Questions on finding integer/rational solutions of polynomial equations.

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0answers
22 views

coin problem with two coins, inductive proof

I want to ask something about the coin problem with two coins. Let $a,b$ be to numbers in $\mathbb{N} \setminus \{0\}$ (elsewhere I include zero) which have no prime factors in common. I will write $$ ...
-7
votes
0answers
42 views

Solving a system of polynomial equations in three variables (x^2-yz=18, y^2-zx=8, z^2-xy=-7)

Solving a system of polynomial equations in three variables (x^2-yz=18, y^2-zx=8, z^2-xy=-7 I've tried rearranging each equation to isolate for one variable ex: z^2-xy=-7 --> z= x^2-18/y after, I ...
0
votes
1answer
60 views

Solve $i^3j-j^3i=x^3y-y^3x$

Do anyone have an idea about how to solve this kind of equation: $i^3j-j^3i=x^3y-y^3x$ where $i,j,x,y$ are distinct natural numbers and $i>j$ and $x>y$ Regards
0
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0answers
23 views

What are some [mostly trivial] Pell transformations?

Euler looked at some transformations which turned one Pell[-type] equation into another. Example 1: $$u^2-av^2=-1 \quad\iff\quad (2u^2+1)^2-a(2uv)^2=1.$$ Example 2: $$u^2-av^2=-2 \quad\iff\quad ...
2
votes
1answer
42 views

solve the diophantine equation: $x^3-3xy^2=z^3$

Let $ x,y,z$ be 3 integers greater than 1,if $x$ and $y$ are relatively prime, solve the diophantine equation: $x^3-3xy^2=z^3$.
-1
votes
1answer
19 views

How can we generate a $2$-digit number $XY$ on base $B$, such that $BX+Y=Y^X$?

For example, $25$ on base $10$ is equal to $5^2$. This should be pretty easy to solve using fairly simple arithmetic. But I'm finding it hard to generate any other solutions besides the one ...
1
vote
1answer
23 views

“Descent” on binary quadratic forms?

Let's say I have the Diophantine equation $$ x^2+3n^2 = y^2+3z^2. \tag{$\star$} $$ where $n$ is a known integer, and we're trying to determine solutions in integers $x,y,z \ge 1$. Rewrite ($\star$) ...
0
votes
0answers
25 views

Superelliptic curves

I'm trying to find information on superelliptic curves and how to solve them over the integers. The equation is $$y^k = f(x)$$ where $k=3$ and $f$ has degree $d=3$. Does anyone know any ...
1
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0answers
21 views

Powerful numbers in Pell solutions (or, more generally, any Lucas sequence)

There are several definitive results regarding perfect powers in the Pell numbers — e.g., the only perfect power is $P_7=169=13^2$. On the other hand, when it comes to powerful numbers, I've only ...
5
votes
1answer
39 views

Conditions on solutions of a diophantine equation.

I wanted to list all the natural number solutions $(d_1,d_2,...,d_n)$ to the equation: $$\sum_1^n \frac1{d_i} = 1$$ I could not succeed. I noted that for $n=4$, $(2,4,8,8), (3,3,6,6), (2,3,12,12), ...
3
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0answers
68 views

Looking for help with this elementary method of finding integer solutions on an elliptic curve.

In the post Finding all solutions to $y^3 = x^2 + x + 1$ with $x,y$ integers larger than $1$, the single positive integer solution $(x,y)=(18,7)$ is found using algebraic integers. In one of the ...
1
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0answers
40 views

Given a Pell “solution” in [integer] polynomials, what can be said about the components?

Let $x,y$ be integers, and $f(x,y)$, $g(x,y)$, and $h(x,y)$ be polynomials in $x$ and $y$ with integer coefficients such that $$ f(x,y)^2 - g(x,y)h(x,y)^2 = 1. \qquad(\star) $$ Furthermore, assume it ...
2
votes
1answer
56 views

The Diophantine equation $x^p - 4y^p = z^2$ with $(x, y) = 1$ and $x, y, z > p.$

If $p \geq 5$ is a prime, are there any integers $x, y, z > p$ such that $(x, y) = 1$ and $$x^{p} - 4y^{p} = z^{2}$$
0
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0answers
39 views

The Diophantine equation $x^2 - y^2 = 4z^n$

If $n \geq 3$ is an integer, then under what conditions on $x, y, z$ does the equation $$x^2 - y^2 = 4z^n$$ have no solution in integers? (If there is any known result, please feel free to share with ...
0
votes
2answers
73 views

Find the non-trivial solutions of the diophantine equation: $a^3+3a^2b=c^3$

If $ a$ and $b$ are co-prime integers >2, can $a^3+3a^2b$ be a cube?
4
votes
3answers
107 views

Find all positive integers satisfying $\frac{2^n+1}{n^2} =k $

Find all positive integers satisfying $$\frac{2^n+1}{n^2} =k $$ where $k$ is a integer. I can't just come up with a solution.
9
votes
2answers
124 views

