Questions on finding integer/rational solutions of polynomial equations.

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-1
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0answers
22 views

Diophantine Equation with gcd. [duplicate]

Find all positive integers $a,b$ such that $\gcd(3^a+1,3^b+1)$ is a multiple of $ab$. I've given this problem many attempts but I can't seem to make any progress, there doesn't appear to be any way ...
-2
votes
1answer
35 views

Solve for b and d

Solve for b and d in the following equation. A triangle with sides $(a, a, b)$ has the same area and the same perimeter as a triangle with sides $(c, c, d)$ where $a, b, c$ and $d$ are positive ...
3
votes
2answers
111 views

Showing there is no triplet of positive integers $(a,b,c)$ satisfying $a^7+b^7=7^c$ [duplicate]

Show that $$a^7+b^7=7^c$$ has no positive integer solutions $(a,b,c)$. I've posted a general and way too long approach as an answer. How may one prove the claim more briefly and specifically?
0
votes
1answer
66 views

Are all even numbers the difference of prime powers

Does there exist an even positive integer greater than $100$ (to eliminate trivial cases) that cannot be expressed in the form: $p^2-q$ $p-q^2$ $p^2-q^2$ $p^3-q^3$ where $p$ and $q$ are primes.
0
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1answer
61 views

Can the solvability of a single Diophantine equation be undecidable (in any sense of the word)?

Apologies in advance for asking the following "philosophical" question, which falls dramatically short of any reasonable standards of mathematical rigour: Is it possible that there should exist a ...
2
votes
1answer
140 views

Solve $x^n+z^n=(x+1)^n$ for $n\ge 3$ without FLT

Is there a way to prove that for $x,z,n \in \mathbb{Z}$, $x > 0$, $z > 0$, $n > 2$, the equation $$ x ^ n + z ^ n = (x + 1) ^ n $$ has no solution, without using Fermat's Last Theorem? ...
4
votes
1answer
62 views

Substitutions that transform Fermat Equations to Elliptic Curves

I was reading Chapter 1 of Elliptic Curves - Number Theory and Cryptography by Lawrence C Washington. He was considering Fermat equations $$a^4+b^4=c^4\text{ and }a^3+b^3=c^3.$$ For the 1st equation, ...
3
votes
1answer
95 views

To find positive integers $n$ such that $\dfrac {n(n+1)(n+2)}6$ is a perfect square [duplicate]

How many positive integers $n$ are there such that $\dfrac {n(n+1)(n+2)}6$ is a perfect square ? I know $n=1 , 2$ works ; are there any more ? Are there only finitely many such $n$ ?
6
votes
1answer
134 views

To solve $n(n+1)(n+2)=6m^3$ in positive integers $m,n$

How to find all positive integers $m,n$ such that $n(n+1)(n+2)=6m^3$ ? I can see that $m=n=1$ is a solution , but is it the only solution ?
1
vote
2answers
50 views

Square numbers in the form $1+4y$

I want to solve the equation $y+x=x^2$: $$ x^2-x-y=0 \\ x_{1;2}=\frac{1\pm \sqrt{1+4y}}{2} $$ However I want the solutions to be only natural numbers; the question then turns to find values of $y$ ...
1
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2answers
44 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 ...
1
vote
2answers
49 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 ?
1
vote
1answer
25 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 $\theta$ ...
0
votes
2answers
35 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?
1
vote
3answers
76 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: ...
0
votes
0answers
37 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 ...
9
votes
0answers
82 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} ...
4
votes
3answers
129 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.
1
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5answers
138 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 ...
2
votes
2answers
63 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 ...
1
vote
1answer
52 views

How to solve a bivariate quadratic (not necessarily 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 ...
0
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0answers
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 ...
3
votes
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 ...
3
votes
2answers
61 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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0answers
42 views

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

Find all pairs of prime numbers $(p,q)$ such that $$p^3-q^5=(p+q)^2$$
3
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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 ...
3
votes
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 ...
9
votes
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 ...
3
votes
2answers
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.
2
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2answers
128 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 ...
3
votes
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.
2
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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 ...
7
votes
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 ...
6
votes
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 ...
4
votes
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 ...
2
votes
0answers
69 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.
1
vote
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 ...
1
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0answers
57 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: $x=\frac{1}{2^{a}-3}$ ...
1
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2answers
60 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 ...
2
votes
5answers
174 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.
2
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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 ...
1
vote
1answer
36 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 ...
0
votes
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 ...
2
votes
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.
6
votes
2answers
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 ...
3
votes
1answer
110 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 ...
3
votes
0answers
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 ...
1
vote
2answers
27 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 = ...
0
votes
0answers
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 ...
1
vote
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 ...