Questions on finding integer/rational solutions of equations.

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If $a$ and $b$ are whole numbers from $1$ to $100$, how many pairs of numbers $(a,b)$ are there which satisfy $a^{\sqrt{b}}=\sqrt{a^b}$

If $a$ and $b$ are whole numbers from $1$ to $100$, how many pairs of numbers $(a,b)$ are there which satisfy $a^{\sqrt{b}}=\sqrt{a^b}$ This was from a math contest I did earlier today and I was ...
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1answer
23 views

How can you solve this type of (not quite linear) diophantine equation in 2 variables?

Is there a general technique to find solutions of this type of equation? 555555=t+2rt+r I'll provide the only answer I know in the comments. Thanks.
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0answers
35 views

How to find the first integer making two progressions have gcd $> 1$

Is there a technique to efficiently find the first positive integer, $r$, that makes: $$\gcd(97+r, 106-r) > 1\text{?}$$
6
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1answer
92 views

Sum and product of rational numbers is unity

Consider the system of equations: $$\sum_{i=1}^n X_i = 1$$ $$\prod_{i=1}^n X_i = 1$$ It is reasonably simple to show that for $n\ge 4$, this system admits a rational solution $(x_1, \dots, x_n) \...
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3answers
65 views

Integer solutions to $x^3+y^3+z^3 = x+y+z = 8$

Find all integers $x,y,z$ that satisfy $$x^3+y^3+z^3 = x+y+z = 8$$ Let $a = y+z, b = x+z, c = x+y$. Then $8 = x^3+y^3+z^3 = (x+y+z)^3-3abc$ and therefore $abc = 168$ and $a+b+c = 16$. Then do I just ...
3
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2answers
142 views

Solving $2^x+2^y+1=3^z$ in integers

I have reasons to believe that there should be an elementary, relatively simple way to find all solutions of the equation in the title in positive integers $x>y$ and $z$. Any ideas?
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2answers
72 views

Find all triples satisfying an equation

Another question I saw recently: Find all triples of positive integers $(a,b,c)$ such that $\frac{1}{a}+\frac{1}{b}+\frac{1}{c}=1$. Can someone help me with it?
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2answers
49 views

Find all possible three digit numbers whose digits' sum equals 12

How many 3 digit numbers exist such that the sum of their digits equals 12? I ran a little program and found that there are $66$ of such numbers. I feel that this type of problem is similar in style ...
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3answers
61 views

Solve $391x + 253y = 2760$ integer $x, y$

Solve $391x + 253y = 2760$ integer $x, y$ I took some mods: $138x \equiv 230 \pmod{253}$ this means $3x \equiv 5 \pmod{11} \implies x \equiv 9 \pmod{11}$ Thus $ \implies x = 9 + 11k$. So, $253y =...
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0answers
11 views

Perfect powers by Oblath's result [duplicate]

What do you mean by this statement? Obl\'ath proved that the only perfect powers all of whose digits are equal to a fixed one $ a \neq 1$ in decimal representation are 4, 8 and 9. This is equivalent ...
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0answers
22 views

Application of the theorem about diophantine equations having either infinite or finite solution.

How can i apply the theorem below in an equation like \begin{equation} \label{eq:(4)} 10^{n+3} a - 10^3 a + 999b = (3y)^2. \end{equation} that equation is actually from letting $m = 3$, from the ...
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0answers
52 views

Right angled triangle and Pythagorean triplet

Show that there exists a right angled triangle with rational sides and area $d$ if and only if $x^2,y^2$ and $z^2$ are squares of rational numbers and are in arithmetic progression with common ...
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1answer
44 views

Why does the modulo affect other terms in the equation? [closed]

i just want to ask if why does the modulo affect the other terms in an eqution? Why does the 4th equation has to be multiplied by $a^2$? Then as the modulo becomes $n≡1(mod3)$ in the 5th eq. then the ...
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1answer
38 views

Probability of solution amongst a set of Diophantine equations

I have a set of Diophantine equations which I know only one equation has a single solution. I am trying to find a way to give probabilities to which equation contains the solution. For example, ...
6
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1answer
198 views

$m^2+2017=n^3$ has no solutions

Show that $m^2+2017=n^3$ has no solutions for positive integers $m,n$. I'm having trouble tackling this one, especially since $\mathbb{Z}[\sqrt{-2017}]$ isn't a UFD. We can write the equation as $m^...
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5answers
93 views

Integer solutions to $5m^2-6mn+7n^2 = 1985$

Are there integers $m$ and $n$ such that $$5m^2-6mn+7n^2 = 1985?$$ Taking the equation modulo $3$ gives $n^2-m^2 \equiv 2 \pmod{3}$. Thus, $3 \mid n$ but $3 \nmid m$. How can I use this to find a ...
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2answers
122 views

Matrix decomposition into square positive integer matrices

This is an attempt at an analogy with prime numbers. Let's consider only square matrices with positive integer entries. Which of them are 'prime' and how to decompose such a matrix in general? To ...
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0answers
15 views

