Questions on finding integer/rational solutions of equations.

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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 ...
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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": ...
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1answer
42 views

Given the conditions above, find when x,y,z satisfy below: $ (x^2-1)(y+1)=\frac{z^2+1}{y-1}$

Let $x,y,z \in \mathbb{Z^+}$ and $x \neq y \neq z$ Given the conditions above, find when x,y,z satisfy below: $$ (x^2-1)(y+1)=\frac{z^2+1}{y-1}$$ What I did was I factored the numerator to: ...
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2answers
66 views

How to prove indirectly that there exist no positive integers $x$ and $y$ for which $x^2 - y^2 = 1$?

I already did it with a contraposition. But how to do it indirectly? It has to be something like "there exist positive integers $x$ and $y$ for wich $x^2 - y^2 = 1$". Tried to use $(x+y)(x-y)=1$ and ...
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2answers
91 views

Exponential Diophantine equation $7^y + 2 = 3^x$ [closed]

Find all positive integer solutions to $$7^y + 2 = 3^x.$$
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1answer
155 views

Integer solutions of $x^3+y^3=z^3$ using methods of Algebraic Number Theory

I'm asked to prove that the famous equation $$x^3+y^3=z^3$$ has no integer (non-trivial) solutions, i.e. FLT for $n=3$ I'm aware that on this website there are solutions using methods of Number ...
2
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3answers
53 views

Finding values that make $xy=x+y+z$ true.

Let $x,y,z \in \mathbb{Z^+}$ and $x \neq y \neq z$ Find all values that make this true: $$xy=x+y+z$$ I am trying to solve a way to make this true but clearly, I cannot find any values that make ...
2
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1answer
35 views

Integer solutions $d = \frac{ab}{a + b + 2\sqrt{ab}}$ [duplicate]

$$d = \frac{ab}{a + b + 2\sqrt{ab}} = \frac{ab}{(\sqrt{a} + \sqrt{b})^2}$$ What are the positive integer solutions? The majority of solutions are when $a=b$, so that $a = 4d$.
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3answers
62 views

Find two solutions of $299x+247y=13$

Applying Euclidean algorithm I got $299=247+52$ $247=4\cdot 52+39$ $52=39+13$ $39=3\cdot 13+0$ Applying this in reverse order I got $52-39=13$ $52-(247-4\cdot 52)=13$ $5(299-247)-247=13$ ...
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2answers
57 views

Are all Mordell equations $y^2=x^3+k$, for any integer $k$, solvable

Are all Mordell equations $y^2=x^3+k$, for any integer $k$, solvable? Not that there are solutions $x,y$ for every $k$, but that you can determine for every $k$ if there are solutions, and if there ...
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1answer
29 views

Number of solutions to Pell-type equation $k(x^2-2y^2) = d$

Given equation $$k(x^2-2y^2) = d$$ Where d is a constant. k,x,y are variables. All are positive integers. Is there some characterization for the values of d for which there are unique solutions, ...
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1answer
85 views

For which $n,p$ does $\cfrac {5n^2-3n}{2}=2^p-1$ [closed]

For which $n,p$ does $\cfrac {5n^2-3n}{2}=2^p-1$, where $n,p$ are positive integers
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3answers
148 views

Help solving $ax^2+by^2+cz^2+dxy+exz+fzy=0$ where $(x_0,y_0,z_0)$ is a known integral solution

Help solving over the integers: $$ax^2+by^2+cz^2+dxy+exz+fzy=0$$ where $(x_0,y_0,z_0)$ is a known integral solution and $a,b,c,d,e,f$ are integral coefficients. I found in Tito Piezas' identities the ...
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2answers
33 views

Prove diophantine equation $S^2+R^2+(r_1-r_2)^2 = 2R(r_1+r_2)$ has at most one solution

Given this diophantine equation: $$S^2+R^2+(r_1-r_2)^2 = 2R(r_1+r_2)$$ $S,r_1,r_2$ are variables. $R$ is a given constant. all values are positive integers. How do I prove that there's at most one ...
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0answers
26 views

linear Diophantine equation with real coefficients

I am studying linear Diophantine equations with real coefficients: \begin{equation} ax+by=1\qquad a,b\in\mathbb{R} \end{equation} I have no clue about where to begin; could someone please point me to ...
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0answers
23 views

Solution to a quadratic diophantine

Given $b,d,e,M\in\Bbb Z$ how can we solve for $x,y\in\Bbb Z$ such that $$by+xd+xy e=M?$$
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3answers
40 views

number theory for finding value of $k$

How do I find what is the smallest positive integer $k$ such that $(3^3 + 4^3 + 5^3)\cdot k = a^n$ for some positive integers $a$ and $n$, with $n > 1$?
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1answer
92 views

Quintic diophantine equation $x^5+y^5=7z^5$

Are there any non-zero integer solutions to the equation $x^5+y^5=7z^5$? I am unsure how to approach this.
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0answers
23 views

How many quadruples are there?

