Questions on finding integer/rational solutions of polynomial equations.

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3
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1answer
51 views

Cubic Congruence Solutions

While I was reading a paper on number theory, there was a claim which wasn't prove there and I couldn't find a way to justify it. The claim is as follows For a prime $p$, when $p\nmid a$, the number ...
6
votes
4answers
801 views

Correct statement of Fermat's Last Theorem

I'm looking at the wikipedia page on Fermat's Last Theorem In the statement it requires $a,b,c$ to be positive integers. Is that correct? I always took it to be no solutions in non-zero integers. ...
1
vote
1answer
83 views

Solving to find the general equation with a “mod” equation

They probably aren't called "mod" equation but i couldn't think how else to word them, so I have this equation $8x + 10y ≡ 8 \pmod 7$ And have been tasked with finding the general solution, I know ...
0
votes
1answer
23 views

Integer solutions of a degree 3 curve

Suppose you have a square pyramid made out of rigid balls and all these balls are equal. Suppose now that you want to fill a square with the same number of balls that the pyramid is made. If $x$ ...
2
votes
2answers
59 views

Types of elliptic curves

I'm trying to research elliptic curves, and I always get the generic equation $$y^2 = a_0 x^3 + a_1 x^2 + a_2 x + a_3.$$ However, I'm looking for information on an equation like $$y^3 = a_0 x^3 + a_1 ...
3
votes
1answer
42 views

Solving Mordell Equations

I am looking at the solution provided in my lecture notes for solving this particular mordell equation: $$y^2 = x^3 − 2$$ which factors into: $$ (y- \sqrt {-2})(y+ \sqrt {-2}) = x^3 $$ In the ...
0
votes
0answers
62 views

Finite solution of Power Diophantione Equation.

Given an equation $x^2+k=y^3$ where k is a constant and $y=f(x)$,$f(x)$ is differentiable and algebraic. for which- $$\frac{d}{dx}x^{2} \neq\frac{d}{dx} f(x)^3$$ 1. Can I infer that the ...
0
votes
1answer
76 views

equation $x^4 + y^4 = z^4$

Diophantine equations that are insoluble in $\mathbb{Z}$ may become soluble in finite integral domains. Show that \begin{equation*} x^4 + y^4 = z^4 \end{equation*} is soluble (as a congruence) in ...
1
vote
2answers
230 views

How to solve 3 variable in 2 equation?

This paper is abstracted from 2007 British Mathematics Olympiad Round 1 Question 2. I am currently practicing grade 8 (Singapore Secondary 2) for the upcoming Singapore Mathematics Olympiad(SMO). ...
4
votes
1answer
87 views

30th problem of the fifth book of Diophantus;

Is there a complete answer to this problem? I have found Saunderson's answer, but I believe it is missing a few answers. The problem states: $a^2+b^2=d^2 \\ a^2+c^2=e^2 \\ b^2+c^2=f^2$ Saunderson ...
2
votes
0answers
41 views

Diophantine eqution with odd prime

HOW to find all possible set of solutions of an equation type $y^p \pm 2 = x^2$, where $p$ is any odd prime High regards to one and all
2
votes
1answer
59 views

Solvability of the equation $2a_{1}^{2} = a_{2}^{n} + a_{3}^{n} + a_{4}^{n}$ when $n \geq 5$ is prime?

As a natural extension of the question titled Solvability of $a_{1}^{2} = a_{2}^{n} + a_{3}^{n} + a_{4}^{n}$ when $n \geq 5$ is prime?, I wonder if the equation $$2a_{1}^{2} = a_{2}^{n} + a_{3}^{n} + ...
11
votes
0answers
88 views

Congruence properties of $x_1^6+x_2^6+x_3^6+x_4^6+x_5^6 = z^6$?

