Questions on finding integer/rational solutions of polynomial equations.

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2
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2answers
5k views

How many solutions are there to the equation $x + y + z + w = 17$?

How many solutions are there to the equation $x + y + z + w = 17$? I don't know if I'm doing this right, but I guessed that the solution would be $\binom{20}{3}$, which equals $1140$. Am I doing ...
4
votes
1answer
141 views

A quartic diophantine equation

Here is the statement: Let $a,b \in \mathbb{Z}$ positive integers such that $a^2=b^4+b^3+b^2+b+1.$ Prove $b=3.$ I've tried is the following: Let $\Sigma=b^4+b^3+b^2+b+1$. If $a\equiv 0\mod 3$, then ...
7
votes
4answers
667 views

Right triangle where the perimeter = area*k

I was doodling on some piece of paper a problem that sprung into my mind. After a few minutes of resultless tries, I advanced to try to solve the problem using computer based means. The problem ...
4
votes
5answers
2k views

Count the number of integer solutions to $x_1+x_2+\cdots+x_5=36$

How to count the number of integer solutions to $x_1+x_2+\cdots+x_5=36$ such that $x_1\ge 4,x_3 = 11,x_4\ge 7$ And how about $x_1\ge 4, x_3=11,x_4\ge 7,x_5\le 5$ In both cases, ...
5
votes
4answers
193 views

Use induction to prove a product of sums of squares is a sum of squares

For any natural number $n\ge 1$, given pairs $(a_1,b_1),(a_2,b_2),...,(a_n,b_n)$ of integer numbers, there exist integer number $c$ and $d$ such that $$\prod_{i=1}^{n}(a_i^2+b_i^2) = c^2+d^2$$ My ...
0
votes
3answers
182 views

Using induction to find a general pattern of $x,y$

Let $m$ be an even natural number. Find natural numbers $x$ and $y$ such that $$ m=(x+y)^2+3x+y$$ Try a few cases to find pattern and then use induction to prove that the pattern works. P.S I saw ...
5
votes
1answer
761 views

Integer coordinate set of points that is a member of sphere surface

I have a graphic application to develop which involve many spheres. I should determine then on run time. Supposing that I have a sphere of radius r, how can I determine the sub set of the sphere ...
3
votes
3answers
511 views

Finding all positive integer solutions to $(x!)(y!) = x!+y!+z!$

The equation is $(x!)(y!) = x!+y!+z! $ where $x,y,z$ are natural numbers. How to find out them all?
1
vote
0answers
210 views

Quadratic fields and solving Diophantine equations

I would like to learn to solve Diophantine equations and I think my next step would be quadratic fields or number fields. What are kind of methods there are to use those on solving equations? And what ...
1
vote
1answer
94 views

Proving $\frac{m-n}{(m+1)(n+1)}=\frac{1}{k}$ for every $k>1$

How can we show that for any integer $k>1$ there are positive integers $m$ and $n$ such that $$\frac{1}{k}=\frac{m-n}{(m+1)(n+1)}.$$ (Thanks to Arthur Fischer for the reformulation!)
4
votes
3answers
2k views

$\mathbb Z[\sqrt 3]$ contains infinitely many units

I'm asked to show that there are infinitely many units in the ring $\mathbb Z[\sqrt 3]$. But I don't really see a good approach to this one, so far. Some thoughts: The inverse of $a+\sqrt3 b$ ...
2
votes
4answers
258 views

Form of rational solutions to $a^2+b^2=1$?

Is there a way to determine the form of all rational solutions to the equation $a^2+b^2=1$?
2
votes
2answers
106 views

Find a couple of integers such that the third power of a given natural can be written as the difference of the squares of those integers

Given a natural number $n$, find inegers $a, b$ such that $n^3=a^2-b^2$. I've tried, but I'm a bit rusty. Please Help
2
votes
2answers
94 views

Integer points of a circumference which radius in $n^{3/2}$

The question is: with a fixed integer $n$, what are the points with integer coordinates $(a,b)$ so that $a^2 + b^2 = n^3$? The equation is symmetric in $a$ and $b$, so if $(x,y)$ is a solution, then ...
0
votes
2answers
340 views

How to find solutions for linear equation?

