Questions on finding integer/rational solutions of equations.

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1answer
51 views

is A an even number?

Let $a,b,c,d$ be positive integers such that $(3a+5b)(7b+11c)(13c+17d)(19d+23a)=2001^{2001}$ hence, prove that $a$ is even. I tried to approach this problem reducting it modulo 6. From which we ...
3
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0answers
124 views

the system of diophantine equations: $x+y=a^3$; $xy=\dfrac{a^6-b^3}{3}$ has only trivial solutions.

Without using Fermat's Last Theorem, how can one prove that the following system of diophantine equations has only trivial solutions: $$x+y=a^3$$ $$xy=\dfrac{a^6-b^3}{3}$$ We suppose of course that $\...
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1answer
32 views

Representations of some primes as $x^2-2y^2$?

I am looking for (elementary) proofs (idea of the proofs is also OK) or references to proofs of the followings: $$ p\equiv\pm1(\mod8)\longrightarrow p=x^2-2y^2 $$ Any help appreciated.
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1answer
49 views

Hyperbolic Diophantine Equations: Application of Euclidean Algorithm?

I'm trying to determine whether or not I can find the integer solutions to $(x+a)$$(x+b)$ $=$ $x(x-1)$ + $x(a-b)$ (with a known $x$ value you choose, i.e. $707$). Plugging in my example value on ...
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0answers
39 views

Problem about Diophantine's equation and congruences

This problem is from Niven's, 5.4.8. Let $\,f(\mathbf{x})=f(x_1,x_2,x_3) = x_1^4+x_2^4+x_3^4-x_1^2x_2^2-x_2^2x_3^2-x_3^2x_1^2-x_1x_2x_3(x_1+x_2+x_3).$ Show that $f(\mathbf{x}) \equiv1$ $($mod$\,4$$)$...
3
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1answer
76 views

How to solve $p^n+12^2=m^2$

Find all triples $(m,n,p) \in \mathbb{N}^3$, with $p$ prime, which satisfy $$p^n+12^2=m^2$$
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0answers
44 views

A farmer bought some chickens and cows from a local rancher…

A farmer bought some chickens and cows from a local rancher. If a chicken costs 2 dollars and a cow costs 5 dollars, how much of each can he purchase if the total cost is 38 dollars and he purchases ...
4
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2answers
90 views

Solve in positive integers the equation $a^3+b^3=9ab$

Solve in positive integers the equation: $$a^3+b^3=9ab$$ I try to: $$\dfrac{a^2}{b}+\dfrac{b^2}{a}=9\Longrightarrow a^2<9b,b^2<9a$$ Of course, I can't solve it. Can anyone help?
0
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1answer
117 views

Solutions to simultaneous Diophantine equations $2y^2-3x^2=-1$ and $z^2-2y^2= -1$

I am looking for integer solutions for the following set of equations: $2y^2-3x^2=-1$ $z^2-2y^2= -1$ I know that there are the solutions (1,1,1) and (-1,-1,-1) for this set of simultaneous ...
3
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2answers
94 views

proof - $(x,y) = (4,6)$ is the only solution for $x^3 + x^2 - 16 = 2^y$

I saw this question and on seeing the answers I believed it did not have to be so complicated. The pair $(x,y) = (4,6)$ only fits the equation. Find all possible $(x, y)$ pairs for $x^3 + x^2 - 16 ...
2
votes
2answers
219 views

Find integers $(w, x, y, z)$ such that the product of each two of them minus 1 is square.

In the case of $(5, 442, 541)$, the product of each two of them minus 1 is a square: $$5 \times 442 - 1 = 47^2, 5 \times 541 - 1 = 52^2, 442\times541 - 1 = 489^2$$ What are the integer-solutions $(w,...
5
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1answer
91 views

Consecutive smooth triplets

Consecutive $n$-smooth triplets with no common factors are possible. The sequence 64, 120, 324, 2024, 17576, 248676, 314432, 6571774, 7496644, 116026274, 196512876 isn't in OEIS, but they do appear in ...
2
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0answers
21 views

Reformulating diophantine inequalities

Assume we are given inequalities $x \not\equiv a_i\text { (mod }b_i)$ for $i=1,\ldots,n$ where $1 \leq a_i \leq b_i, x \in \mathbb{Z}$. Can we somehow reformulate the problem as $x \not\equiv a_i'\...
5
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1answer
70 views

