Questions on finding integer/rational solutions of equations.

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2answers
38 views

Solve $x + \frac{ 1 }{y+1/3}=38/3$ in the set of natural numbers

The following equation should have a solution with $x,y$ being natural numbers. I cannot find it. Is there such solution? $$x + \frac{ 1 }{y+1/3}=\frac{38}{3}$$
13
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1answer
74 views

Pythagorean triplets of the form $a^2+(a+1)^2=c^2$ and the space between them

I was searching for pythagorean triples where $b=a+1$, and I found using a python program I made the first 10 integer solutions: $0^2+1^2=1^2$ $3^2+4^2=5^2$ $20^2+21^2=29^2$ $119^2+120^2=169^2$ ...
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1answer
23 views

Is solvability of diophantine equations over a p-adic field decidable?

As far as I understand, the decidability of solvability of diophantine equations over the rationals is an open problem. What about the decidability of solvability over a given p-adic field?
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2answers
38 views

Diophantine equation: $\frac1x=\frac{a}{x+y}-\frac1y$

For how many different natural values of $a$ the Diophantine equation: $\frac1x=\frac{a}{x+y}-\frac1y$ has natural roots? I rearranged the equation as: $xy+x^2+y^2=(a-1)xy$ , hence I said we must ...
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0answers
26 views

How common are diophantine equations for which the local global principle is invalid?

The local global principle says that in some families of diophantine equations the solvability over the rationals is equivalent to solvability over the reals and in p-adic fields $Q_p$ for each prime ...
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2answers
52 views

Find the smallest positive value taken by $a^3+b^3+c^3-3abc$

Find the smallest positive value taken by $a^3+b^3+c^3-3abc$ for positive integers $a,b,c$. Find all integers $a,b,c$ which give the smallest value. Since it is generally hard to find the minimum ...
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1answer
37 views

Find all the numbers $a$ such that the number $an(n+2)(n+4)$ is an integer for all $n \in \mathbb{N}$

Find all the numbers $a$ such that the number $an(n+2)(n+4)$ is an integer for all $n \in \mathbb{N}$ It's trivial to see that if $a$ is irrational, we get no solution. Thus $a \in \mathbb{Q} ...
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0answers
28 views

Integral solution of equation $Ax + By = z$ with contraints on $x, y, z$

Given $x$, $y$ and $z$, how can I check if there exists integral solution of $$Ax + By = z$$ Such that : if $x > y$, $A$ must be positive and $z \geq y$ ( Given ) if $y > x$, $B$ must be ...
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2answers
64 views

Is there a general formula to solve $a^2+b^2+c^2=d^2+e^2+f^2$?

Is there a general formula to solve the Diophantine equation $$a^2+b^2+c^2=d^2+e^2+f^2?$$ If so, can I please have the reference? Thanks.
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2answers
39 views

Solve the following congruence: $x(x+1)(x+2) \equiv 0 \pmod{221}$

Find the first five solutions for, $$x(x+1)(x+2) \equiv 0 \pmod{221}$$ I am very confused. By CRT, $x(x+1)(x+2) \equiv 0 \pmod{13}$ and $x(x+1)(x+2) \equiv 0 \pmod{17}$ But these two ...
10
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4answers
136 views

How many answers to $|3^x-2^y|=5$?

How many answers are there to the equation $|3^x-2^y|=5$ such that $x$ and $y$ are positive integers? Are there infinite? I've found $(2,2)$, $(3,5)$, and $(1,3)$. It seems to explode with larger ...
1
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1answer
67 views

Solve $x^2 = 2^x$. [duplicate]

One can see that the solutions are $x=2, 4$ and $x=-0.77$(approximately) seen from the graph. I am posting this to find if there is a way to solve this and find solutions like polynomial equations. ...
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3answers
114 views