It is known that given primitive (co-prime) integer solutions to, $$x_1^4+x_2^4+x_3^4+x_4^4 = z^4$$ then there is one $x_i$ such that $z^4-x_i^4$ is divisible by $d_4=5^4$. Additionally, Ward ...
0
votes
1answer
71 views

Determining all the positive integers $n$ such that $n^4+n^3+n^2+n+1$ is a perfect square.

I successfully thought of bounding our expression examining consecutive squares that attain values close to it, and this led to the solution I'll post as an answer, which was the one reported. ...
0
votes
2answers
54 views

Solving the equation in natural numbers

How can I find the solutions in natural numbers for the following equation? $$a_{1}x_{1}+a_{2}x_{2}+\cdots +a_{n}x_{n}=b$$ Where $x_{1},...,x_{n}$ are unknown. I want to find the whole of solutions ...
1
vote
2answers
62 views

Solvability of $a_{1}^{2} = a_{2}^{n} + a_{3}^{n} + a_{4}^{n}$ when $n \geq 5$ is prime?

Aware of a Darmon-Merel theorem that asserts that if $n \geq 5$ is prime then the equation $a_{1}^{2} = a_{2}^{n} + a_{3}^{n}$ has no solution in relatively prime integers $a_{1}, a_{2}, a_{3},$ I ...
0
votes
0answers
48 views

prove that the number of solutions is finite

Prove that \begin{equation*} \frac{1}{x_1}+\frac{1}{x_2}+\frac{1}{x_3}+\cdots +\frac{1}{x_n}=1,~∀i,x_i\in \mathbb{Z^+} \end{equation*} has a finite number of integer solutions. I tried to solve ...
4
votes
2answers
85 views

Find a non-trivial solution for the Diophantine Equation $17a^4 + 5b^4 = 35c^4$, or show that no non-trivial solutions exist

This is a problem on my practice exam for number theory, and we haven't had an example like this in class yet. The question is looking for a solution in $\mathbb{Z}$ for $a,b,c \in \mathbb{Z}$. I've ...
4
votes
1answer
75 views

Solve in positive integers: $5x^2+6x^3=z^3$

Solve in positive integers: $5x^2+6x^3=z^3$. $x^2(6x+5)=z^3$ If $(x,5)=5$, let $x=5k$. So $k^2(6k+1)=\left(\frac{z}{5}\right)^3$, we're left with solving $6n^3+1=m^3$. If $(x,5)=1$, ...
0
votes
0answers
43 views

A nice set of squares. [duplicate]

Are there integers $a, b, c, d$ such that $$a^2+b^2=c^2$$ $$a^2-b^2 = d^2?$$ I have tried by showing that $a^2 = b^2 + d^2$ and thus $a^2+ b^2 = 2b^2+d^2 = c^2$ But how do I show that there are no ...
0
votes
1answer
34 views

Exponential diophantine equation

Need some help regarding the equation $$2^a-3^b=(2^c-1)\cdot d >0$$ where $a,b,c,d$ are integers; $a,b$ are fixed; and $c>2$. Can we show that $c,d$ exist? Thank you!
1
vote
2answers
56 views

Show that there are exactly 16 pairs of integers $(x,y)$ such that $11x+8y+17=xy$.

Original problem Show that there are exactly 16 pairs of integers $(x,y)$ such that $11x+8y+17=xy$. My work From case by case analysis I come to know that the equation will hold if and only if $x$ ...
3
votes
3answers
42 views

Split 16 Consecutive Integers into Two Subsets of 8 Integers

Show that any given set of sixteen consecutive integers {$x+1,x+2,\ldots,x+16$} can be divided into two eight element subsets with the properties that they have the same sum, the sums of the squares ...
6
votes
2answers
101 views

Only finitely many $a, b$ such that $2+3^n+5^{n^2}=2^a7^b$ for some $n$?

