFC+large cardinal that $D$ has a zero, ...
- 650
3
votes
1
answer
220
views
Prove that if $d^k = (3^n - 1)/2$, then $d^{k+1}$ cannot be of the form $(2^m + 1)/3$
Suppose that $d^k = \frac{3^n - 1}{2}$ for positive integers $d,n,k$, with $d,k \geq 2$. Prove that it is impossible for $d^{k+1}$ to be of the form $(2^m + 1)/3$.
I originally asked a similar ...
- 303
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votes
2
answers
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views
Prove that if $d^k = (3^n + 1)/2$, then $d^{k+1}$ cannot be of the form $(2^m + 1)/3$?
I wish to prove that if $d^k = \frac{3^n +1}{2}$ for positive integers $d,k,n$, with $d, k \geq 2$, then it is impossible that $d^{k+1}$ be of the form $\frac{2^m +1}{3}$.
Some simple observations I ...
- 303
4
votes
4
answers
320
views
Two variable equation: $x^3+y^3=x^2+18xy+y^2$
I am at a math minicamp right now, and we were given mock tests. In particular, one of the problems is the following:
Find all ordered pairs $(x,y)$ of positive integers which satisfy the equation $x^...
- 1,113
1
vote
1
answer
85
views
Smallest $k,n \in \mathbb{Z}$ such that $x,y \in \mathbb{R}$ are contained in $[kn, (k+5)n]$
Find the smallest integers $k,n$ such that the real numbers $x,y$ are contained in the interval $[kn, (k+5)n]$.
For instance, if $x=11$, $y=23.5$, we can observe that the difference $2\cdot5 < |y-x|...
- 5,517
2
votes
3
answers
223
views
Solving the Diophantine equation $ 100a+b = (a+b)^2 $ without a computer
After solving project Euler problem 932, I started searching for a pen-and-paper solution for a simple version of that problem, i.e. the following Diophantine equation:
$$
100a+b = (a+b)^2,
$$
where $...
- 4,746
1
vote
0
answers
115
views
Quadratic forms vanishing simultaneously
From this question : Need help solving system of Diophantine equations
A discussion of existence theory for a system of this type Simultaneous vanishing of quadratic forms
So far, I have run brute ...
- 147k
3
votes
1
answer
159
views
Solve Diophantine Equation given a: $k^2 = a + \sum_{n=0}^{N} n^2$
We can define a function on the integers that takes an integer input and keeps adding successive terms of a sequence to that input until the output is another number of the sequence. Let's use the ...
- 113
2
votes
3
answers
209
views
How many integer solutions are there $(a,b)$ such that $(a+b+3)^2+2ab=3ab(a+2)(b+2)$?
Problem: How many integer solutions are there $(a,b)$ such that $(a+b+3)^2+2ab=3ab(a+2)(b+2)$?
My try: I tried expanding both sides and use divisibility. For these types of long Diophantine equations, ...
- 499
7
votes
1
answer
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views
Solutions to $2^n + n = m^2$
Find all solutions to the diophantine equation $2^n + n = m^2$ in positive integers.
First we can prove that n must be odd, if $n= 2t$,$(2^t)^2 + 2t = m^2$
obviously $m> 2^t$ and $m$ must be even, ...
- 353