Questions on finding integer/rational solutions of polynomial equations.

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41 views

Concluding three statements regarding $3$ real numbers.

$\{a,b,c\}\in \mathbb{R},\ a<b<c,\ a+b+c=6 ,\ ab+bc+ac=9$ Conclusion $I.)\ 1<b<3$ Conclusion $II.)\ 2<a<3$ Conclusion $III.)\ 0<c<1$ Options By ...
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44 views

Write $x(a^2+b^2)+(2ab)y$ as a product of factors.

Let $a,b,c,x,y \in\mathbb{Z}>1$ and $\gcd(a,b)=\gcd(x,y)=\gcd(a,b,x,y)=1$, Can $$x(a^2+b^2)+(2ab)y$$be factorized ?
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1answer
24 views

for which values of $\theta$.does this equation: $x_1^{\sin\theta}+\cdots+x_n^{\sin\theta}=1$ have rational solutions for all $n$?

I'd surprised if this equation:$$x_1^{\sin\theta}+\cdots+x_n^{\sin\theta}=1 $$ have rational solutions for all $n$ and for all values of $\theta$. My question here is: for which values of $\theta$ ...
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2answers
34 views

How many non-negative integer solutions does $x_1+x_2+\cdots+x_n=A$ have?

If I have the Diophantine equation $\displaystyle{\sum_{i=1}^n x_i =A}$, is there a function $f(n,A)$ that will yield the number of non-negative integer solutions of the equation?
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3answers
75 views

Maximize the following sum

Let $a, b, c, d, e$ be nonnegative integers such that $625a + 250b + 100c + 40d + 16e = 15^3$ . What is the maximum possible value of $a + b + c + d + e$? Quick arithmetic gives: ...
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0answers
36 views

The amount of the third degree.

Often have to deal with such a cubic Diophantine equation. $$q(a^3+b^3)=t(x^3+y^3)$$ $q,t - $ are specified for the problem. Interesting - in all the values of the coefficient of solutions are ...
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73 views

Generalizing the Pell equation $x^2-61y^2 = 1$

In a table of fundamental solutions $f_1(x,y)$ to Pell equations, $$x^2-dy^2=1\tag1$$ with $d<110$, two will stand out, $$(U_{61})^6 = \big(\tfrac{39+5\sqrt{61}}{2}\big)^6 = x+y\sqrt{61} ...
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128 views

Solve in positive integers: $5^x 7^y +4=3^z$

Solve in positive integers: $5^x 7^y +4=3^z$. I tried to solve it with log but I couldn't complete.
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5answers
122 views

Finding integers of the form $m+n$ that satisfies $m+n+mn=118$

Let $m$ and $n$ be two positive integers such that $m+n+mn=118.$ My question is: Can the value of $(m+n)$ be uniquely determined? I find by inspection that the pair $(m,n)=(16,6)$ (or the ...
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2answers
61 views

Are there any primes for which $a^2 = pb^2 + 1$ does not exist?

The smallest solution to the above equation for various primes are: $(p=2)$ $3^2 = 2*2^2 +1$ $(p=3)$ $2^2 = 2*1^2 +1$ $(p=5)$ $9^2 = 5*4^2 +1$ $(p=7)$ $8^2 = 7*3^2 +1$ Is there at least one ...
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1answer
48 views

A Pell-type equation

Simple Pell equations often have solutions that can be found with little work given certain conditions. These are of the form $x_{n}^{2} - A y_{n}^{2} = \pm 1$. There are harder equations that involve ...
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41 views

For which values of $k$ does: $ y^2 = x^3+(2^{2^k}+1)x$ have solutions in integers?

let $E_D$ be elliptic curve and $k$ is integer number $$E_D: y^2 = x^3+px. $$ When $p = 2^{2^k}+1$ is prime fermat . my question is :For which values of $k$ does:E.d $$ y^2 = x^3+px. $$ have ...
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1answer
23 views

Hensel lifting when not a power of a prime

Say you have the equation $x^2 + x + 47 = 0$ and that you want to determine the solutions in $\mathbb{Z}/1715 \mathbb{Z}$. Note that $1715 = 7^3 \cdot 5$. Then, using Hensel's lemma, one can find the ...
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2answers
60 views

Show that there are infinitely many integer solutions to the equation $x^3+y^5=z^7$

Show that there are infinitely many integers such that $$x^3+y^5=z^7$$ and where $x^3,y^5$ and $z^7$ are all non-zero and distinct. The hint suggests to look at solutions of simultaneous equation ...
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42 views

Number theory problem of finding prime values p and q [duplicate]

Find all pairs of prime numbers $(p,q)$ such that $$p^3-q^5=(p+q)^2$$
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1answer
37 views

Fruit vendor selling fruit

This is a very interesting word problem that I came across in an old textbook of mine. So I know its got something to do with simultaneous diophantine equations, but other than that, the textbook gave ...
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1answer
36 views

Ratio of fruits

This is a very interesting Diophantine equation word problem that I came across in an old textbook of mine. It is quite nice and I decided I would share it with MSE for future reference and a fun ...
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1answer
115 views

Integer solutions to the equation $a_1^2+\cdots +a_n^2=a_1\cdots a_n$

What is the general solution to the equation $$\sum_{j=1}^n a_j^2=\prod_{j=1}^n a_j,$$ $n\in \mathbb N$ , $n \ge 2$ over $\mathbb N_0$ ? WLOG, we can assume $0\le a_1 \le a_2\le \cdots \le a_n$ For ...
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50 views

Proving that an equation has no solution in the set of integers.

