Questions tagged [diophantine-equations]

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

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51 views

What is the importance of the Diophantine inequality $\frac{1}{p} + \frac{1}{q} + \frac{1}{r} > 1$?

In Humphreys' book Introduction to Lie algebras and Representation Theory(3rd printing, p.62), the Diophantine inequality $$\frac{1}{p} + \frac{1}{q} + \frac{1}{r} > 1$$ appeared while classifying ...
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1answer
225 views

What are the integer solutions to $5x^3=y^2+1$?

I want to find the integer solutions to this Diophantine equation: $$5x^3=y^2+1$$ I have seen a lot of problems with monic variables, but not with a constant on the $x^3$ such as this. I know I can ...
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1answer
43 views

How many ways can $2^{2012}$ be expressed as the sum of four (not necessarily distinct) positive squares?

How many ways can $2^{2012}$ be expressed as the sum of four (not necessarily distinct) positive squares? Thanks! For those curious, the solution which I have trouble comprehending is item 2 from the ...
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44 views

How to avoid nested loops intelligently with the computer when solving Diophantine equations with finite range of variables?

I have 59 quadratic equations in 59 variables $\{x_1, ..., x_{59}\}$ and I'm interested in the integer solutions of this equation system, where all $x_i$ can have values only between $-100$ and $+100$....
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How do you get the $6/5$ in Baker's explicit $abc$ conjecture?

I can't find Baker's "Experiments on the ABC conjecture" in which he gives the $6/5$ as the absolute constant. How do you get the $6/5$ as the constant? Baker's conjecture: For $a+b=c$, $(a,b,c)=1$...
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1answer
66 views

Show that $x^3+3y^3+9z^3-9xyz=1$ has infinitely many integer solutions. [duplicate]

Show that $x^3+3y^3+9z^3-9xyz=1$ has infinitely many integer solutions. I have found that (1,0,0) and (1,-18,12) are two solutions and tried (1,-18+n,12-n). There is a hint saying that I should try ...
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1answer
52 views

$x^2 + 4xy = 10^{100}$ how many solutions [closed]

$x^2 + 4xy = 10^{100}$ how many integer solutions there? Can't find the answer, because there is too big number. Help me please I know, how to solve it if I have 100 instead of $10^{100}$: 100 is $2^...
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1answer
46 views

Find solutions of $f(2020)x+f(2019)y=1$ where $f$ is Fibonacci sequence [closed]

I need to find at least one solution of $$f(2020)x+f(2019)y=1$$ with $x,y\in\mathbb{Z}$, where $f(n)$ is the $n^{th}$ Fibonacci number, starting at $f(0)=0$, so that: $$f(0)=0,\qquad f(1)=1,\qquad f(...
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Quickly Finding Positive Solutions to Diophantine Equations

I'm wondering if someone can help point me to the fastest available methods for solving problems like the following: Given positive integers $C, D,$ find the smallest positive integers $x$ and $y$...
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“Factoring” Diophantine approximations

I’ve never tried approximating Diophantine solutions before, and have a beginner question. In a certain system of equations, I have positive integers $a,b,c,d$ that by assumption satisfy the system. ...
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55 views

A system of quadratic Diophantine equations with six variables

In 1918, Norman Alliston noted that the following system of quadratic Diophantine equations \begin{cases} \begin{split} a^2\,\quad+c^2&=u^2\\ b^2\,\quad+c^2&=v^2\\ (a+b)^2+c^2&=w^2 \...
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82 views

Let $x,y\in\Bbb R$ such that $x ^2 + y^2 = 2x – 2y + 2$. What is the largest possible value of $x^2 + y^2 -\sqrt{32}$??