Find all pairs of nonzero integers $(a,b)$ such that $(a^2+b)(a+b^2)=(a-b)^3$

Find all pairs of nonzero integers $(a,b)$ such that $(a^2+b)(a+b^2)=(a-b)^3$ My effort Rearranging the equation I have \begin{array} \space (a^2+b)(a+b^2)-(a-b)^3 &=0 \\ ...
21
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3answers
2k views

Find a thousand natural numbers such that their sum equals their product

The question is to find a thousand natural numbers such that their sum equals their product. Here's my approach : I worked on this question for lesser cases : \begin{align*} &2 \times 2 = 2 + ...
1
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1answer
53 views

Proof that $ax+by+cz=0$ has infinitely many solutions. [closed]

For all non-zero integers $x,y,z$ clearly there exist infinitely many non-zero integers $a,b,c$ such that $$ax+by+cz=0$$ How can I prove this simple statement?
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1answer
109 views

Integer solutions for $\frac{1}{x^2}+\frac{1}{y^2}=\frac{1}{z^2}$?

Find all triples $(x, y, z)$ where $x, y, z$ are coprime integers such that $$\dfrac{1}{x^2}+\dfrac{1}{y^2}=\dfrac{1}{z^2}$$ I did the following: ...
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1answer
30 views

Solving $\sqrt[3]{7x^2-13xy+7y^2}=|x-y|+1$ in integers

Find all pairs $(x,y)$ of integers such that $$\sqrt[3]{7x^2-13xy+7y^2}=|x-y|+1.$$ Since $x,y$ are symmetric, we can assume $x\geq y$, so the right-hand side is $x-y+1$. If $x=y$, the equation ...
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0answers
16 views

Solving equation with sin and exp

I'm trying to solve the following equation but getting nowhere. $\pi*\sin(\pi*x/b)/(2*b)+m*\exp(n-m*x) = 0$ With b,m,n constants. I remember from a long time ago that there are some types of ...
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0answers
67 views

Integer solutions to $y^2 = \frac{x^5-1}{x-1}$

$$y^2 = \frac{x^5-1}{x-1}$$ has integer solutions. How many pairs $(x,y)$ are there? My Work If $\sqrt{x^4+x^3+x^2+x+1}$ is an integer then there is a solution. But what to do now. Note: This ...
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2answers
59 views

A triple of pythagorean triples with an extra property

I'm trying to prove the non-existance of three positive integers $x,y,z$ with $x\geq z$ such that\begin{align} (x-z)^2+y^2 &\text{ is a perfect square,}\\ x^2+y^2 &\text{ is a perfect ...
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2answers
32 views

Find all $n$ such that $m = an$ or $m =\dfrac{n}{a}$

$a$ is the 1st digit (from the left) of a $3$-digit number $n$. We get the number $m$ by removing a from $n$ and putting it on the right of the unit-digit. For example, the number $123$ becomes $231$. ...
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1answer
13 views

When does:$(p^y+1 )\bmod (p^x+1)=0 $ if $(y,x)=1$ and $p $ is a prime number?

I'm interesting to look the solution of this equation :$$(p^y+1 )\bmod (p^x+1)=0 $$ at a least to see an example of the two coprime $y, x$ for which $(p^y+1 )\bmod (p^x+1)=0 $ but i don't succed , ...
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2answers
63 views

$X^4 - 4Y^4 = -Z^2$ has no solutions in non zero integers

I am trying to prove that $X^4 - 4Y^4 = -Z^2$ has no solutions in non zero integers. I know there are similar questions on MS, but that minus signs before the $Z$ gives me a hard time. For the ...
2
votes
1answer
81 views

Solve in integers $b^{11}-1=a^{2016}+a^{2015}+\dots+1$

Find all integers $(a,b)$ satisfying $$b^{11}-1=a^{2016}+a^{2015}+\dots+1.$$ Obviously, we can get the factorisation $(b-1)(b^{10}+\dots+1)=a^{2016}+a^{2015}+\dots+1$, but I'm not sure how to ...
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3answers
34 views

