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Questions tagged [diophantine-equations]

Use for questions about finding integer or rational solutions to polynomial equations.

4
votes
1answer
88 views

Is Fermat's last theorem provable in Peano arithmetic?

The sentence $S$ which Gödel in his proof of the incompleteness theorem proves to be be unprovable in the system of Peano arithmetic can be proved (as a true theorem of PA) outside PA (and necessarily ...
1
vote
1answer
94 views

The equation $x^4 − y^4 = z^p$

In Darmon's paper The equation $x^4 − y^4 = z^p$, he says: "Factorizing the left hand side of $a^4 − b^4 = c^p$, the assumption $\gcd(a, b) = 1$ forces the three factors $a + b, a − b, a^2 + b^2$ to ...
0
votes
1answer
26 views

Linear Diophantine Equation Signs

How to determine the coefficient signs for the solution of a linear diophantine equation? Take $24x + 69y = 33$ for example. I know the solution is $x = 33 − 23k , y = −11 + 8k$, and I understand ...
2
votes
2answers
67 views

Solve the following Diophantine equation.

Find all integral pairs (x,y) such that - $$( xy - 1)^2 = (x +1)^2 + ( y+1)^2$$ My Approach : I just expanded this equation and wrote it in another form - $$\frac{(xy+1)(xy-1)}{(x+y)}-2=x+y$$ and ...
3
votes
1answer
73 views

Find solutions for $\lambda$ in this (diophantine?) equation?

I am interested in finding the solutions to the equation below: $$\lambda = \frac{2^c+3b}{2^d-3^a}$$ $$\lambda, a, b, c, d \in \Bbb Z^+$$ $$d>a,c$$ Edit: My original version was actually: $$\frac{...
10
votes
2answers
131 views

Solve $ \binom{a}{2} + \binom{b}{2} = \binom{c}{2} $ with $a,b,c \in \mathbb{Z}$

I am trying to solve the Diophantine equation: $$ \binom{a}{2} + \binom{b}{2} = \binom{c}{2} $$ Here's what it looks like if you expand, it's variant of the Pythagorean triples: $$ a \times (a-1) + ...
16
votes
5answers
725 views

Integer solutions to $x^3=y^3+2y+1$?

Find all integral pairs $(x,y)$ satisfying $$ x^3=y^3+2y+1.$$ My approach: I tried to factorize $x^3-y^3$ as $$(x-y)(x^2 + xy + y^2)=2y+1,$$ but I know this is completely helpless. Please help me in ...
2
votes
2answers
40 views

Algebra - Solving for three unknowns.

Find all possible solutions of $$2^x + 3^y = z^2.$$ My approach. First I substituted $x = 0$, and got the solution, then for $y = 0$. And for $x > 0$ and $y > 0$ , I just know the ...
2
votes
0answers
85 views

Solving equations in $\mathbb{Z}/p\mathbb{Z}$ versus $\mathbb{Q}_p$.

According to Silverman: In order to show that an algebraic set $V/\mathbb{Q}$ has no $\mathbb{Q}$-rational points, it suffices to show that the corresponding homogeneous polynomial equations ...
4
votes
5answers
71 views

Find all pairs of intergers satisfying $x^2+11 = y^4 -xy$ and $y^2 + xy= 30 $

Find all pairs of intergers $(x,y)$ that satisfy the two following equations: $x^2+11 = y^4 -xy$ $y^2 + xy= 30 $ Here's what I did: $x^2+11 +(30) = y^4 -xy +(y^2 + xy)$ $x^2+41 = y^4 +y^...
2
votes
0answers
36 views

What are the properties of abundancy numbers?

Define abundancy numbers as the rational numbers that are equal to the abundancy index of some integer (not to be confused with «abundant numbers», which are natural numbers with abundancyindex ...
1
vote
2answers
48 views

Solving in Integer sequences

Each $x_n$ comes from the set $\{2,3,6\}$, these statements are true $x_1 + x_2 + x_3+\cdots+x_n = 633$ $\frac{1}{{x_1}^2} + \frac{1}{{x_2}^2} + \frac{1}{{x_3}^2}+\cdots+\frac{1}{{x_n}^2} = \frac{...
-5
votes
1answer
51 views

Can you solve this integer equation? [closed]

${( - 4{x^2} - y + 1)^5} + {(y + 12x + 15)^5} = {(4{x^3} + 4y - 4)^5}$
7
votes
1answer
126 views

A nicer form for $\sum_{r=1}^{m}\sum_{d|r}(-1)^{r+d} d^3$?