$N=(x^2-1)(y^2-1)$ has more than one solution

Given that $N=(x^2-1)(y^2-1)$ where $N,x,y,a,b$ are positive integers, find with proof the smallest value of $N$ such that $N=(x^2-1)(y^2-1)=(a^2-1)(b^2-1)$, where $a$ is not equal to either $x$ or $y$...
5
votes
2answers
90 views

If $\frac1x-\frac1y=\frac1z$, $d=\gcd(x,y,z)$ then $dxyz$ and $d(y-x)$ are squares

Let $x, y, z$ be three non negative integer such that $\dfrac{1}{x}-\dfrac{1}{y}=\dfrac{1}{z}$. Denote by $d$ the greatest common divisor of $x, y, z$. Prove that $dxyz$ and $d(y-x)$ are squares ...
0
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1answer
53 views

Find minimum of the $n$ such $x+11y+11z=n$ has $16653$ triples of postive integers solution

I wish to solve following problem $$x+11y+11z=n(n\in N^{+})$$ has $16653$ triples $(x,y,z)$ of postive integers. Find $n_{\min}$ Of course, I can't solve it by Now, so there any solution? Problem 2:...
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2answers
55 views

Find all integer solutions for $x*y = 5x+5y$

For this equation $x*y = 5x + 5y$ find all possible pairs. The way I did it was: $x=5y/(y-5)$ And for this I wrote a program to brute force a couple of solutions. If it helps, some possibilities ...
14
votes
1answer
192 views

Integral solutions $(a,b,c)$ for $a^\pi + b^\pi = c^\pi$

We know that $a^n + b^n = c^n$ does not have a solution if $n > 2$ and $a,b,c,n \in \mathbb{N}$, but what if $n \in \mathbb{R}$? Do we have any statement for that? I was thinking about this but ...
3
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2answers
148 views

Can only the middle school math knowlegde help to find solutions for $2013 y^2 -xy -4026 x=0$?

I found the following equation form an answer written for a question. $$2013 y^2 -xy -4026 x=0$$ But I'm confused that can I really learn how to find the positive integer solutions for $x,y$ with ...
6
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1answer
120 views

On a remarkable system of fourth powers using $x^4+y^4+(x+y)^4=2z^4$

The problem is to find four integers $a,b,c,d$ such that, $$a^4+b^4+(a+b)^4=2{x_1}^4\\a^4+c^4+(a+c)^4=2{x_2}^4\\a^4+d^4+(a+d)^4=2{\color{blue}{x_3}}^4\\b^4+c^4+(b+c)^4=2{x_4}^4\\b^4+d^4+(b+d)^4=2{x_5}...
4
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2answers
78 views

Find the solutions of the diophantine equation $(x^2-y^2)(z^2-w^2)=2xyzw$

Let $x,y,z,w$ be postive integers. Find all solutions of: $$(x^2-y^2)(z^2-w^2)=2xyzw$$ This gives: $$\left(\dfrac{x}{y}-\dfrac{y}{x}\right)\left(\dfrac{z}{w}-\dfrac{w}{z}\right)=2$$ $$\left(p-\...
5
votes
0answers
96 views

Looking for the most elementary proof that $48X^4+12X^2+1=Y^2$ has no non-trivial integer solution.

As relayed in this question of mine (which is more general in scope), I believe I have found a relatively easy, and completely elementary, way to show that the equation $$48X^4 + 12X^2+1 = Y^2$$ has ...
1
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1answer
36 views

Weight 2 Newforms of large level computations.

I am stuck with some weight $2$ newform computations of large level. For example I want to compute newforms of level $11520$. Can anyone suggest me a way to do it? I need it to solve some diophantine ...
1
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1answer
44 views

Simultaneous congruences to different moduli

For some natural numbers $a,b$, let it be known that $b\leq 8$: $$b\cdot 99a = x68y - 8$$ where we read $x68y$ as x-thousand sixhundred and eighty-y. What I have established is: $$\begin{cases}b\mid (...
2
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0answers
37 views

solutions for the diophantine equation $x^2+y^2=n$ [duplicate]

Are there any solutions for the diophantine equation $x^2+y^2=n$ ? For $n \in \mathbb{P} \wedge n \equiv1\pmod4$ solutions are widely known. Can we generalize a bit?
10
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4answers
215 views

Does the equation $x^2+23y^2=2z^2$ have integer solutions?