I tried to make perfect squares by adding 2 on both sides and I got something which looked like a Pythagorean Triplet but I am stuck there. Also is there a well defined way or an algorithm which gives ...
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4answers
77 views

If two positive integers m and n, both bigger than 1, satisfy the equation 2005^2 + m^2 = 2004^2 + n^2 , find the value of m + n – 200 [closed]

I tried to use the difference of two squares and was able to get product of two quantities on either side of the equation ,but i am stuck there .
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1answer
50 views

Proving Diophantine Equation has no solution using Legendre Symbol

Given that $\left(\frac{10}{23}\right)=-1$. How would I go about showing that $9x^2-46(y^3+3y+1)=10$ has no integer solutions? I believe it has something to do with Quadratic Reciprocity. For ...
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187 views

Infinite descent argument in the case of $x^5+y^5=z^5$.

Euler proved that: $$2u(u^2+3v^2)=(2a(a^2+3b^2))^3$$ had no non-trivial solutions. Several other mathematicians have proven that: $$x^5+y^5=z^5$$ Using a similar argument, it is easy to see when $x$ ...
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76 views

What are the integer solutions to $a^{b^2} = b^a$ with $a, b \ge 2$

I saw this in quora. What are all the integer solutions to $a^{b^2} = b^a$ with $a, b \ge 2$? Solutions I have found so far: $a = 2^4 = 16, b = 2, a^{b^2} = 2^{4\cdot 4} =2^{16}, b^a = 2^{16} $. $...
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49 views

For an exceptional quality $q$, can $q-1$ be part of an infinite family of qualities of $ABC$ triples?

Consider the $ABC$ conjecture in the form: For every $\epsilon>0$ $q>1+\epsilon$ in a finite number of triples $(a,b,c)$ where $a+b=c$ $gcd(a,b,c)=1$ and $q=log(c)/log(radical(abc)).$ An ...
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Diophantine equation with negative numbers

Good morning, I can't solve this diophantine equation through the Euclidean division: $45x - 8y = 231$ $45x'-8y' = 1$ Euclidean division $45 = -5*(-8)+5$ $-8 = -2*5+2$ $5 = 2*2+1$ $2 = 2*1+0$ $...
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1answer
50 views

Insights for proving $(x^2-1) = l(l+1)(l+2)(l+3)$ [closed]

I am looking for some insights on how to prove that for every integer $l$, there exists an integer $x$ such that: $$x^2-1 = l(l+1)(l+2)(l+3).$$ So far, what I did was: $$x^2-1=(x-1)(x+1).$$
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Solving a diophantine equation by finding a parametrization of the solutions given several conditions

The diophantine equation that needs to be solved is $kx^2-y^2=3$ with the added conditions that: \begin{cases}k>1\\k\text{ not a square}\\ x>2\\ x\equiv 0\mod 2\\ x/2 \not| \ k \\ x/2 \text{ is ...
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Quadratic Diophantine equation $x^2+6y^2-xy=47$ has no solutions.

I am trying to show that $x^2 + 6y^2 - xy = 47$ has no integer solutions. I know that the an efficient way is to look at this equation modulo $n$; other equations can be easily be solved this way. I ...
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26 views

Linear equation with two unknowns

Why the indeterminate equation, which has coprime coefficients, gives the same solution when we set the value of one variable to equal the value of the other variable? I will clear it with an example: ...
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107 views

Rigid pentagons and rational solutions of $s^4+s^3+s^2+s+1=y^2$

Gerard 't Hooft, Nobel Prize in Physics laureate, wrote three articles on what he called "Meccano math" (1, 2, 3) – rigid constructions following rules quite similar to my earlier question on doubling ...
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1answer
57 views

Integer solutions to $(a^x - b^y)/(a - b)=c$ [closed]

I would like to know if all integer solutions to $\frac{a^x -b^y}{a - b} = c$, where $c$ is also an integer, are known.
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Diophantine Equations involving primes [duplicate]

I wish to prove that if $\exists x,y \in Z$ such that $x^2 + 5y^2 = p$ then there cannot exist $a, b \in Z$ such t hat $a^2 + 5b^2 = 2p$. Here p is a prime natural number. I kind of see why this might ...
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Diophantine equation from “Solving mathematical problems” by Terence Tao