Diophantine-equation problem

In a diophantine equation ax+by=c, where b>a, can we assume the equation as by+ax=c and then use the Euclidean Algorithm to find the solutions?
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1answer
40 views

Prove that there exists solution for $x^2-Dy^2 = kp$, $p$ prime, $D$ quadratic residue

Let $p$ be a prime, and let $D$ be a quadratic residue modulo $p$. Show that for some $1\leq k\leq |D|$, there exists an integer pair $(x,y)$ that solve $x^2-Dy^2 = kp$ I have managed to prove this ...
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2answers
95 views

Finding integer solutions to $y^2=x^3-2$

I have the equation: $$y^2=x^3-2$$ It seems to be deceivingly simple, yet I simply cannot crack it. It is obviously equivalent to finding a perfect cube that is two more than a perfect square, and a ...
3
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2answers
57 views

solutions of the equation $x^3-y^3=z!-18$

What are the solutions of the equation $x^3-y^3=z!-18$? Here $x,y,z$ are non-negative integers. I have tried brute force but is there a better method?
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3answers
85 views

For which integers $a,b,c,d$ does $\frac{a}{b}+\frac{c}{d} = \frac{a+c}{b+d}$?

For which $a,b,c,d \in \mathbb{Z}$ does $\frac{a}{b}+\frac{c}{d} = \frac{a+c}{b+d}$? This is actually the question I meant to ask in a previous question that I asked here. What about $a,b,c,d \...
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5answers
239 views

Find a positive integer solution to $xyzw=504(x^2+y^2+z^2+w^2)$

Find positive integer values of $x,y,z,w$, such that $$xyzw=504(x^2+y^2+z^2+w^2)$$ I found it at some point and now I am unable to find the solution anymore, maybe this equation isn't satisfiable? ...
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1answer
65 views

Need help with proof about Diophantine equations

The way I am planning to arrange this is by providing fragments of the proof, so I can understand what's going on before forging ahead, so if you are going to help me, keep in mind that I am going to ...
3
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1answer
112 views

For which integers $a,b,c,d$ does $\frac{a}{b} + \frac{c}{d} = \frac{a+b}{c+d}$?

A long time ago one of my professors gave me this question. He didn't know the answer and has since passed away. For which $a,b,c,d \in \mathbb{Z}$ does $\frac{a}{b} + \frac{c}{d} = \frac{a+b}{c+d}...
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1answer
38 views

Number of Rational Solutions $\mathbf{x}\in[0,1)^n$ to the Matrix Condition $\mathbf{A}\,\mathbf{x}\in\mathbb{Z}^n$

Let $n$ be a positive integer and $\mathbf{A}$ an $n$-by-$n$ matrix with integer entries. Suppose that $k:=\big|\det(\mathbf{A})\big|$ is nonzero. How many $n$-by-$1$ column vectors $\mathbf{x}\in\...
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2answers
125 views

How does $x^4+y^4=z^2 \implies x^4+y^4=z^4$?

Why is the statement "the following cannot be satisfied" for $x^4+y^4=z^2$ more strong than for $x^4+y^4=z^4?$ More specifically, how does $x^4+y^4=z^2 \implies x^4+y^4=z^4?$ This statement was ...
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0answers
92 views

Is a finite number of quadratic equations in two variables sufficient to solve for the two variables?

Let's say I’m trying to solve a Diophantine problem in two positive integers, $y$ and $q$. Furthermore, let’s say I can derive an extremely large (read: arbitrary) number of equations of the form $$ay^...
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1answer
57 views

Given integers $m,n$, find integers $a,b,c$ such that $a^3+b^3+c^3-3abc=m n$

With a³+b³+c³-3abc=m (m-random integer) And a³+b³+c³-3abc=n(n-another integer) How to find a³+b³+c³-3abc=mn(m and n are Co prime) I came across this in an online math contest. Hint was given as ...
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1answer
112 views

Siegel's Lemma for two solutions

Consider the homogeneous diophantine equation $$ax_1+bx_2+cx_3=0$$ over $\mathbb{Z}$ with a, b, c coprime. (A version of) Siegel's Lemma states, that there exists a non-trivial solution $x$, such ...
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3answers
166 views

Find all integer roots of: $x^2(y-1)+y^2(x-1)=1$

Find all integer roots of: $x^2(y-1)+y^2(x-1)=1$ Obviously $(2,1)$ and $(1,2)$ are two answers. But I was unable to manipulate the equation algebraically giving a useful form for finding all other ...
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1answer
27 views

Integer solution to an hyperbola equation

Given the general equation of an hyperbola $Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0 $ where $B^2-4AC>0$ is it possible to find all integers solutions $(x,y)$ as a function of $A, B, C, D$ and $ F $ ?...
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2answers
59 views

Multiple of 3 as Sum of 4 Cubes

Prove that every multiple of 3 can be expressed as sum of four integer cubes. When working with numbers of form $6n$, a very clear pattern emerged, which the equation below proves elegantly - $\...
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2answers
101 views

Find all positive integers satisfying: $x^5+y^6=z^7$

Find all positive integers satisfying: $x^5+y^6=z^7$ No algebraic method came into my mind,just tried to find some answers and failed! Of course it's very simple to write a computer program finding ...
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0answers
39 views

Is there a general solutions to min(Floor(N/a), Floor((N ± a)/r))?