Solve $x^2+2xy-782y=0$ diophantine equation

I'm trying to solve $x^2+2xy-782y=0$ diophantine equation. With these steps: a) $(*4)$; $4x^2+8xy-3128y=0$ b) $(+/-4y^2)$; $4x^2+8xy+4y^2-3128y-(4y^2)=0$ c) Reducing; $(2x+2y)^2+(-3128y)-(4y^2) = ...
2
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2answers
45 views

A fundamental tool used in the study of Diophantine equations…

Notations: $K$ is a perfect field, $\overline K$ an algebraic closure and $V\subseteq \mathbb P^n(\overline K)$ is a projective variety on $\overline K$. If $V$ is defined over $K$, in symbols ...
5
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3answers
102 views

Solutions to $x+y+z-2 = (x-y)(y-z)(z-x)$

Show that the equation $$x+y+z-2 = (x-y)(y-z)(z-x)$$ has infinite solutions $(x,y,z)$ with $x, y,z$ distinct integers. In my attempt to solve the problem only found solutions form $x=y, z=2-2x$. ...
3
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5answers
111 views

Prove $a/b+b/a$ for $a$ and $b$ natural is only natural for $a=b$ [closed]

Is it possible to prove that for any natural $a,b$ the value of $a/b+b/a$ will not be natural with exception $a=b$?
5
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1answer
54 views

Proving minimum number of chairs is $567$

Hints only please! I am trying to figure this out somehow. A row can have as many girls, and a column can have as many boys. Proof by contradiction seems like a good technique, but I am not sure ...
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1answer
18 views

Finding other solutions to diophantine equations

I understand how to find the first solution to these equations but can't grasp how the other solutions are found. E.g. $102x\equiv 12 \pmod{174}$ So I can find the $gcd(174,102)=6$ (showing that ...
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0answers
47 views

Solving for 2 unknowns in a perfect square

Basically was looking at a sequence of perfect squares for a given constant integer A, where in the first instant we can easily and trivially generate a sequence of perfect squares using the ...
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2answers
40 views

Integral Solutions in Diophantine Equation

How do you solve this problem: Describe the integral solutions to the equation $317a + 241b = 9.$ I know the answer is $(a, b) = (35 + 241k, −46 − 317k)$ for integers k but I don't know how ...
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1answer
43 views

Find integers $x$ and $y$ such that $\frac{27^{x+y}}{9^{xy}}=27$ and $\frac{4^{2xy}}{8^{x+y}}=512$ .

Find all the integers $x$ and $y$ such that : $$\frac{27^{x+y}}{9^{xy}}=27$$ and :$$\frac{4^{2xy}}{8^{x+y}}=512$$ I'm in Algebra two and I feel like there are certain types of math I haven't ...
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0answers
42 views

Generate All Triangular Square Numbers Recursively?

We define a triangular number as follows: $$\sum_{n=1}^{n} x_{i}$$ As in $T_3$ = $3+2+1$, or $6$. Generating these triangular numbers is rather simple and done by the equation: $$T_n ={(n^2 + ...
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0answers
14 views

Find all numbers $x, y\in\mathbb{Z}$ satisfying certrain equations

I'm looking for hints on solving such equations as $(x+2)^4=y^3+x^4$ or $x^2+y^2=1997(x-y)$. They cannot be solved using typical techniques. Again, I'm looking for hints.
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0answers
11 views

Reference Request For Hermite normal form of non full row rank matrix

Could someone recommend me some references which discuss the problem of the reduction of a matrix which is not full row rank into its Hermite normal form?
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1answer
43 views

Is every integral point of an arithmetic scheme contained in an affine open set?

