Questions on finding integer/rational solutions of equations.

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Strategies for solving rational Diophantine equations

Are there any strategies for solving Diophantine equations where the solutions can be any rational number, not just an integer, besides substituting $x=p/q$ and $y=r/s$, with $p,q,r,s$ integers with ...
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309 views

Three pythagorean triples

Are there any solutions for $a, b, c$ such that: $$a, b, c \in \Bbb N_1$$ $$\sqrt{a^2+(b+c)^2} \in \Bbb N_1$$ $$\sqrt{b^2+(a+c)^2} \in \Bbb N_1$$ $$\sqrt{c^2+(a+b)^2} \in \Bbb N_1$$
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2answers
52 views

What are all the concordant forms $n$ such that $a^2+b^2 = c^2,\,a^2+nb^2=d^2$ for $n<1000$?

Part I. The list of congruent numbers $n<10^4$ such that the system, $$a^2-nb^2 = c^2$$ $$a^2+nb^2 = d^2$$ has a solution in the positive integers is known (A003273) $$n = 5, 6, 7, 13, 14, 15, ...
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112 views

$abx^2+bcy^2+acz^2=(xyz)^2+2abc$ has no integral solutions if $a,b,c,x,y,z >1$?

let $a,b,c,x,y,z$ be all pairwise coprime integers . Show that: $$abx^2+bcy^2+acz^2=(xyz)^2+2abc$$ has no integral solutions if $a,b,c,x,y,z >1$. I tried to confirm the results in wolfram but I am ...
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2answers
77 views

The diophantine equation $m(n-2016)=n^{2016}$

How many natural numbers, $n$, are there such that $$\frac{n^{2016}}{n-2016}$$ is a natural number? HINT.-There are lots of solutions HINT.-$\frac{n}{n-2016}=m \iff \frac{2016}{n-2016}=m-1$ and if, ...
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1answer
115 views

Finding integer solutions to $y^2=x^3+7x+9$ using WolframAlpha

I am an unconditional admirer of WolframAlpha and for this reason I want to let the people of this error (or is it really the fault of mine?). If I'm not mistaken, I would be very happy to contribute, ...
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3answers
99 views

How can one show the inequality

Let $a,b,n$ be natural numbers (in $\mathbb{N}^*$) such that $a>b$ and $n^2+1=ab$ How can one show that $a-b\geq\sqrt{4n-3}$, and for what values of $n$ equality holds? I tried this: We suppose ...
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1answer
40 views

What is the Diophantine Prime-Representing Polynomial with the Least Variables?

Recently I was reading Jones et al.'s famous paper "Diophantine Representation of the Set of Prime Numbers." They present a Prime-Representing Polynomial in 26 variables, and outline the construction ...
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1answer
117 views

Find all primes $p$ such that $ p^3-4p+9 $ is a perfect square.

Find all primes $p$ such that $ p^3-4p+9 $ is a perfect square. I tried a few different values for $p$, namely $2,3,5,7,$ and $11$. The prime $p =2,7,11$ all worked but $p =13$ didn't so it ...
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2answers
86 views

two questions involving $x^3+y^3+z^3-3xyz$ factorization

(1) Given that $x^3+y^3+z^3=3xyz+1$, determine the minimum of $x^2+y^2+z^2$. I know that Lagrange multiplier can solve this but I believe there is a way out using the factorisation: ...
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1answer
51 views

Using FLT for $ n=3$

Using FLT for exponent $3$, I need to show that if n positive integer is divisible by $3$, then there are no $x,y,z$ positive integers such that $x^n+y^n=z^n$. This is what I did: if $3/n$ then ...
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2answers
87 views

On $p^2 + nq^2 = z^2,\;p^2 - nq^2 = t^2$ and the “congruent number problem”

(Much revised for brevity.) An integer $n$ is a congruent number if there are rationals $a,b,c$ such that, $$a^2+b^2 = c^2\\ \tfrac{1}{2}ab = n$$ or, alternatively, the elliptic curve, $$x^3-n^2x = ...
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2answers
91 views

Simulating Bernoulli processes using several random binary strings

I recently stumbled upon a mathematical puzzle while trying to work with the limitations of a certain program. Said program effectively has the ability to simulate Bernoulli processes, but only at ...
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1answer
63 views

How to solve the equation of $\sqrt{x}+\sqrt{y}=\sqrt{2205}$ in integers? [closed]

How to solve the equation of $\sqrt{x}+\sqrt{y}=\sqrt{2205}$ in integers? How in general to solve the similar equations?
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3answers
94 views

Is it true that the equation $27x^2+1=7^3y^2$ has infinitely many solutions in positive integers $x,y$ ?