Solving $n! + 3n = k^2$

Let $n$ and $k$ be integers. Need to solve $n! + 3n = k^2$, where $n!$ denotes $n$ factorial. I do not have any ideas about this equation, except I suppose the only $6$ roots are $(0,1), (0, -1), ...
0
votes
1answer
36 views

Solution to Diophantine equation with constraint.

solve the following equation over $z_x,z_y$ \begin{align} &az_x=bz_y\\ &\text{s.t. } a,b,z_x,z_y \in \mathbb{Z} \text{ and } 1 \le z_x \le N \text{ and } 1 \le z_y \le N \end{align} How ...
4
votes
1answer
48 views

Find integral solutions for $2x^2+y^2=2\times(1007)^2+1$

Find integral solutions to the equation $$2x^2+y^2=2\times(1007)^2+1$$ I tried: I rewrote the equation as $2x^2+y^2=2028099$. I found that $y_{max}=1424$ and $y$ must be odd, so I set ...
1
vote
2answers
44 views

Given a Pell solution $(u_k,v_k)$, is there a closed form “descent” to $(u_{k-1},v_{k-1})$?

Given: a solution $(u_k,v_k)$ to the Pell equation $$U^2-dV^2=1, \qquad(\star)$$ where $d$ is a non-square integer, and $k \ge 1$ is an arbitrary integer. There are well-known recurrences to ascend ...
2
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0answers
50 views

The Diophantine equation $x^n - y^n = z^2$

Darmon-Merel theorem (DMT) ensures that if $n \geq 4$ is an integer and $x, y, z > 0$ are integers such that $(x, y, z) = 1$ then $x^n + y^n \neq z^2.$ The question is: Does DMT apply to the ...
4
votes
0answers
65 views

Binomial triplets

Solutions to the equation $$\dbinom{a}{n}+\dbinom{b}{n}=\dbinom{c}{n}$$ I will refer to as 'Binomial triplets of order $n$'. These triplets describe simplicial $n$-polytopic numbers that can be ...
2
votes
1answer
55 views

Hard Simultaneous Diophantine Equations

Find all positive integers $a,b,c,d,e,f$ such that : $de^2=ab^2+1$ and $df^2=ac^2+1$. I tried subtracting them, it factors quite nicely. But after that, haven't a clue. I'm not sure if it's even ...
0
votes
3answers
51 views

how can I find Integer solutions for the two variables equation without searching factors?

If $(30X+7)(30Y+1) = 50437$, then what are the integer solutions? Any way to solve it without searching $50437$ factors?
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1answer
35 views

how can I find Integer solutions for the two variables equation?

how can I find Integer solutions for the following equation: $$900 X \, Y + 210 Y + 30 X - 3 \times 10^{100} + 7 = 0$$
0
votes
2answers
79 views

Find all non-trivial triplets $(a,b,c)$ such that $ a+b^{d}\equiv0\pmod c $ for all $d>0$

Let $a,b,c$ be co-prime integers $>2$ . Find all non-trivial triplets $(a,b,c)$ such that $ a+b^{d}\equiv0\pmod c $ for all $d>0$.
1
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2answers
50 views

Diophantine equation with perfect squares

Find all the integer solutions of the equation: $$(n^2-4)n = 3b^2$$ Progress I tried casework based on what $n$ is modulo $3$ but it didn't work.
1
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0answers
26 views

Characterizing Coprimes

Here's a question about coprimes that I stumbled upon while doing some research. Providing insight into this question would prove quite helpful to me. Choose a pair of coprimes $x, y \in \mathbb Z$. ...
3
votes
2answers
55 views

Which natural numbers can be represented as a sum of natural numbers raised to different powers?

Waring's problem asks about natural numbers that can be represented as a sum of natural numbers all raised to the same power $k$. I'm wondering which natural numbers can be represented as a sum of ...
7
votes
3answers
373 views

How to find natural solutions of an equation?

When I'm solving problems, I'm often confronted to solving equations, and when I'm solving equations, I'm often confronted to find the natural solutions of these equations. My actual personal ...
10
votes
2answers
202 views

Cubic polynomial equal to a cube

I've been researching cubes and I'm trying to solve this Diophantine equation over the integers. $$ax^3 + bx^2 + cx + d = y^3$$where a, b, c, d are parameters for a given $n$. For example, for $n = ...
0
votes
0answers
44 views

Diophantine Equations involving cubes

I'm doing some number theory research and I came across these two Diophantine equations (created under my own transformations): $$y^3 = ax^3 + bx$$ (where $a$ and $b$ are parameters) $$z^3 = x^2 + ...
0
votes
0answers
36 views

Hard Diophantine: $ xy-\frac{(x+y)^2}{n}=n-4 $

Solve in positive integers $x,y$: $ xy-\frac{(x+y)^2}{n}=n-4 $ $n>4$ is a given positive integer. I cannot even solve in the case $n=5$. I have been able to find $x,y$ and construct $n$ using ...
3
votes
2answers
121 views