I want to find the possible values for {$x_i$} for given that I know $y$, {$p_i$} and the sum of $x_i$. In other words, let: $$x_1 \cdot p_1 + x_2 \cdot p_2 + \cdots + x_n \cdot p_n = y$$ ...
1
vote
2answers
129 views

Is there an easy way to determine when this fractional expression is an integer?

For $x,y\in \mathbb{Z}^+,$ when is the following expression an integer? $$z=\frac{(1-x)-(1+x)y}{(1+x)+(1-x)y}$$ The associated Diophantine equation is symmetric in $x, y, z$, but I couldn't do ...
12
votes
5answers
850 views

Another quadratic Diophantine equation: How do I proceed?

How would I find all the fundamental solutions of the Pell-like equation $x^2-10y^2=9$ I've swapped out the original problem from this question for a couple reasons. I already know the solution to ...
2
votes
2answers
560 views

Using recurrences to solve $3a^2=2b^2+1$

Is it possible to solve the equation $3a^2=2b^2+1$ for positive, integral $a$ and $b$ using recurrences?I am sure it is, as Arthur Engel in his Problem Solving Strategies has stated that as a method, ...
4
votes
1answer
347 views

Finding pairs of integers such that $x^2+3y$ and $y^2+3x$ are both perfect squares

Can we find pairs $(x,y)$ of positive integers such that $x^2+3y$ and $y^2+3x$ are simultaneously perfect squares? Thanks a lot in advance. My progress is minimal.
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votes
7answers
2k views

Solutions to Linear Diophantine equation $15x+21y=261$

Question How many positive solutions are there to $15x+21y=261$? What I got so far $\gcd(15,21) = 3$ and $3|261$ So we can divide through by the gcd and get: $5x+7y=87$ And I'm not really ...
2
votes
1answer
393 views

Proving a Diophantine equation has no solutions

I'm trying to show that $7u^2=x^2+y^2+z^2$ has no solutions in $\mathbb{Z}$ when $u$ is odd. If $u$ is even, then it's simple to show that no solutions exists by looking modulo $4$. The odd case looks ...
5
votes
1answer
192 views

A question about the solutions of $y^2 = x^3 - 4$ for $(x,y) \in \mathbf Z^2$

Here's a question from an old examination paper: Find all $(x,y)$ in $\mathbf{Z}^{2}$ where $y$ is odd and $y^2=x^3-4$. Find all $(x,y)$ in $\mathbf{Z}^{2}$ with $y$ even and $y^2=x^3 -4$. ...
1
vote
3answers
139 views

Does method exist to solve Diophantine/Algebraic equation with nearest integer variable?

Can anyone kindly tell me if there is a method (other than trial and error) to solve equations of the form below: $$x^2 + x - 35 - 35[(x^2)/35] = 0$$ where $x$ is an integer and $[y]$ denotes the ...
0
votes
1answer
60 views

Can a^2 = 2b^2 have a solution where a, b are in Z but not zero? [duplicate]

Possible Duplicate: How can you prove that the square root of two is irrational? Can $a^2 = 2b^2$ have a solution where $a, b$ are in $\mathbb{Z}$ but not zero? $\mathbb{Z}$ = positive and ...
1
vote
1answer
228 views

Solution of Diophantine equation

I have seen $3^x$ + $3^y$ = $6^z$ and $4^x$ + $18^y$ = $22^z$ on lecture series of Prof. Gandhi. In my own study, I have constructed the following theorem (I am not sure about solvability) and I am ...
2
votes
1answer
186 views