Is it true that the equation $27x^2+1=7^3y^2$ has infinitely many solutions in positive integers $x,y$ ?
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1answer
268 views

Diophantine problem related to pythagorean triples: prove 2 expressions cannot both be perfect squares

Given 2 primitive pythagorean triples with parameters as per: https://en.wikipedia.org/wiki/Pythagorean_triple#Generating_a_triple First triple has parameters $(m,n)$. Second has parameters $(r,s)$. ...
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0answers
111 views

Solving (n+1)(n+2)…(n+k)−k = x^2

Let $n$ and $k$ be positive integers. Need to find all pairs of $(n,k)$ such that $$(n+1)(n+2) \cdots (n+k)−k = x^2,$$ where $x^2$ is a perfect square.
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1answer
29 views

In $x^2(y-2)-x(y-4)=2^{z+1}-2$ for a given $y$, how many solutions $x,z$ are possible?

In $$x^2(y-2)-x(y-4)=2^{z+1}-2$$for a given $y$, how many solutions $x,z$ are possible? I know that there is a finite amount of them, but is there any way to get exactly how many? Also $x,y,z$ are ...
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1answer
140 views

On products of ternary quadratic forms $\prod_{i=1}^3 (ax_i^2+by_i^2+cz_i^2) = ax_0^2+by_0^2+cz_0^2$

The equation, $$ (ax_1^2+by_1^2)(ax_2^2+by_2^2) = ax_0^2+by_0^2\tag1$$ has the well-known solution when $a=b=1$, $$ (x_1^2+y_1^2)(x_2^2+y_2^2) = (x_1 y_2 + x_2 y_1)^2 + (x_1 x_2 - y_1 y_2)^2$$ ...
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2answers
90 views

Finding the integer solutions of $y^2 = x^3 - 12$

I tried to find the solution of this equation, or conclude there are none. This i what i found out: I noticed that $x \neq -1$ mod $3$. We can write $y^2 + 4 = x^3 - 8 = (x-2)(x^2+2x+4)$ I tried ...
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3answers
73 views

Rational solutions $(a,b)$ to the equation $a\sqrt{2}+b\sqrt{3} = 2\sqrt{a} + 3\sqrt{b}$

Find all rational solutions $(a,b)$ to the equation $$a\sqrt{2}+b\sqrt{3} = 2\sqrt{a} + 3\sqrt{b}.$$ I can see that we have the solutions $(0,0), (2,0), (0,3), (3,2), (2,3)$, and I suspect that ...
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0answers
31 views

Find all solutions to $a^b = b^a$ [duplicate]

Find all ordered pairs $(a, b)$ such that $a$ and $b$ satisfy $a^b$ = $b^a$ and $a$ and $b$ are integers. The only way I can think of solving this question is by trial and error, but there must ...
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6answers
120 views

If $a_{n+1}=\frac {a_n^2+5} {a_{n-1}}$ then $a_{n+1}=Sa_n+Ta_{n-1}$ for some $S,T\in \Bbb Z$.

Question Let $$a_{n+1}:=\frac {a_n^2+5} {a_{n-1}},\, a_0=2,a_1=3$$ Prove that there exists integers $S,T$ such that $a_{n+1}=Sa_n+Ta_{n-1}$. Attempt I calculated the first few values of ...
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Solutions of diophantine equation: $s^2 = (ad)^2+ (bc-ad+4ac)^2$

Given diophantine equation: $$s^2 = (ad)^2 + (bc-ad+4ac)^2$$ $s,a,b,c,d$ are all variables. They are all odd. a and b are coprime. c and d are coprime. How do you parametrize all the solutions? ...
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Solve $a^n - b^n = 8$ with $a,b \in \mathbb{Z}$ and $n \in \mathbb{N}_{\geq 2}$.

I already solved the question myself, but there is something bottering me. In the exersice is told "Solve, by easy estimations". I couldnt find a boundary for $n$ or something. I started with $a^n = 8 ...
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2answers
56 views

Diophantine equation $x^2+y^2=z^2+t^2$?