Specific Question Let $R$ be a whole number. Is it possible for any whole number $c>0$ that there is a nicer form for $\sum_{r=1}^{R^c}\sum_{d|r}(-1)^{r+d} d^3$? I can hope that we can just ...
0
votes
3answers
38 views

Area of the polygon with coordinates that satisfy the equation

Determine the area of the polygon formed by the ordered pairs (x, y) where x and y are positive integers which satisfy the equation $\frac{1}{x} + \frac{1}{y}= \frac{1}{13}$ I got : $(x-13)(y-13)=...
1
vote
1answer
105 views

Is it possible to narrow down a domain of possible counter-examples to the Collatz Conjecture?

First of all, I am not trying to prove the Collatz Conjecture. I want to know if it is possible to rule out certain values of a counter-example. Suppose $k \in \Bbb Z^+$ is the lowest counter-...
1
vote
1answer
48 views

Is there a solution to the diophantine equation $a^3 + b^3 = c^2$ other than $1^3 + 2^3 = 3^2$ or scalings thereof? [duplicate]

My question concerns the diophantine equation $a^3 + b^3 = c^2$. I know one solution: $1^3 + 2^3 = 3^2$. But this is special in (at least) two ways: the $a$ and $b$ are not coprime; the solution is a ...
1
vote
1answer
131 views

Help prove $p^4 + 4q^4$ is never a perfect square?

Given that $p$ and $q$ have no common factors, how can I prove that $p^4 + 4q^4 \not =z^2$, if $p,q$ and $z$ have to be positive integers? From the comments: I'm working on the congruent number ...
1
vote
1answer
46 views

simple hyperbolic Diophantine equation [closed]

How can we show that there are infinitely many integers $C$ such that the simple hyperbolic diophantine equation: $$6xy ± x ± y = C$$ gives a non-integer solutions for $x, y$, except at $(0, ±C), (±C,...
0
votes
1answer
58 views

Goldbach's conjecture and Diophantine equations.

I was re-watching some Numberphile videos on the Goldbach conjecture and other things, and in on of them it mentioned something about primes being used in Unique Factorisation. In particular, what ...
1
vote
1answer
79 views

Solution to Mordell's Equation $y^2=x^3+4$

Here is my question: Find all solutions to $y^2=x^3+4$. My attempt: Rewrite the equation as $(y-2)(y+2)=x^3$. Notice that if $y$ is odd, then $(y-2,y+2)=1$. Hence they are both cubes, but no ...
1
vote
0answers
48 views

Simplest nontrivial example of an L-function yielding information about a Diophantine equation

I got excited while reading Langlands' essay, REPRESENTATION THEORY: ITS RISE AND ITS ROLE IN NUMBER THEORY, because he appears to provide concrete motivation for the study of L-functions: We have ...
5
votes
4answers
126 views

Integer points on a hyperbola

Why I'm here I have the following problem in probability from a book: You have a bag with red and white balls and you draw two balls without replacing. If the probability of drawing 2 white balls ...
6
votes
1answer
117 views

How to prove that $x^2+1=5^y$ has no positive integer solutions for $y\geq 2$? [duplicate]

I am sure I saw a similar question like this one before but I can't find it now. I tried using "order" but failed. It is quite obvious when $y$ is an even number. The real problem is when $y$ is odd. ...
1
vote
1answer
63 views

Why doesn't Buchberger's algorithm solve Hilbert's tenth problem?