A diophantine equation

I'm trying to determine the ideal class group of $\mathbb{Q}(\sqrt{223})$ using elementary methods. Is there an easy way to show that $a^2 - 223b^2 = - 3$ has no integer solutions? I've tried ...
1
vote
2answers
42 views

diophantine equation of the form $A(x)=a(rx+s)^{2}(x+t)$

It will now be shown that i) if $A(x)= a(rx+s)^{2}(x+t) ; a,r,s,t \in \mathbb{Z}; ar\ne 0$ then there are infinite $(x,y) \in \mathbb{Z}^{2}$ so that $y^{2}=A(x)$. ii) there is exactly 1 solution ...
2
votes
3answers
659 views

Showing that there are infinitely many integer solutions to $x^2+y^2=z^2$

Let $x,y,z \in \mathbb{N}$ with $x,y$ relatively prime, $x$ even and $x^{2}+y^{2}=z^{2}$. Show that there are infinitely many $(x,y,z)$ triplets which satisfy these conditions.
0
votes
4answers
251 views

On the solutions of $x^2+y^2=z^2$

Let $x,y,z \in \mathbb{N}$ where x is even; x,y are relatively prime and $x^{2}+y^{2}=z^{2}$. It will be tried to show that there exist $u,v \in \mathbb{N}$ relatively prime and $u> v$ and ...
3
votes
3answers
667 views

Pythagorean quadruples

Another Project Euler problem has me checking the internet again. Among other conditions, four of my variables satisfy: $$a^2+b^2+c^2=d^2 .$$ According to Wikipedia, this is known as a Pythagorean ...
1
vote
1answer
344 views

Find all solutions of this diophantine equation of the second degree in three variables

Consider the Diophantine equation $Q(x,y,z)=1$, where $Q(x,y,z)$ is the quadratic form $x^2+y^2-z^2$. Let $S \subseteq {\mathbb Z}^3$ denote the set of all solutions. It is rather easy to find several ...
7
votes
4answers
2k views

General formula to obtain triangular-square numbers

I am trying to find a general formula for triangular square numbers. I have calculated some terms of the triangular-square sequence ($TS_n$): $TS_n=$1, 36, 1225, 41616, 1413721, 48024900, 1631432881, ...
4
votes
2answers
4k views

Solving quadratic diophantine equations

I hope it's not inappropriate asking this here. I stumbled upon this site recently while researching a Project Euler problem, now I figure I'd use it to ask about a recurring theme in these problems: ...
13
votes
5answers
910 views

Integral solutions of $x^2+y^2+1=z^2$

I am interested in integral solutions of $$x^2+y^2+1=z^2.$$ Is there a complete theory comparable to the one for $x^2+y^2=z^2?$
3
votes
0answers
82 views

Do the solutions to the unit equation lie dense in the complex numbers

Let $S\subset \overline{\mathbf{Q}}$ be the set of solutions to the unit equation, i.e., $S$ consists of algebraic integers $a$ such that $a$ and $1-a$ are units in the ring of algebraic integers. ...
8
votes
1answer
518 views

Can $x^{n}-1$ be prime if $x$ is not a power of $2$ and $n$ is odd?

Are there any solutions to $x^{n}-1=p$ with p prime, integers $x,n>1$ and $x$ not a power of $2$? $x$ must be even. $n$ is odd since if $n=2m$ then $p=x^{n}-1=(x^{m}+1)(x^{m}-1)$ hence $p=x^{m}+1$ ...
4
votes
1answer
400 views

One sum of squares and two Diophantine equations

This question comes from trying to see why 24 is the only non-trivial value of $n$ for which $$1^2+2^2+3^2+\cdots+n^2$$ is a perfect square. To this end, let $m,n \in \mathbb N$ be such that ...
4
votes
1answer
187 views

Given $XX^\top=A$, solving for $X$

Not equal to this (my) own question. It's more general, probably more easy than the original question. All of the elements of $X$ and $A$ are integers. $XX^\top=A$ and $A$ is a symmetric matrix. ...
3
votes
0answers
365 views

$XX^t=A$, $X=?$. Where $X \in \{0,1\}^{n \times m}$

The problem: $XX^t=A$, $\quad$ ($X_{ij}\in{0,1}$, $\quad$ $\sum_{j=1}^m x_{ij}=2$), $\quad$ $X=?$ Details: $n,m \in N$ $A \in \{0,1,2\}^{n \times n}$ $X \in \{0,1\}^{n \times m}$ $A$ is a ...
1
vote
1answer
131 views

How many $n$'s can make $4m^2-n^2$ a perfect square? And, triple of a perfect square?

