Questions on finding integer/rational solutions of polynomial equations.

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3
votes
2answers
72 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
411 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
218 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
54 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
37 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 ...
4
votes
2answers
130 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
56 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
38 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
61 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
73 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
112 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 $a^4-b^4=c^2$.
3
votes
1answer
75 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 ...
7
votes
3answers
276 views

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
172 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
5answers
61 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
52 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
117 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
26
votes
1answer
484 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
82 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 ...
1
vote
1answer
71 views

Solution to the Diophantine equation $x^4+y^4=2z^2$ [duplicate]

Does there exists a nontrivial positive integer solution with $x\ne y,$ of $$x^4+y^4=2z^2.$$
1
vote
2answers
67 views

Generating all the Pythagorean triples by factorizing using complex numbers

Can anyone help me generate all the triplets solution of the Diophantine equation: $a^2+b^2=c^2$ by factoring using Complex numbers? thanks.
2
votes
2answers
38 views

An arctan problem including a diophantine equation

This is a follow-up question to An equation of the form A + B + C = ABC . I totally messed up with making the equation from the question specification . Actually the question was $$ ...
0
votes
1answer
103 views

An equation of the form A + B + C = ABC

So I was on a SPOJ spree until I came across this question . The question says $$\tan(\frac{1}{A}) = \tan(\frac{1}{B}) + \tan(\frac{1}{C})$$ where we have to find the $\min(B+C)$ for a fix $A$ where ...
2
votes
2answers
96 views

Diophantine equation: $n^p+3^p=k^2$

Find all solutions to the Diophantine equation $n^p+3^p=k^2$, where $p\in \mathbb{P}$ and $n,k$ positive integers. I have tried everything, from mods to bounding to LTE; nothing seems to work on ...
0
votes
1answer
40 views

Are there any parametric solution for this Pythagoras type quadratic Diophantine equation?

I wont to find a parametric solution for the Diophantine equations $x^2+y^2=m(m+1)$ and $x^2-y^2=m(m+1).$ I can simplify them up to $(2x)^2±(2y)^2+1=(2m+1)^2,$ but after that I'm stuck. How can I ...
-2
votes
2answers
345 views

Is this a solution for the problem: $\ a^3 + b^3 = c^3\ $ has no nonzero integer solutions?

Is this a solution for the problem: $\ a^3 + b^3 = c^3\ $ has no nonzero integer solutions? Suppose $\ a^3 + b^3 = c^3,\ a,b,c \in \mathbb Z^*,\ $then: $c^3 - b ^ 3 = (c - b)((c - b) ^ 2 + 3cb) = ...
2
votes
3answers
60 views

Diophantine Equatiοn $x^3=2^y+15$

I would like some help with the diophantine equation $x^3=2^y+15$ I have tried working with last digits and modular arithmetic but that hasn't got me anywhere.
2
votes
3answers
208 views

Diophantine equation $ax + by = c$ has an integer solution $x_0, y_0$ if and only if $\gcd(a,b)|c$

Let $a,b,c$ be positive integers. Verify that Diophantine equation $ax + by = c$ has integer solution $x_0, y_0$ if and only if $GCD(a,b)|c$. Attempt Diophantine $ax + by = c$ has integer solution ...
1
vote
1answer
45 views

how to show that $\mathbb{Q}[\sqrt[3]{2}]$ is a field? (by elementary means)

To be very concrete, I want to show that every element of the form $1/(p+qx+rx^2)$ where $x=\sqrt[3]{2}$ where $p,q,r$ are rationals can be written in the form $a+bx+cx^2$ where again $a,b,c$ are ...
3
votes
3answers
149 views

$x^2+1$ is almost always square free

It seems like $x^2+1$ is almost always square free. Any research or heuristics why? I tried breaking the problem into solving $$x^2-ky^2=1$$ For various $k$, and I conjecture that for every $k$ there ...
1
vote
2answers
42 views

Diophantine solution for a quadratic in two unknown variable

How do we determine integral solutions to the following equation: $$324x^2-8676x + 56700 = y^2$$ Where $x$ and $y$ are positive integers.
0
votes
3answers
138 views

number of ordered pairs of integers (x,y) satisfying the equation

i need to find number of ordered pairs of integers(x,y) satisfying below equation. $$x^2 + 6x + y^2 = 4$$ i have tried and i think x<0 . is there a specific way to solve such equations?
1
vote
3answers
34 views

number of solution to the given equation.

a,b,c, are all non-negative integers such that a + b + c=100 and 1000a + 300b + 50c = 10000 How many such triplets are possible? i have tried to reduce ...
3
votes
1answer
80 views

How to calculate the number of integer solution of a linear equation with constraints?