I am looking for a general solution to $\min\left({\left\lfloor{\frac{N}{a}}\right\rfloor, \left\lfloor{\frac{N \pm a}{r}}\right\rfloor}\right)$ where $N$, $a$, and $r$ are positive integers with the ...
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1answer
94 views

Find all positive integers $k,m,n$ satisfying: $\frac1k+\frac1m+\frac1n=\frac{1}{1996}$

Find all positive integers $k,m,n$ satisfying: $\frac1k+\frac1m+\frac1n=\frac{1}{1996}$ The trivial answer is: $k=m=n=3*1996$ $kmn=1996(km+mn+nk)=499\times4\times(km+mn+nk)$ , now $kmn$ must be ...
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0answers
41 views

Find the general solutions of $x^2+qy^2-z^2=4n$

I have this diophantine equation and i need the general solution form for it the equation $$x^2+qy^2-z^2=4n$$ some conditions $y\le 0,x\ge 0,z\ge 0$ x,z are even or odd together $n=5y+x-3xy$
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positive integer solutions of $6xy+5x+5y=n$

Can any one help me with this? Determine all Positive integer $(x,y)$ such that $2 \le x \le y$ and $6xy+5x+5y=n$ Please do not solve as $(6x+5)(6y+5)=6n+5^2$, I need a more helpful method. we have ...
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55 views

software for the general integer solutions of quadratic diophantine equations

Does anyone know of software (either commercial or free/open) that implements the general algorithm to solve quadratic diophantine equations as shown in this paper by Grunewald and Segal (2004), ...
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1answer
46 views

Number of integral roots?

The number of integral roots of the equation $x^8-24x^7-18x^5+39x^2+1155=0$ I tried to figure out by considering the changes in signs which tell us that there are atmost $2$ positive real roots ...
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1answer
54 views

Find all nonnegative integers $m$ and $n$ such that $m!+1=n^2$. [duplicate]

This question is inspired by Subgroup of Order $n^2-1$ in Symmetric Group $S_n$ when $n=5, 11, 71$. Find all nonnegative integers $m$ and $n$ such that $m!+1=n^2$. We know that $(m,n)=(4,5)$, $(...
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27 views

Elementary literature on Group theoretic Power Diophantine Equation

I am looking for an elementary books/pdf notes on group theory related to Power Diophantine Equation. I have read elementary group theory. Please advise some books/pdf notes. Also, it would be ...
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1answer
69 views

Mordell curves with many integral points

For $k\in{\mathbb Z},k\neq 0$, denote by $f(k)$ the number of integral points on the Mordell curve $y^2-x^3=k$. According to the data at http://tnt.math.se.tmu.ac.jp/simath/MORDELL , the largest value ...
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1answer
37 views

Solving diophantine equations of the form $x^2 - ay^2 = b$

Solving $x^2 - ay^2 = b$ $a, b$ are given integers with a squarefree and $b$ prime, and I need to find pairs of integers $(x,y)$ that is a solution. I try writing $a$ as $(\sqrt{a})^2$ , ...
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1answer
174 views

Prove that $(2^n-1)(3^n-1)$ is not a perfect square

Prove that $(2^n-1)(3^n-1)$ is not a perfect square. I have tried this problem for a few days already and I feel I am really far from solving it. Most of my approaches have been analyzing how many ...
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0answers
20 views

how do we get the given diophantine equation and it's value

\label{thm:3} The only squares of the form $${\overline{aa \ldots ab \ldots b}}_{(10)}$$ in decimal representation are the trivial infinite families $10^{2i},\; 4\cdot 10^{2i}$, $9\cdot 10^{2i}$ with $...
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3answers
89 views

Help with Diophantine equation

Prove that the equation $$x^2 - x + 1 = p(x+y)$$ has integral solutions for infinitely many primes $p$. First, we prove that there is a solution for at least one prime, $p$. Now, $x(x-1) + 1$ is ...
2
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3answers
60 views

Distribution of Primitive Pythagorean Triples (PPT) and of solutions of $A^4+B^4+C^4=D^4$

If we define a $PPTCountingFunction(n)$ as a function that returns the number of PPF with $c < n$ and $a>b$, then up to first $n=100,000$ it is near linear and $\dfrac{n}{PPTCountingFunction(...
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2answers
44 views

Counting integer solutions for a system of (in)equalities

I wish to enumerate the number of solutions of the system of equations and inequalities for 3 non-negative integer unknowns $x,y,z \ge 0$: ($a$,$b$ specified) \begin{align} x+y+z&=a\\ x+y&>...
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4answers
138 views

Prove that $x^4-y^4=1996$ has no integer root.

Prove that $x^4-y^4=1996$ has no integer root. $LHS=(x-y)(x+y)(x^2+y^2)=1996$ Now we have to consider all possible decompositions of $1996$ resulting in a non Linear system of equations seemingly ...