I want to show that $m^3+14n^3 = 12$ has no solution in the set of integers. Could anyone provide any insight on how to do this? Thanks.
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2answers
127 views

Why is ${n\choose k}$ is always a product of the primes of $n$ for all $n>k$? [closed]

Let $n, k$ be two positive integers such that $n>k$. Why is ${n\choose k}$ always divisible by a prime dividing $n$ (or even a product of such primes)? Please help me understand why. I cannot seem ...
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1answer
44 views

If $a^p+b^q+c^r=a^q+b^r+c^p=a^r+b^p+c^q$ .prove that $ a=b=c$ or $p=q=r$?

Let $$a,b,c,p,q,r$$ be positive integers such that : $$a^p+b^q+c^r=a^q+b^r+c^p=a^r+b^p+c^q$$ How do I prove :$ a=b=c$ or $p=q=r$ ? Thank you for any kind of help.
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1answer
35 views

Solving a system of polynomials in $N$ variables

Suppose I am given some non-negative constants $(C_p)_{p=1, ..., l}$ and I would like to find an integer $N$ and vector $v \in R^N$ such that $$ \sum_{i=1}^N (v_i)^p = C_p $$ for $p=1, ..., l$. Can ...
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1answer
82 views

On the solvability of the negative Pell equation $x^2-2py^2 = -1$

Given prime $p=8n+1$. Then $$x^2-2py^2 = -1\tag1$$ is not solvable for, $$p_1= 17, 73, 89, 97, 193, 233, 241, 257, 281, 337, 353, 401, 433, 449, 577, 593,601, 617, 641,\dots$$ but is solvable ...
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1answer
63 views

Is there any general method for solving $(a_1+a_2+..a_n)^2=a_1^3+a_2^3+…+a_n^3$ in positive integers $a_1,a_2,…a_n$?

We know the identity $(1+2+...+n)^2=1^3+2^3+...+n^3$ . So I was thinking , for given $n\in \mathbb N$ , is there any general method for solving $(a_1+a_2+..a_n)^2=a_1^3+a_2^3+...+a_n^3$ in positive ...
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0answers
98 views

When is $8x^2-4$ a square number?

I asked an earlier question on when $32x+32$ is a square number (here) and I got a very clear answer. Now I am looking to solve for which $x$ the equation $8x^2-4$ results in a square number. When I ...
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0answers
68 views

Pairs of $x$ and $y$

Here is my problem: Find all pairs of integers $(x, y)$ for which $x^2 - y$ and $y^2 - x$ are squares. Thanks for your and your suggestions.
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1answer
21 views

Positive solutions to a system of linear diophantine equations

A system of equations is $$x_1+x_2+x_3+x_4=b_1$$ $$x_1+x_5+x_6+x_7=b_2$$ $$x_2+x_5+x_7+x_8=b_3$$ $$x_3+x_6+x_8+x_{10}=b_4$$ $$x_4+x_7+x_9+x_{10}=b_5$$ $b_1,...,b_5$ and $x_1,...,x_{10}$ are positive ...
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56 views

How do I approach these Diophantine equations?

I'm a high school student, so I think my question will be an easy one. I would like to know if there is an easy way of approaching these Diophantine equations: $x=\frac{1}{2^{a}-3}$ ...
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2answers
59 views

if three integer such diophantine equation How find $x+y+z$

following Diophantine equation $$xy^2+yz^2+zx^2=x^2y+y^2z+z^2x+x+y+z$$ ie:$(x-y)(y-z)(z-x)=x+y+z$ where $x,y,z$ are integers. can find $x+y+z$ I tried some values and got some near ...
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5answers
169 views

$x^2-y^2=196$, can we find the value of $x^2+y^2$?

$x$ and $y$ are positive integers. If $x^2-y^2=196$, can we know what the value of $x^2+y^2$ is? Can anyone explain this to me? Thanks in advance.
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0answers
30 views

Finding solutions to a symmetric divisibility condition $x\mid p(y),\;y\mid p(x)$

In general, are there strategies for finding all integers $x$ and $y$ such that $x \mid p(y)$ and $y \mid p(x)$ for some polynomial $p$ with integer coefficients? For example, could we find all ...
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1answer
35 views

Counting the integer soultions to this parametric inequality

hello I am looking for an efficient way, hopefully a formula or a somewhat tight upper bound, for the number of integer solutions to the following let $k$ be a fixed integer and $\lambda \ge 1$ and ...
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2answers
37 views

Arithmetic sequence of exponentials

There is an arithmetic sequence $2^a, 3^b, 4^c$ such that a,b and c are positive integers. The question posed is to find ALL possible ordered triplets $(a,b,c)$. The constant difference d between ...
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1answer
56 views

Prove (or provide a counterexample): no pair of primitive Pythagorean triples (a,b,c) and (2a,k,c) exists.