Let $(a,b)$ be a pair of positive integers such that $$2+3^n+5^{n^2}=2^a7^b$$ for some positive integer $n$. Is it true that there are only finitely many such pairs? I don't know the answer to such ...
0
votes
0answers
25 views

Solving the inverse loop

I was making playing around with the solutions of the equation of a circle in the origin $\{ (x^2)+(y^2) = (r^2) \mid\text{ solutions }y = \pm((r^2)-(x^2))^{1/2} \}$, when I tried to find the inverse ...
4
votes
2answers
89 views

$2^x+7^y=19^z$ has no solution in positive integers $x$, $y$, $z$

How do I show that the diophantine equation $2^x+7^y=19^z$ has no solution in positive integers $x$, $y$, $z$
1
vote
1answer
77 views

Trouble forming general solution for linear congruence

I was given $$ 6x+14y=4 \space \mod 5 $$ I took this approach: $$ 6x+14y-5z=4, \space \text{ for some } z $$ Let $$ w=\frac{6}{(6,14)}x+\frac{14}{(6,14)}y $$ Then, $$ (6,14)w+5z=4 \quad , \quad ...
11
votes
2answers
161 views

Diophantine equation $x^2 + xy + y^2 = \left({{x+y}\over{3}} + 1\right)^3$.

Solve in integers the equation$$x^2 + xy + y^2 = \left({{x+y}\over3} + 1\right)^3.$$
1
vote
1answer
79 views

Solving a Diophantine equation: $y^x=x^{2007}$, $x$ and $y$ integers.

I found this Diophantine equation and to solve it I used the definition of logarithm but the solution doesn't require the use of logarithmic rules. I solved it in this way: $$y^x=x^{2007}$$ ...
1
vote
1answer
39 views

Splitting Field of a real cubic

Let $f(X)$ be a cubic with 3 real roots, integer coefficients irreducible over $\mathbb{Q}$. Let $\alpha$ be one of these roots, and consider the number field $\mathbb{Q}(\alpha)$. Dirichlet's unit ...
2
votes
0answers
61 views

Finding all solutions: $a^2 + b^2 = c^2 + d^2$

I want to find all solutions to the problem of two squares equaling two other squares. $$a^2 + b^2 = c^2 + d^2 \qquad b \le N$$Clearly, without loss of generality, I can assume that $$gcd(a,b,c,d) = ...
3
votes
1answer
29 views

Solvability of the Diophantine equation $x^{2} - y^{2} = 4z^{n}$?

It is known that for every integer $z$ there are integers $x, y$ such that $x^{2} - y^{2} = z^{3}.$ In fact, given an integer $z$, taking $x := z(z+1)/2$ and $y := z(z-1)/2$ suffices. But how is the ...
0
votes
1answer
35 views

Solve in positive integers $a^2-b^2+4a=0$

Solve in positive integers $$a^2-b^2+4a=0$$ I tried considering the residues in mod4 but not so helpful. Any help/hint on how to approach this problem ? Thanks !
0
votes
1answer
41 views

The genus of a certain kind of cubic

I have a cubic curve that looks like $$ a_0 x^3 + a_1 x^2 y + a_2 xy^2 + a_3 y^3 = b $$ with $a_0, a_1, a_2, a_3$, and $b$ all integers, and $a_0$ and $b$ nonzero. I'm not sure but I think in my ...
3
votes
1answer
55 views

Is there any solution to this quadratic Diophantine 3 variables equation?

Is it possible to find all positive integer triplets $(x,y,z)$ satisfying the parametric equation : $$x^2 + 2ax + y^2 + 2by = z^2 + 2cz$$ Here $a, b, c$ are fixed positive integers.
45
votes
3answers
758 views

Finding triplets $(a,b,c)$ such that $\sqrt{abc}\in\mathbb N$ divides $(a-1)(b-1)(c-1)$

When I was playing with numbers, I found that there are many triplets of three positive integers $(a,b,c)$ such that $\color{red}{2\le} a\le b\le c$ $\sqrt{abc}\in\mathbb N$ $\sqrt{abc}$ divides ...
9
votes
2answers
125 views