As the title reads. Given an integer $m\ge1$, how to calculate the number of integer $n$'s ($1\le n\le 2m$) such that $4m^2-n^2$ is a perfect square? Thank you~ Update: Further, how many pairs ...
0
votes
1answer
255 views

How to prove $\sqrt3$ is irrational? [duplicate]

How to prove $\sqrt3$ is irrational using Fermat's infinite descent method? Like says in Carl Benjamim Boyer's book. Isnt the same prove to $\sqrt2$, in Boyer's book says something like this. ...
11
votes
3answers
186 views

For which $n$ are there primitive Pythagorean triples with legs of lengths $a$ and $a+n$?

For which n can $a^{2}+(a+n)^{2}=c^{2}$ be solved, where $a,b,c,n$ are positive integers? I have found solutions for $n=1,7,17,23,31,41,47,79,89$ and for multiples of $7,17,23$... Are there ...
14
votes
2answers
862 views

What is the simplest ellipse that goes through exactly 13 lattice points?

The ellipse $-30 x + 3 x^2 - 10 y - 3 x y + 4 y^2$ goes through exactly 11 lattice points. Another such ellipse is $4 - 30 x + 2 x^2 - 5 y - x y + 3 y^2$. What is the simplest ellipse that goes ...
18
votes
2answers
346 views

All positive integer solutions to $\frac{1}{x_1}+\frac{1}{x_2}+\cdots+\frac{1}{x_n}+\frac{1}{x_1 x_2 \cdots x_n}=1$

As the title states, how would I go about finding the positive integer solutions of $$\frac{1}{x_1}+\frac{1}{x_2}+\cdots+\frac{1}{x_n}+\frac{1}{x_1 x_2 \cdots x_n}=1$$? Thank you for your help. ...
4
votes
2answers
707 views

Determine if equation will generate perfect squares

Given the following quadratic equations: $4n^2 + 128n - 131$ $4n^2 + 16n - 11$ $4n^2 + 24n - 3$ Is it possible to determine how many values of n will generate a perfect square? Or better yet, is ...
15
votes
4answers
1k views

Linear diophantine equation $100x - 23y = -19$

I need help with this equation: $$100x - 23y = -19.$$ When I plug this into Wolfram|Alpha, one of the integer solutions is $x = 23n + 12$ where $n$ is a subset of all the integers, but I can't seem ...
3
votes
1answer
317 views

How to find a three digit number which,when reversed, becomes equal to $17$ times the square of it's cube root?

How to find a three digit number which,when reversed, becomes equal to $17$ times the square of it's cube root? If we assume that the three digit number is of the form $100x+10y+z$,where $x \in ...
1
vote
1answer
420 views

Number of solutions of Frobenius equation

I have one problem which needs to count the number of solution of the equation $$2x+7y+11z=42$$ where $x,y,z \in \{0,1,2,3,4,5,\dots\}$. My attempt: I noticed that that maximum value of $z$ could ...
0
votes
1answer
82 views

what is $x^2+y^2+z^2=w^2$ when $x^2/y^2=y^2/z^2=z^2/x^2$?

What is the answer of $x^2+y^2+z^2=w^2$ when $x^2/y^2=y^2/z^2=z^2/x^2$ and $x, y , z, w\in\mathbb{N}$.
1
vote
1answer
87 views

Given a function $f(x)$, is there an analytic way to determine which integer values of $x$ give an integer value of $f(x)$?

Basically, I have some function $f(x)$ and I would like to figure out which integer values of $x$ make it such that $f(x)$ is also an integer. I know that I could use brute force and try all integer ...