A primitive Pythagorean triple is an ordered set of coprime integers (a,b,c) such that $a^2+b^2=c^2$. Show that the system of Diophantine equations $$a^2+b^2=c^2$$ $$4a^2+k^2=c^2$$ have no solutions.
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125 views

Is $y^2=x^3+7$ unsolvable modulo some $n$?

The equation $y^2=4x^3+7$ has no integral solution since $y^2\equiv4x^3+7\pmod4$ has no solution (i.e. has no solution in $\Bbb{Z}/4\Bbb{Z}$). It is well known that $y^2=x^3+7$ has no integral ...
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1answer
109 views

When is $c^4-72b^2c^2+320b^3c-432b^4$ a positive square?

In trying to solve a certain [third-degree] Diophantine equation, I have used the quadratic equation to determine that $$c^4-72b^2c^2+320b^3c-432b^4$$ must be a positive integer square, where $c$ and ...
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48 views

About pythagorean triples

In the circle of diameter $AB$ it is well known each point $C$ determines a right triangle $\Delta ABC$ and so it is with every point $D$ on the circle of diameter $AC$ determining a right triangle ...
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26 views

Quadratic Diophantine Problem in two variables

I have a quick question in regards to solving a quadratic two-variable diophantine problem. The equation is $6x^2 - 2xy + 3y - 17x = 6$. My attempt thus far starts by making y the subject: $$y = ...
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34 views

What is known about the Heronian primes?

A Diophantine equation $$x^3 - Dy^3 = 1$$ always has a trivial solution $x = y^3 + 1$. It appears that a non-trivial (that is those with $x$ smaller than trivial) solution exists iff $y$ is a Heronian ...
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1answer
46 views

Find all natural solutions to $x^2+2y^2 = z^2$ [duplicate]

I need to find all natural solutions to $x^2 + 2y^2 = z^2$ What I tried: I did $\pmod 2$ to the equation receiving $z^2 - x^2 \equiv 0 \pmod 2$. Then there are two possibilities: $x^2 \equiv 0 ...
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46 views

find all natural solution that satisfy $x^2+y^2 = 3z^2$

I need to find all natural $x,y,z$ that satisfy the following $x^2+y^2 = 3z^2$ $(0,0,0)$ is an answer of course. What I tried: I tried solving with congruences. I know that every square number ...
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23 views

Postive integer solution to this equation $a^2+b^2+c^2+1=kabc$

Frobenius and Hurwitz( in 1880) prove this theorem: For any positive integer $k$ other than 1 or 3, the equation $a^2+b^2+c^2=kabc$ has no integral solution except (0,0,0). My Question,How to solve ...
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2answers
125 views

Diophantine Equation $ x^n + y^n =z^n (x<y, n>2) $

I am looking for simple college level algebraic solution to prove that $x$ and $y$ ($x$ < $y$) for the above equation can't be prime numbers. (I know more complex and involved solution using high ...
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1answer
80 views

Solving a little Diophantine equation:$(n-1)!+1=n^m$ [duplicate]

How can I solve this Diophantine equation: $$(n-1)!+1=n^m$$ with $n,m$ positive integers? From Wilson's theorem we can note that $n$ is a prime number. I proved to rewriting the equation ...
2
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3answers
85 views

Almost extended Euclidean algorithm - $ax+by=\gcd(a,b)+2$

So I have this equation: $$\eta+2=2g+1n,$$ where $g,n \in \mathbb{N}_{\geq 0}$ and $\eta \in \mathbb{N}_{>0}$. I want to find all possible integer-valued 2-tuples $(g,n)$ that satisfy this ...
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1answer
43 views

When do three cubics form an arithmetic progression?

Are there any solutions to the diophantine equation $x^3+y^3=2z^3$ other than the trivial ones? What about $x^4+y^4=2z^4$? I think I remember these equations in one of Euler's work, but having ...
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0answers
51 views

Solving a diophantine equation in an elementary way

I was trying to solve $x^2+1=y^3$ and found this answer: Does an elementary solution exist to $x^2+1=y^3$? but I'm having trouble understanding it. In the second last paragraph, why is there a ...
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1answer
50 views

How do I prove this claim?

Claim :Let $p$ be a prime and $m \geq 2$ be an integer. Prove that the equation $ \frac{ x^p + y^p } 2 = \left( \frac{ x+y } 2 \right)^m $ has a positive integer solution $(x, y) \neq (1, 1)$ if and ...
2
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3answers
61 views

Conjecture about linear diophantine equations

I've been dabbling with linear Diophantine equations and came across a rather interesting pattern that I would like to conjecture as true but I have no idea how about to come up with a proof. Let ...
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1answer
34 views

Solving a Diophantine equation with LTE

Show that only positive integer value of $a$ for which $$4(a^n+1)$$ is a perfect cube for all positive integers $n$, is $1$. Rewriting the equation we obtain: $$4(a^n+1)=k^3$$ It is obvious that $k$ ...