Find all integers $n$ such that the equation $\frac{1}{a} + \frac{1}{b} = \frac{n}{a+b}$ is satisfied for some non-zero values of $a$ and $b$ (with $a + b \neq 0$). I'm reading "Solving mathematical ...
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3answers
114 views

Prove that the Pell's Equation $x^2 −Dy^2 = 1$ always has a solution where $y$ is a multiple of $41$

$D$ is a positive integer that is not a perfect square Recently I am taking a introductory number theory course and I met this question right after we learned Pell's equation and Diophantine ...
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2answers
37 views

For all integers $n$, there are coprime $x,y$ such that $n=px+qy$ where $p,q$ are. also coprime.

For all integers $n$, can we prove there exist coprime $x,y$ such that $n=px+qy$ where $p,q$ are coprime as well?
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175 views

Is there a well known proof that shows solutions of $y^2=3x^4+1$ are only (1,2), (2,7) over positive integers?

I found a theorem from a book 'Diophantine equations', L. J. Mordell, which says The equation $y^2 = Dx^4+1$ where $D>0$ and is not a perfect square, has at most two solutions in positive integers....
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Is Integer Solution of Cubic Negative?

I have been trying to solve a problem where OP seeks Integer roots to cubic equation. There are similar problems ( see 1, 2, 3, 4, 5, 6, 7) in this forum. To solve the general cubic$$x^3+ax^2+bx+c=0\...
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Why does the author mean by “The $(p+1)/2$ numbers to $x^2$, for $0\leq x\leq (p-1)/2$, …”

I couldn't get the proof for the given theorem. How is the $(p+1)/2$ numbering to $x^2$, how is the following relation valid??
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Proof that $x^2+y^2+z^2=7$ does not have a rational solution

Proof that $x^2+y^2+z^2=7$ does not have a rational solution. I figured out that I have to prove it by reductio ad absurdum, but I try and can't still found how to prove it. Or using proof by ...
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1answer
28 views

For which modulos are all but one $n$th power congruent (or have few possible values)?

I recently tried to show that there are no integer solutions to the equation $x^4+ 131 = 3y^4$. After stumbling with some algebraic attempts I reached the conclusion that all forth powers (except of ...
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3answers
30 views

Find the sum of all positive integers $k$ for which $5x^2-2kx+1<0$ has exactly one integral solution.

Find the sum of all positive integers $k$ for which $5x^2-2kx+1<0$ has exactly one integral solution. My attempt is as follows: $$\left(x-\dfrac{2k-\sqrt{4k^2-20}}{10}\right)\left(x-\dfrac{2k+\...
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68 views

Question regarding the resolution of a quadratic Pell-like diophantine equation

We would like to solve the diophantine $$7x^2-5y^2=18 \tag{E}$$ We first solve the linear diophantine $$7x-5y=18$$ Solutions are couples $(7k+2,5k+4)$ where $k$ is an arbitrary integer. thus $(x,...
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87 views

Find all positive integer $a,b,c$ that $a^3+b^3+c^3$ can be divided by $a^2b,b^2c,c^2a$ [closed]

Find all triplets of positive integers $(a,b,c)$ for which $$a^3+b^3+c^3$$ is divisible by $a^2b$, $b^2c$ and $c^2a$. I just found that $a=b=c$ satisfies the problem. Are there any other ...
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2answers
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What are the number of ordered m-tuples of integers,such that sum of square of elements is a given integer?

Given a non-zero integer $n$,the problem is to find a m-tuple of integers,$(x_1,x_2,x_3,...,x_m)$such that the following equation is satisfied---$$\sum_{i=1}^mx_i^2=n$$ I have no idea how to approach ...
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101 views

Integer solutions to the equation $7x^2 = y^2+y+1$

While investigating the related equation $7^n = m^2 + m+1,$ I managed to quite quickly handle the case that $n$ is even. If $n$ is odd, we may let $x = (n-1)/2.$ This now reduces to the question in ...
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3answers
109 views

$N=\underbrace{111 . . . 1}_{n\space times}\space\underbrace{222 . . . 2}_{n+1\space times}\space5$ is always a square?.