If an equation is given like this , $$x_1+x_2+...x_i+...x_n = S$$ and for each $x_i$ a constraint $$0\le x_i \le L_i$$ How do we calculate the number of Integer solutions to this problem?
2
votes
3answers
85 views

solve for three unknowns with two equations

Apple cost 97 dollars. Orange cost 56 and lemon cost 3. The total amount spent is 16047 dollars and total fruits bought is 240 and each one is bought atleast one. Calculate how many of each have been ...
4
votes
2answers
279 views

Solutions to the Mordell Equation modulo $p$

It is well known that for any nonzero integer $k$ the Mordell Equation $x^2 = y^3 + k$ has finitely many solutions $x$ and $y$ in $\mathbf Z$, but it has solutions modulo $n$ for all $n$. One proof of ...
1
vote
2answers
48 views

Diophantine solution to a fraction

How can we find solutions to the following equation: $$ y=\dfrac{x^2-1085}{14718-2x}$$ where $x,\ y$ are integers.
2
votes
1answer
37 views

Primitive-recursive functions and polynomial equations

I am looking for examples of primitive-recursive functions $f:\mathbb{N}\rightarrow\mathbb{N}$ that can not be written as a pair of polynomials, i.e. $$f(n) = m \Leftrightarrow P(n,m) = Q(n,m)$$ ...
3
votes
3answers
149 views

Find all integer solutions of $1+x+x^2+x^3=y^2$

I need some help on solving this problem: Find all integer solutions for this following equation: $1+x+x^2+x^3=y^2$ My attempt: Clearly $y^2 = (1+x)(1+x^2)$, assuming the GCD[$(1+x), (1+x^2)] = ...
3
votes
2answers
139 views

A transcendental number from the diophantine equation $x+2y+3z=n$

Let $\displaystyle n=1,2,3,\cdots.$ We denote by $D_n$ the number of non-negative integer solutions of the diophantine equation $$x+2y+3z=n$$ Prove that $$ \sum_{n=0}^{\infty} ...
0
votes
2answers
71 views

Solving $4a+5b=27$ where $a,b$ are two different positive integers

This problem is making my head spin: The costs of equities of symbol A and symbol B (in dollars) are two different positive integers. If $4$ equities of symbol A and $5$ equities of symbol B together ...
-2
votes
3answers
166 views

Equation $a^{n}+b^{n}=2008$ has no integers solutions. [closed]

Prove that the equation $a^{n}+b^{n}=2008$ has no solutions for $a,b,n\in\mathbb{Z}, n\geq2.$
3
votes
4answers
200 views

Diophantine equation abc + abd + acd + bcd= 1

Is there a reference which classifies or at least gives an infinite family of integer solutions to the above equation? A solution to the problem would also be great obviously.
0
votes
1answer
41 views

integral point on conics

Suppose we have a conic $ax^2 + bxy + cy^2 + dx + ey + f = 0$ where $a,b,c,d,e,f \in \mathbb{Q}$. Is there a way of computing the integer points on this curve. Since it is affine an not projective we ...
0
votes
0answers
43 views

Solving a general Diophantine Equation

For "normal" equations in one variable we have several techniques for solving equations, such as $\sin(5x) = 5\pi\cos(5x)$ or $\ln(x + 2) = 4$. However, imagine we have the following equations: ...
0
votes
0answers
46 views

Looking for solutions to $xy^2 = (1 + z)^2 (5 + 8z)$ in integers

I've been reading about Weierstrass equations and shifted Weierstrass equations and Mordell curve and elliptic curves, but so far I haven't been able to transform my equation to any of this type. ...
1
vote
0answers
62 views

How to solve the diophantine equation $x^y = y^x $? [duplicate]

I know this might be an obvious question as we all know that the answers (beside $(x,y)=(1,1)$) are $(x,y)=(2,4)$ but the problem is, how is this exactly solved? Tags might be inaccurate so feel free ...
2
votes
2answers
92 views

Integer solutions of the equation: $x^2+y^2+z^2=kxyz$

Given the equation: $$x^2+y^2+z^2=kxyz$$ with: $(k,x,y,z)\in\mathbb{N}$, the only solution for $k=2$ is: $x=0,y=0,z=0$. For what values of $k$ the equations has solutions in which $x,y,z$ are ...