Solving $x^3+y^3=x^2y^2+1$ in non-negative integers

I wanted to solve $x^3+y^3=x^2y^2+1$ in non-negative integers. First I set $a=x+y$ and $b=xy$ to get $b^2+3ab+1=a^3$. View as a quadratic in $b$, the discriminant = $4a^3+9a^2-4$, which needs to be a ...
0
votes
1answer
52 views

Solve for x,y: $x^2+1=2y^2$

Solve for integers $x,y$ such that $x^2+1=2y^2$? I tried factoring as $(x-y)(x+y)=(y-1)(y+1)$ but couldn't continue from here, I would appreciate any help. Thanks!
2
votes
2answers
25 views

Prove that the Diophantine equation $ax+by=c$ has no integer solutions if $\gcd(a,b)$ does not divide $c$

Prove that the Diophantine equation $ax+by=c$ has no integer solutions if $\gcd(a,b)$ does not divide $c$ there was a hint which is use use contradiction.
2
votes
2answers
57 views

Diophantine: $px^2+2=y^2$ where $p\in \mathbb{P}$

Solve the Diophantine Equation: $px^2+2=y^2$, where $p$ is a prime number and $x,y$ integers. I tried this for ages but didn't get anywhere, but I don't know any advanced machinery since I am only in ...
3
votes
5answers
59 views

Determine variables that fit this criterion…

There is a unique triplet of positive integers $(a, b, c)$ such that $a ≤ b ≤ c$. $$ \frac{25}{84} = \frac{1}{a} + \frac{1}{ab} + \frac{1}{abc} $$ Just having trouble with this Canadian Math ...
5
votes
1answer
105 views

Fermat: Prove $a^4-b^4=c^2$ impossible

Prove by infinite descent that there do not exist integers $a,b,c$ pairwise coprime such that $a^4-b^4=c^2$.
4
votes
1answer
68 views

Solving $x^2 - 11y^2 = 3$ using congruences

I'm looking to find solutions to $x^2 - 11y^2 = 3$ using congruences. The question specifically asks "Can this equation be solved by congruences (mod 3)? If so, what is the solution? (mod 4) ? (mod ...
8
votes
6answers
1k views

Are there finitely many Pythagorean triples whose smallest two numbers differ by 1?

Has it been shown whether there is a finite or infinite number of Pythagorean triples whose smallest two numbers differ by 1? In either case I’d appreciate a link to the proof. Edit: thank you all ...
6
votes
3answers
225 views
+150

System of Diophantine Equations

I'm working on this problem I came across on the internet but I have no solution yet. The problem states: Find all prime numbers p that are such that $p+1=2x^2$ and $p^2+1=2y^2$ where x and y are ...
5
votes
1answer
162 views

How to prove that $3ab(a+b)$ cannot be a cube?

Consider the diophantine equation: $ 3ab(a+b)=c^3 $ where $a,b,c$ are non-zero integers,how do you prove that this equation has no integral solutions?
3
votes
0answers
76 views

Prove the equality

Given $a,b,c,d$ are positive integers such that $a^2+b^2+c^2+d^2-ab-bc-cd-da$ is divisible by $abcd$. Prove that $a=b=c=d$.
1
vote
4answers
46 views

Linear Diophantine equation $3x + 5y = 11$

Solve the Diophantine equation $3x + 5y = 11$ I know how to calculate GCD $$5 = 1\cdot 3 + 2$$ $$3 = 1\cdot 2 + 1$$ $$2 = 2\cdot 1 + 0$$ But how do I use this theorem to derive the correct ...
1
vote
2answers
47 views

Solving Multiple Equations with Many Variables

Here's a problem I have stumbled upon, which may have a straightforward solution with linear algebra. If so, I cannot see it. Choose $n > 0 \in \mathbb N$, and consider the sequence of equations: ...
0
votes
6answers
105 views

Solve $f^2-e^2=d^2-c^2-b^2+a^2$

I'm looking for a solution or some clarifications for this equation: $f^2-e^2=d^2-c^2-b^2+a^2$ with $f>e$, $d>c$, $b>a$ and $f, e, d, c, b, a$ natural numbers. Regards
27
votes
1answer
459 views

For integers $a\ge b\ge 2$, is $f(a,b) = a^b + b^a$ injective?

Given two integers $a \ge b \ge 2$, can we encode them as a unique integer $a^b + b^a$? This question was asked a few weeks ago, but did not rule out the trivial cases. For example, if we ...
2
votes
1answer
77 views

Exponential Diophantine: $2^{3x}+17=y^2$

Is there a way of solving the following equation, in integers $(x,y)$, by hand? : $2^{3x}+17=y^2$. You can also try: $2^{2x}+17=y^2$ or more generally $2^x+17=y^2$; each of these has at least 1 ...
0
votes
0answers
29 views

How to solve this class of diophantine forms

I found a class of equations with the following form. $$A (Bm)^k | (Cm^2 + Dm + E)^n$$ $ m \ge 12$ can be any rational number, $n > k$ are natural numbers. $ 0 < A < 1$ is fixed and the ...