How to prove the cubic formula without root extraction

I'm trying to prove the cubic formula, in the following form: Given a field $F$ and $x,p,q\in F$, define $m=\frac p3$ and $n=\frac q2$, and suppose also that $\gamma,\tau$ are given such that ...
5
votes
2answers
384 views

Diophantine Equations : Solving $a^2$ $+$ $ b^2$ $=$ $2c^2$

I was working through some number theory problems , when I came across the following question : Find all solutions of $a^2$ $+$ $b^2$ $=$ $2c^2$ My Solution (Partial) : We can rewrite the ...
0
votes
3answers
41 views

Diophantine Equations : Solve $a^2 + b^2 = 4c + 3$

I was working my way through some number theory problems , when I came across the following question : Find all solutions to the equation $a^2 + b^2 = 4c + 3$ My Solution (partial) : If ...
2
votes
3answers
69 views

$a^2 = 2b^3 = 3c^5$ Find the smallest value of $abc$.

We have following equation: $a^2 = 2b^3 = 3c^5$ Where $a, b, c$ are natural numbers. Find the smallest possible value of product $abc$.
2
votes
2answers
68 views

Find the probability of solutions of an equation.

Let $x+y+z=20$. What is the probability that all the solutions are distinct? (No two variables have the same value). Assuming that the solutions are only positive integers or zero. I have tried- ...
0
votes
2answers
38 views

Strategy to find the most money to use.

As a reward for a week of good behavior, Tommy was given 7 dollars to spend at the canteen. By the time Tommy got to the canteen, there were only chocolate bars, meat pies and pizza pieces left. The ...
-2
votes
1answer
41 views

Trouble with two equations with 4 unknowns [closed]

I was wondering if I could receive assistance for the following system: $$\begin{cases}(x/a)^{3.2}+(y/b)^{3.2}=1\\ a/b = 174.1/86\end{cases}$$ I'm looking for integer solutions or how to find them ...
2
votes
2answers
44 views

How to find the number of values for $x$ and $y$?

I have come across numerous questions where I am asked for example, if $x$ and $y$ are non-negative integers and $3x + 4y = 96$, how many pairs of $(x,y)$ are there? Usually, I just use trial and ...
1
vote
2answers
25 views

Proving expressibility of integers as the difference of two squares.

I'm given the task: Prove that a positive integer is expressible as the difference of two squares of integers if and only if it is not of the form $4n+2, n\in\mathbb{Z}$ I was given a hint that I ...
1
vote
0answers
35 views

The exponent on Thue's theorem

I have been reading about Runge's theorem on diophantine approximation Theorem. Let $\xi$ be an algebraic real number of degree $d\geq 3$. For every $\epsilon >0$ there is a number $\gamma >0$ ...
1
vote
2answers
68 views

Intersection of integer sets

This is probably a trivial question for mathematicians but I am not seeing how to approach the following problem: Imagine two sets defined by: ...
0
votes
1answer
85 views

Gorgeous diophantine equation [closed]

How to find all integer solutions of the following equation? $$y^7=14 \cdot 3^{100}x^6 + 70 \cdot 3^{300} x^4 + 42 \cdot 3^{500} x^2 + 2 \cdot 3^{700}$$
10
votes
1answer
300 views

How prove this systems-equation has least two postive integers solution

Show that: for any $k\ge 100,(k\in N^{+})$, there exsit $p\in N^{+}$, such $$\begin{cases} a+b+c=k\\ abc=p\\ a>b>c \end{cases}$$ has at least two postive integers solution $(a,b,c)$ ...
6
votes
2answers
106 views

Pythagorean Triples : Is every positive integer $\gt$ $2$ part of at least one Pythagorean triple?

I was doing some basic number theory problems from Rosen and came across this problem: Show that every positive integer $\gt$ $2$ is part of at least one ...