I've been reading a bit about Buchberger's algorithm and Groebner bases. I'm not really trying to understand the math behind it at this point, just trying to get an idea of how the method could be ...
0
votes
0answers
160 views

Positive even numbered integer solutions of $y=n^2-m^2-x^2$

Prove that no integer $x$ exists where $y=n^2-m^2-x^2$ has solutions: For all even integer values of $y$ in the range $2\le y \le 2x+1$ where $x$ is odd. For all odd integer values of $y$ in the ...
0
votes
1answer
50 views

Elementary solution to the diophantine equation $n(n+1)=4m(m+1)$?

Recently I tried to solve a diophantine equation $n(n+1)=4m(m+1)$ with $n,m\in\mathbb{Z}$ which resulted from an other equation. But how can one show, that there are no non-trivial solutions. ...
0
votes
2answers
111 views

Does this qualify as a prime-representing Diophantine equation?

Given the below coefficients, if the Diophantine equation $Axy + Bx + Cy + D = \lfloor\frac{n}{3}\rfloor$ has exactly one solution, then $n$ is prime, otherwise $n$ is composite. In a sense, this ...
9
votes
2answers
142 views

Finding solutions to the diophantine equation $xy(x+y) = n$.

I initially wanted to prove that there are no integer solutions for the equation $xy(x+y) = 4$, but I got intrigued by the general case as I noticed that there tends to be solutions when $n$ is in the ...
2
votes
1answer
89 views

Check whether $k \in [0, p]$ in the equation: ${a^k + b^k \equiv c^k}\mod{p}$ has no solutions under the following conditions

Check whether the equation: ${a^k + b^k \equiv c^k}\mod{p}$ has no solutions where, $ p $ is a prime $ > 3 $, $k \in [0, p]$ and the condition $ 0 < a, b, c < p$ holds. Can we determine ...
0
votes
0answers
26 views

Integer solutions of the general polynomial $a_k x^k + … + a_1 x + a_0 = y(c-x)$

I'm trying to show the existence of and find the solutions to the equation $(a_k x^k + ... + a_1 x + a_0)mod(c-x)=0$, with $a_k,...,a_0,c,k,x \in \Bbb N$, and $a_k,...,a_0,c,k$ are known, and $x<c$...
3
votes
2answers
93 views

Solve $\frac{1}{x}+\frac{1}{y}= \frac{1}{2007}$

The number of positive integral pairs $(x<y)$ such that $\frac{1}{x}+\frac{1}{y}= \frac{1}{2007}$ The answer is 7 where as i am getting 6. The ordered pair are (2676,8028),(2230,20070),(2016,...
8
votes
2answers
197 views

If $ab \mid c(c^2-c+1)$ and $c^2+1 \mid a+b$ then prove that $\{a, b\}=\{c, c^2-c+1 \}$

If $ab \mid c(c^2-c+1)$ and $c^2+1 \mid a+b$ then prove that $\{a, b\}=\{c, c^2-c+1 \}$ (equal sets), where $a$, $b$, and $c$ are positive integers. This is math contest problem (I don't know the ...
2
votes
4answers
86 views

The diophantine equation $5\times 2^{x-4}=3^y-1$

I have this question: can we deduce directly using the Catalan conjecture that the equation $$5\times 2^{x-4}-3^y=-1$$ has or no solutions, or I must look for a method to solve it. Thank you.
1
vote
3answers
61 views

Parameterization to the Diophantine equation $x^2+4y^2=5z^2$ [closed]

I'd like to ask how to obtain all integer solutions to the Diophantine equation $x^2+4y^2=5z^2$ using parameterization? Thanks.
14
votes
3answers
200 views

triangles in $\mathbb{R}^n$ with all vertices in $\mathbb{Q}^n$

In the context of this post, a triangle will mean a triple $(a,b,c)$ of positive real numbers which qualify as side lengths of a triangle (i.e., the triangle inequalities are satisfied). Call ...
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vote
0answers
36 views

Solving OR bounding sums of solutions to certain linear diophantine equations

(This question arose during group work on classifying modular tensor categories.) Let $p$ and $q$ be two distinct primes. We seek integral solutions $\lbrace x_{i,j} \rbrace$ to the equation \begin{...
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votes
2answers
38 views

Prove this diophantine equation [closed]