How many quadruples are there $(x_1,x_2,x_3,x_4) \in \mathbb{Z}^{+}_{0}$ such that $(x_1+x_2)(2x_2+2x_3+x_4)=95$? My attempt. We have that $x_1+x_2 = 19$ and $2x_2+2x_3+x_4 = 5$ or $x_1+x_2 = 5$ and ...
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2answers
52 views

Frobenius Coin Problem: Can anyone prove my closed form for n = 3? [on hold]

The closed form for $n = 3$ where $3! <= p < q < (r > 2p$ and $r > 2q)$ is $g(p,q,r)= (p -2)(q -2)(r -2) -2[(p -1)(q -1) -1 +(p -1)(r -1) -1 +(q -1)(r -1) -1] +3[p +q +r +1] -1.$ It ...
3
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2answers
77 views

For which integer $x,y$ does $x^2+23=3(2^y)$

How do I solve this Lesbegue-Ramanujan-Nagell type equation ($x^2+D=AB^y$): $x^2+23=3(2^y)$ I have been trying for quite some time now, to no end. Any suggestions/help would be greatly appreaciated.
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30 views

What does it means to find an solution for an elliptical equation?

What does it means to find integer solutions for $(x-2y)^2+2(y-6)^2=102$ , for example.
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2answers
86 views

Which integers $a, b, c$ satisfy the equation $a \sqrt 2 − b = c \sqrt5$?

This is a non-calculator question: Which integers $a, b, c$ satisfy the equation $a \sqrt 2 − b = c \sqrt5$? I've tried solving it through trial and error and the only solution I seem to be ...
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1answer
125 views

Prove that $x^3-x = y^2+1$ has no integer solution

Prove that $x^3-x = y^2+1$ has no integer solution: I began the proof by case distinction considering the cases if x,y are both even, if x,y both odd, if x even, y odd and the last one if x odd and y ...
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1answer
20 views

How to show a certain integer solution does not exist for a system of linear equations?

I'm having some trouble with the following problem: Considering an independent system of linear equations in $x$ and $y$ with integer coefficients \begin{equation} ax + by = c_1\\ cx + dy = c_2. ...
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1answer
30 views

Three diophantine equations

I have three Diophantine equations that I need to solve. However, I am struggling to find the values of $x$, $y$, and $z$. To clarify, all equations equal the same thing. The approach that I took was ...
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1answer
31 views

$x^2+(a^2+b^2)x+ab=0$ where $a,b,x\in \mathbb Z$

Trivial solutions include $a=0, x=-b^2$ and $b=0,x=-a^2$. Then $a=b=1,x=-1$ or $a=b=-1, x=-1$. Is there any other? How could you prove there aren't? $(a^2+b^2)^2-4ab=(a^2+b^2)+2ab(ab-1)$ needs to be a ...
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2answers
67 views

Find all integer solutions to $x^2+y^2+z^2=2xyz$ [closed]

I am working on some of these types of problems for fun, just want to see a couple solved as examples.
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2answers
105 views

Solve $y^2 = x^3 - 4$ where $x,y \in \mathbb{Z}$ [duplicate]

Solve $y^2 = x^3 - 4$ So far i've tried to look at this: $x^3 = (y-2i)(y+2i)$. I think we need to prove that $y+-2i$ are coprime, so suppose not then: $d = u+iv$ divides both numbers. So also d ...
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0answers
67 views

Which positive integers satisfies $a^{b^2} = b^a$

How one can find all integers satisfying $a\geq 1,b\geq 1,a^{b^2} = b^a$? I think that the solutions are $ (a,b)=(1, 1), (16, 2),(27, 3)$.
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53 views

Original proof of Ljunngren's equation

The equation $$x^2=2y^4-1$$ was studied and solved by Ljunngren, who showed that 1,1 and 293,13 are the only integer solutions.However, his proof was very difficult and L.J.Mordell thought there must ...
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2answers
43 views

Trivial case for Pell's equation

I was reading Pell's equation, and in the beginning there is a statement: Notice that if $D = d^2$ is a perfect square, then this problem can be solved using difference of squares. We would have ...
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3answers
66 views

Solve modular arithmetic equation $\frac{1}{24} \cdot n(n+1)(n+2)(n+3) \equiv 1 \pmod{10}$