The schemes $$X = Proj \mathbb{Z}[s,t] = \mathbb{P}^1_{\mathbb{Z}}$$ and $$Y = Proj \mathbb{Z}[x,y,z]/(x^2 + y^2 - z^2)$$ both have isomorphic generic fibers as schemes over $\mathbb{Z}$, and there is ...
2
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1answer
62 views

On $4n+1 = x^2, 5n+1 = y^2$ and the Fibonacci numbers

While answering this question, I decided to look at the particular case, $$4n+1 = x^2\\5n+1 = y^2$$ to be solved simultaneously. The solutions for $n,x,y$ are A157459, A007805, and A049629, ...
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0answers
93 views

Find all $x,y,z \in \mathbb{N}$ that $x^xy^y=z^z$ [duplicate]

I was curious about the following diophantine equation $$x^xy^y=z^z$$ The equation seemed to have no positive integer solutions except for when $x$ or $y$ was $1$. Though I attempted to solve the ...
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2answers
65 views

Solve in positive integers

Solve the following equations in postivie integers: 1) $x^2+3y^2=z^2$ 2) $x^2+y^2=5z^2$ I have solved them using this as a reference, but I am interested in other solutions, more "elegant" ones. ...
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2answers
54 views

Is there a systematic way to solve in $\bf Z$: $x_1^2+x_2^3+…+x_{n}^{n+1}=z^{n+2}$ for all $n$?

Is there a systematic way to solve in $\bf Z$ $$x_1^2+x_2^3+...+x_{n}^{n+1}=z^{n+2}$$ For all $n$? It's evident that $\vec 0$ is a solution for all $n$. But finding more solutions becomes harder ...
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2answers
93 views

Find all triples of non-negative Integers $a,b,c$ such that $a!b!=a!+b!+c!$ [duplicate]

Exactly what it says in the Title; not much development from there :/
1
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1answer
19 views

Linear Diophantine equation solving

Find all the positive solutions in integers of $x+y+z=31$ $x+2y+3z=41$ For the first, I have subtract second equation from first , and by that I have found that $y=2k-10$ ,$z=k$,is it possible?
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1answer
48 views

Solution to $p^3-p+1=a^2$

What are the solutions to $p^3-p+1=a^2$ where $p$ is prime and $a$ is natural? I found the solutions: $p=3$ and $a=5$ $p=5$ and $a=11$ and one solution when $p$ is not a prime: $p=56$ and $a=419$, ...
8
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1answer
87 views

Diophantine equations for polynomials

I know that there has been work on diophanitine equations with solutions in poynomials ( rather than integers ) of the Fermat and Catalan type $x(t)^n+y(t)^n=z(t)^n$ ; $x(t)^m-y(t)^n=1$ and these have ...
1
vote
3answers
110 views

Solving $x^2 + y^2 = 1 + z^4$ with (x,y,z) = 1 and z < x < y

I have a computer programming problem where I need to find n many sets of integers that meet the condition $x^2 + y^2 = 1 + z^4$ with (x,y,z) = 1 and z < x < y I can do this relatively easily ...
3
votes
0answers
81 views

Symmetry and trivial solutions to Pell equations

Below is a representation of the solutions to the equation $x^2-Dy^2=1$ for $6(6-1)\leq D \leq 6(6+1)$: \begin{array}{c} & 30 & 31 & 32 & 33 & 34 & 35 & 36 & 37 & ...
2
votes
1answer
22 views

How to deal with an additive constant in a linear congruence equation?

I am trying to solve the following equation: $10x+3 \equiv 2 \pmod{17}$. The problem I am having is that I don't know what to do with the number $3$. This is what I have done so far: $10x+3 = ...
23
votes
2answers
349 views

A diophantine equation with only “titanic” solutions

I made a note some time ago that I had read in some book that the equation $$313(x^3+y^3)=t^3$$ has positive integer solutions, but that these are so large that it would be absolutely hopeless to ...
11
votes
4answers
241 views

Find all natural numbers $x,y$ such that $3^x=2y^2+1$.

Find all natural numbers $x,y$ such that $$3^x=2y^2+1$$ solutions are $(1,1)$, $(2,2)$, $(5,11)$. I found that parity of both is same and If $x$ Is odd it is of the form $4k+1$.