Show that for non-negative $n$ and $m$ there is not solution for following equation $n^2-13=2^m$.
7
votes
1answer
63 views

number of different ways to represent a positive integer as a binomial coefficient

For each positive integer $x$, let $f(x)$ be the number of different ways that $x$ can be expressed in the form $x={\large{\binom{n}{k}}}$, where $n,k$ are integers with $2 \le k \le {\large{\frac{n}{...
2
votes
3answers
57 views

Diophantine with non-zero solutions

Find all non-zero integers $a,b$ such that: $(ab)^2 + ab - (a-b)^3 + a^3 + b^3 = 0$ I have tried using parity, got nowhere. Tried finding some bounding inequality, got nowhere. I took a small look ...
3
votes
1answer
188 views

System of Diophantine equations $a^2+b^2=2x^2+1,c^2+d^2=2y^2+1,ac-bd=1$ has no natural solutions.

How to prove that system $$ a^2+b^2=2x^2+1, \\ c^2+d^2=2y^2+1, \\ a\cdot c-b \cdot d=1 $$ has no natural solutions? It can be proved that system equal to the equation $$(2х^2+1)(2у^2+1)=4z^2+1$$ In ...
1
vote
2answers
253 views

Diophantine equation $a^2+b^2+c^2=a^2b^2$

I am trying to find all non trivial integers for which $a^2+b^2+c^2=a^2b^2$. As suggested I have tried working (mod 4). This is what I've gotten so far: Squares can have a remainder of 0 or 1 (mod 4)....
0
votes
4answers
77 views

In the Diophantine equation, 5x + 17y = c

In the Diophantine equation, 5x + 17y = c where c is the smallest value of the equation where x and y are both positive integers and every number larger than c can also be written in the form of 5x + ...
2
votes
0answers
37 views

Other diophantine equations solved thanks to modularity theorem

The diophantine equation \begin{equation} x^n+y^n=z^n \end{equation} is an example of an equation that was not solved before the work of Wiles, and has now been solved through his and other's ...
2
votes
2answers
76 views

Find number of integers satisfying $x^{y^z} \times y^{z^x} \times z^{x^y}=5xyz$

Find number of integers satisfying $$x^{y^z} \times y^{z^x} \times z^{x^y}=5xyz$$ My try: we can rewrite the equation as $$x^{y^z-1} \times y^{z^x-1} \times z^{x^y-1}=5$$ Then since all are ...
0
votes
2answers
33 views

If $z$ is not divisible by $r$, where $r$ is a prime of the form $(4n+3)$, then there exists an integer $z_{1}$ such that $zz_{1} \equiv 1 (mod r) $ [closed]

I was reading Diophantine Analysis (Robert Carmichael), and on Page No.-34 of the book, it says : If $z$ is not divisible by $r$, where $r$ is a prime of the form (4n+3), then there exists an integer ...
2
votes
2answers
59 views

Generating equation for Diophantine equation

Given a triangular tower of bricks where the bottom row has $N$ bricks, the next row has $N-1$ bricks, etc, we can ask the question: For which $N$ does there exist a smaller tower with $M \lt N$ ...
0
votes
5answers
117 views

Formula to generate primitive quadruples to Diophantine equation $x^2+y^2=2(u^2+v^2)$

I'd like to ask whether there exists a formula which generates the primitive integer solutions to the Diophantine equation $x^2+y^2=2(u^2+v^2)$? Primitive means $x$, $y$, $u$ and $v$ are coprime. ...
3
votes
2answers
65 views

Number of solutions to $x^2+ax=y^2+by$

I'm trying to find the number of positive integer solutions, in terms of $a$ and $b$, to the equation: $$x^2+ax=y^2+by$$ I have tried many approaches but couldn't seem to get an answer. Is this ...
10
votes
0answers
266 views

Prove the equation $\left(2x^2+1\right)\left(2y^2+1\right)=4z^2+1$ has no solution in the positive integers

Prove the equation $$\left(2x^2+1\right)\left(2y^2+1\right)=4z^2+1$$ has no solution in the positive integers My work: 1) I have the usually problem $$\left(nx^2+1\right)\left(my^2+1\right)=(m+n)z^...