From another problem, I have reduced it to: This is the last step in solving: Solve $\frac{1}{24} \cdot n(n+1)(n+2)(n+3) \equiv 1 \pmod{10}$ How should I begin, a major problem is the $1/24$
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12 views

Performing Multivariable Polynomial Ling Division

What are the limitations of performing long division for polynomials in more than 1 variable? For instance, if I have a degree 4 diophantine equation of two variables and I know one solution, is ...
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1answer
47 views

Equation $x^2=y^p+1$

can you help me please for solving this dophantine equation $$x^2=y^p+1$$ and if you can give me a general method to studying such equation $$x²=y^p+t$$ Thanks
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1answer
33 views

Polynomial of degree two in two variables and rational points

This question of mine and the comments on the answer led me to the following more general problem: Suppose that we have polynomial $f(x,y)=a_1xy+a_2x^2+a_3y^2+a_4x+a_5y+a_6$: Consider now the ...
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143 views

Are there infinitely many rational pairs $(a,b)$ which satisfy given equation?

I saw on some facebook page this concrete example: $1.2^2+0.6^2=1.2+0.6$ The question that immediately arises is: Are there infinitely many pairs $(a,b)$ of rational numbers such that we have ...
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1answer
67 views

Theorem: Odd positive integer N is a prime number if …

Is the following theorem well known? Theorem Odd positive integer N is a prime number if and only if there is no non-trivial solution for Diophantine equation $x^2−y^2=N$ (trivial solution: ...
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55 views

Employment of Diophantine equation $x^2-y^2=N$ for primality checking

Could the following theorem be useful for number theory? Theorem Odd positive integer $N$ is a prime number if and only if there is no non-trivial solution for ...
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1answer
23 views

how to solve this equation with the unknown at the powers

Please help me to solve this equation: Find $n \in \mathbb{N}$ such that: $\sqrt{1+5^n+6^n+11^n} \in \mathbb{N}$. $0$ is a particular solution, and does it have other one ?
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132 views

Can a sum of three fifth power of integers be 8?

By congruence computation we get that $n= a^5+b^5+c^5$ implies $n \not \equiv 4,5,6,7 \pmod{11} $ (with $a,b,c \in \mathbb{Z}$) For $a,b,c \in \{-100,-99, \dots , 99, 100\}$, the set of integers ...
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1answer
31 views

How many ways can $3$ regular polygons meet at a vertex?

This is equivalent to the positive integer solutions to $$\frac{a-2}{a} + \frac{b-2}{b} + \frac{c-2}{c} = 2$$ with $3 \le a \le b \le c$. Small solutions like $(6, 6, 6)$ and $(4, 8, 8)$ can be ...
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1answer
21 views

Diophantine equations: $x_1y_1+x_2y_2 = x_3y_3+x_4y_4$

Given 3 diophantine equations: $$x_1y_1+x_2y_2=x_3y_3+x_4y_4$$ and $$x_1+x_2 = x_3+x_4$$ and $$y_1+y_2 = y_3+y_4$$ We're interested in solutions to this system of equations when all variables ...
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1answer
61 views

The sum of three cubes

I'm reading an article about numbers that can be expressed as the sum of three cubes: http://www.ams.org/journals/mcom/2007-76-259/S0025-5718-07-01947-3/S0025-5718-07-01947-3.pdf It states in the ...
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2answers
74 views

Is every integer a mixed sum of three squares?

Lagrange's four-square theorem states that every natural number can be represented as the sum of four integer squares $n = a^2 + b^2 + c^2 + d^2$. Question: Is every integer a mixed sum of three ...
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1answer
44 views

Primes whose mirror is prime

I am interested in finding a method to determine all four-digit primes (notation $ p = xyzw $) such that (its mirror) $ q = wzyx$ is also a prime in other words, to solve the system with digits ...
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1answer
78 views

Maximum number of pythagorean triples on a circle not centered on the origin

Suppose we two equations $$x^2+y^2=r^2$$ and $$(x-a)^2+(y-b)^2=2g^2$$ Where x,y and r are integer variables greater than 0. a,b and g are integer constants greater than 0. I conjecture that for any ...
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1answer
47 views

Prove that the equation $y^2 = x^5 + 1$ has exactly p solutions modulo p.

Condition on p: $ p \ncong 1(mod5) $ I understand that I'm trying to count the number of times that $x^5 + 1$ is a quadratic residue modulo p. I just can't quite figure out exactly why the number of ...
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1answer
29 views

Is there a test to determine whether a Diophantine equation has a solution in the positive integers?

Is there a test to determine whether the Diophantine equation, $$ ax + by = z $$ with $a,x,b,y,z$ integers, $a >0, b>0, z>0$, has a solution with $x\geq 0$ and $y\geq 0$? In general we ...