I come to see this problem do not know if it's easy or difficult (I sense it's not hard) and I want to share it with members of MSE. It comes from a French source in which it says it is an Olympics ...
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1answer
61 views

Diophantine equations of three variables

If $M,N,P$ are positive integers such that $$\begin{cases}M+N+P&=2024\\MNP&=2020^2\end{cases},$$ Show that $(M,N,P)=(4,1010,1010)$ is the only solution up to permutation. I've got this ...
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1answer
41 views

prove or disprove that the integer solutions to equation $x^2-2xy+y^2-x+y+1=x^3-2y^3$ are only ( x=1, y=0) and (x=-2, y=3).

integer solutions to equation $x^2-2xy+y^2-x+y+1=x^3-2y^3$ are only ( x=1, y=0) and (x=-2, y=3). This question is related to another question about a system of equations. I showed how these ...
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115 views

Integer solutions to $x^3 + 3x^2y - 3xy^2 - 3y^3 = 1$

Prove that the only integer solutions to $x^3 + 3x^2y - 3xy^2 - 3y^3 = 1$ are $(1,0)$ and $(-2,3)$. My only idea for now is to try to represent this in the form $A^3 + B^3 = C^3$ and apply Fermat's ...
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1answer
44 views

Finding solutions to a diophantine equation

Is there finitely many solutions to the equation: $$(x_1x_2)(x_1^{a}+x_2^{a})=(y_1y_2)(y_1^{b}+y_2^{b})$$ for $x_1,x_2,y_1,y_2,a,b$ all positive integers greater than $1$ and $x_1x_2 < y_1y_2$.
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115 views

System of quadratic Diophantine equations $x^2-xy+y^2=a^2$,$x^2-xz+z^2=b^2$,$y^2-yz+z^2=c^2$

If it is only one quadratic equation $x^2-xy+y^2=a^2$, we can get some integral solutions as follows. \begin{align*} &\left\{ \begin{split} x&=k(2mn-n^2)\\ y&=k(m^2-n^2)\\ a&=...
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3answers
70 views

How many integer solutions are there for the equation $c_1 + c_2 + c_3 + c_4 = 25$, where $c_i \ge 0$ for all $1 \le i \le 4$

Question Statement: How many integer solutions are there for the equation $c_1 + c_2 + c_3 + c_4 = 25$, where $c_i \ge 0$ for all $1 \le i \le 4$. I would like to solve this problem using ...
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4answers
157 views

Best Book/Note for Collection of Power Diophantine Equation

What is a good book that has a large collection of power Diophantine equation?
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4answers
152 views

Integer Solution to $x^3+y^2=z^2$

What is the non-zero integer general solution to $x^3+y^2=z^2$ ? I guess it is already solved in some book or paper, in that case plz help me to find that. Edit: One solution is - $n^3 = [(n)(n+1)/...
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4answers
177 views

For how many integers $n$ is $n^6+n^4+1$ a perfect square? [duplicate]

QUESTION For how many integers $n$ is $n^6+n^4+1$ a perfect square? I am completely blank on how to start. Could anyone please provide tricks on how to get a start on such questions? Thanks for any ...
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1answer
80 views

Properties of three terms of a geometric series

I’m [still!] working on the equation in this question, namely $$(b^2+2)^2=(a^2+2c^2)(bc-a). \tag{$\star$}$$ where $a,b,c$ are integers. Evidently, $(\star)$ implies $$\frac{b^2+2}{bc-a} = \frac{a^...
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1answer
104 views

Is the hexagonal cannonball number the only one that ends with $6$?

I've been searching for cannonball numbers, namely polygonal numbers that are also pyramidal numbers with the same number of sides, patterned after the famous cannonball number $4900$, the square that ...

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