I would like to find some source books or articles which discuss the Diophantine equation $$ x^2+y^2=z^2+t^2,\qquad |y-z|=1 $$ for which $x,z$ are odd positive and $y,t$ are even positive integers. ...
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2answers
60 views

Find $a$, $b$ and $c$ that satisfy the following equations

Find all integers $a$,$b$ and $c$ that satisfy the following equations $a^2=bc+1\tag{i}$ $b^2=ac+1\tag{ii}$ I tried solving came out with following results: $(i)-(ii)$ gives $a+b+c=0$ which ...
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4answers
122 views

find all integers $a,b,c$ such that $a^2=bc+1,$ $b^2=ca+1.$ [duplicate]

This was one of the problems in a math contest in India. This is how I tried it: Subtracting the second equation from the first one gives $a^2-b^2=bc-ca$ or $(a-b)(a+b)=c(b-a)$. $a-b=-(b-a)$. ...
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0answers
56 views

Help solving the quadratic equation $ax^2-4bx+4bc-\frac{d^2}{a}=0$

I have been struggling to solve this quadratic equation in the variable $x$ with integral coefficients: $$ax^2-4bx+4bc-\frac{d^2}{a}=0$$ $a\neq 0$ of course.How do I ensure that $x$ is an integer? ...
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2answers
81 views

On the complete solution to $x^2+y^2=z^k$ for odd $k$?

While trying to answer this question, I was looking at a computer output of solutions to $x^2+y^2 = z^k$ for odd $k$ and noticed certain patterns. For example, for $k=5$ we have $x,y,z$, $$10, 55, ...
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1answer
65 views

Quartic diophantine equation: $16r^4+112r^3+200r^2-112r+16=s^2$

Given this diophantine equation: $$16r^4+112r^3+200r^2-112r+16=s^2$$ Wolfram alpha says the only solutions are $(r,s)=(0,\pm4)$ How would one prove these are the only solutions? Thanks.
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5answers
209 views

Diophantine equation $x^2 + y^2 = z^3$

I have found all solutions to the Diophantine equation $x^2 + y^2 = z^3$ when $z$ is odd. I am having some difficulty finding the solutions when $z$ is even. I am asking for a proof that provides the ...
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1answer
31 views

Diophantine equation $x^{y} - y^{x}=1$

I was thinking about the equation $x^{y} - y^{x}=1$ where $x,y \in \mathbb N$ and the solution $x=3$ and $y=2$ was easy to find. Also $x=2$ and $y=1$ is a solution. I would like to know: Is this ...
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1answer
40 views

Is the diophantine equation $a = x^p - y^p$ sufficient to find $x$ and $y$ in terms of $a$ and $p$?

Let there be a natural number $a,$ that can be expressed as $a = x^p - y^p$ where $x, y$ and $p$ are natural numbers, each two of them being pairwise co-prime and $p$ is an odd prime. Then can $x$ and ...
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0answers
29 views

Impossible form of a triangular number

Show that there are no positive integers $t,i,j$ with $j>i$ such that: $\displaystyle \frac{t(t+1)}2=\frac{2i(j-i)j(j+i)}3$ If possible provide an elementary proof. I believe the statement is ...
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0answers
24 views

Prove 3 diophantine expressions cannot simultaneously be perfect squares

Given s and g are positive integers and $cos\theta$ and $sin\theta$ are rational and not equal to 0 or 1. Show these 3 expressions cannot all be perfect squares: $$s^2+2g^2-2sg(cos\theta-sin\theta)$$ ...
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What determines the number of families to $1-4x-4(1-x^2)z = w^2$?

This is related to this post. First, we have, Theorem: "If $w_0, z_0$ is a solution to, $$1-4x-4(1-x^2)z = w^2\tag1$$ then, $$w = w_0+2(x^2-1)n$$ $$z = z_0+w_0\,n+(x^2-1)n^2$$ is also a ...
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1answer
31 views

Show that a cubic Diophantine equation has no solutions

Show that the Diophantine equation $x^3+117y^3 = 5$ has no solutions. I tried using like an odd and even argument for $x$ but it doesn't seem to work because it doesn't matter if $x$ is odd or even. ...
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1answer
42 views

Prove for $abc$-triples that $c\leq rad(abc)^2$

Prove for $abc$-triples that $c\leq \text{rad}(abc)^2$. $\text{rad}(abc)$ and $\text{rad}(x)$ means here the product of all prime factors of $x$. (edit 2) The above holds for: $16+5=21 \leq ...
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1answer
33 views

Proving a power Diophantine equation has no solutions

Show that the Diophantine equation $2^n-x^m = 1$ with $x,n,m > 1$ has no solutions. How do I show that $x^m$ can never be $1$ less than a power of $2$? I tried factoring it but it doesn't seem ...
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3answers
61 views

$f (x) \in\mathbb Z[x]$ be such that $f$ is a perfect $k$-th power on taking positive integer values . Is $f$ a perfect $k$-th power?

Let $f(x)\in\mathbb Z[x]$ and integer $k>1$ such that $f(n)$ is a perfect $k$-th power for every positive integer $n$. Is it true that there is a $g(x)\in\mathbb Z[x]$ such that ...
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2answers
15 views

Show that if a Diophantine equation has a solution then both $x$ and $y$ must be odd

Show that if the Diophantine equation $y^2=x^3+ 2$ has a solution, then $x$ and $y$ must both be odd. How do I take into account the condition that $y^2=x^3+ 2$ has a solution? How do I take this ...
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2answers
52 views

$a,x,t$ are non-zero integers. $x^2+2ax+at$ is a square for what value(s) of $t$?

For what values of $t$ does the diophantine equation: $$x^2+2xa+at$$ is a square? For me, the obvious solution is: $$t=a$$ However, by assuming that there exists an integer $r$ such that: ...
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1answer
65 views

solution of Diofantine equation

Find number of non-negative integer solutions of equation: $x+y^2+z=x^2z+y$. I have tried to rearrange it like $x^2z-y^2=x-y+z$, but I don't have an idea what to do in the next step. Thanks for any ...
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3answers
46 views

$Dm^2 - n^2D^2$ is a perfect square then $D$ is the sum of two squares

How do I show that if $$Dm^2 - n^2D^2$$ is a perfect square for some integers $m$ and $n$ ($n \neq 0$), $D$ is the sum of two (non-zero) perfect squares? I tried solving for $D$ but that only gives me ...
3
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2answers
95 views

integer solution of $\frac{x^3+y^3+z^3-xy(x+y)-yz(y+z)-xz(x+z)-2xyz}{(x+y+z)(x+y-z)(x-y+z)(x-y-z)}=\frac{1}{2016}$

Let $x,y,z$ be positive integers such that $\frac{x^3+y^3+z^3-xy(x+y)-yz(y+z)-xz(x+z)-2xyz}{(x+y+z)(x+y-z)(x-y+z)(x-y-z)}=\frac{1}{2016}$. How to find all solutions ? I have no any idea. Thanks in ...
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1answer
52 views

Find all values of $n$ for which the Diophantine equation $n=a^2-b^2$ has a solution

Let $n$ be an integer. Find all values of $n$ for which the Diophantine equation $n=a^2-b^2$ has a solution for integers $a$ and $b$. For those values of $n$ found in the previous part find all ...
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1answer
75 views

Find all positive integers $n$ such that $2^8+2^{12}+2^n$ is a perfect square

Find all positive integers $n$ such that $2^8+2^{12}+2^n$ is a perfect square. For $n=2$ and $n=11$, $2^8+2^{12}+2^n$ is a perfect square. How to find a closed form?
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3answers
71 views

Solve the system:$\,\small\begin{eqnarray}a_1P+b_1Q+c_1R&=&0\\ a_2P+b_2Q+c_2R&=&0\\ a_3P+b_3Q+c_3R&=&0 \end{eqnarray}$ where $\small PQR\neq 0$

Consider the system of diophantine equations: \begin{eqnarray} a_1P+b_1Q+c_1R&=&0\\ a_2P+b_2Q+c_2R&=&0\\ a_3P+b_3Q+c_3R&=&0 \end{eqnarray} where $PQR\neq 0$. What is the ...
0
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0answers
42 views

Do these Diophantine equations have solutions?

In this answer/this blog post by Andrej Bauer, he mentions finding these two short Diophantine equations, which "gave a professional number theorist something to munch on for a couple of weeks": ...