I would like to show that the image of the norm map $\text N : \mathbb Z \left[\frac{1 + \sqrt{-23}}{2} \right] \to \mathbb Z$ does not include $2.$ I first thought that the norm map from $\mathbb Q(\...
1
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2answers
47 views

Proving there are infinitely many solutions

Let c be any integer. Prove that if $k$ and $l$ are coprime positive integers, then the linear Diophantine equation $kx-ly=c$ has infinitely many positive integer solutions To start off, I know ...
2
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1answer
110 views

Generalizing Fermat's challenge to Frenicle

In 1643, Fermat asked Frenicle et al to find a special Pythagorean triple $a,b,c$ such that for $n=1$, $$a+nb = r_1^2\\ a^2+b^2 = r_2^4\tag1$$ Equivalently, $$\color{blue}{\big((p^2-q^2)^2-(2pq)^2\...
0
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1answer
39 views

integer points on an ellipse

I have the equation $1=x^2-xy+y^2 = \frac 1 4 (x+y)^2 + \frac 3 4 (x-y)^2$ where I am looking for integer solutions $x,y \in Z$. When you draw this ellipse it is quite obvious that the integer points ...
4
votes
2answers
73 views

Integer solutions for $x^4 + 4xy^3 = z^2$

Find all triplets $(x,y,z)$ of integers so that $$ x^4 + 4xy^3 = z^2. $$ What I've done: Suppose $x=0$. Then we see $z=0$ and hence $(0,y,0)$ is a solution. Suppose $y=0$. Then we see $x^4 = z^2$ ...
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0answers
29 views

Alternative derivation of the second Frobenius number

Let $a,b$ be integers with $\gcd(a,b)=1$. What is then the largest integer $N$ which cannot be written as a linear combination with non-negative integer coefficients of $a$ and $b$? A few days ago, I ...
0
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0answers
22 views

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

Is there a simple test to determine whether the Diophantine equation, $\sum_i a_i x_i = c$ with $a_i, x_i, c$ as integers and $a_i > 0, x_i \geq 0, c > 0 $ has a non-negative integer solution?
3
votes
1answer
94 views

Solving $2(n-1)n(n+1)(n+2)=(m-3)(m+3)$

The question is: Find all pairs $(n,m)\in\mathbb{N}^2$ such that $$2(n-1)n(n+1)(n+2)=(m-3)(m+3)$$ I checked all $n<10000$ and only got $n=1$ and $n=4$ with their corresponding $m$, so I ...
4
votes
4answers
186 views

When is $991n^2 +1$ a perfect square?

What should be the value of $n$ so that the number obtained after adding $1$ to $991$ times its square is itself a perfect square? Can you please give me a few hints on this topic with a few specific ...
3
votes
1answer
107 views

Special Euler bricks and $x^2(y^2-1)^2+y^2(x^2-1)^2=z^2$

Define, $$P_1 := a^2+b^2\\ P_2 := a^2+c^2\\ P_3 := b^2+c^2$$ Let, $$a,b,c = 2xy,\;x(y^2-1),\;y(x^2-1)$$ and $P_1,P_2$ become squares. If we wish to make $P_3$ a square as well, then, $$P_3:=x^2(...
0
votes
1answer
54 views

Diophantine Equation: how to find k such as solutions are natural numbers? [duplicate]

I was trying to solve some algorithms when I found out that one could leverage a simple stupid linear Diophantine equation, but I cannot figure out how solve it (I'm pretty sure I would have been able ...
30
votes
0answers
729 views

How to solve this two simultaneous “divisibilities” : $n+1\mid m^2+1$ and $m+1\mid n^2+1$

Is it possible to find all integers $m>0$ and $n>0$ such that $n+1\mid m^2+1$ and $m+1\,|\,n^2+1$ ? I succeed to prove there is an infinite number of solutions, but I can not progress anymore......
3
votes
2answers
109 views

Primes of the form $x^3+y^3+z^3 - 3xyz$

Do quadruplets $(x,y,z,p)$ of positive integers exist for which $p$ is a prime number and $$x^3+y^3+z^3 = 3xyz + p?$$ I've tried looking for solutions in mathematica for $x,y,z<1000$, without ...
2
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1answer
56 views

Solving diophantine equations

So the equation I am trying to solve is $x^2=y^4-77$ So far I have rearranged and factorised the equation to get: $$(y^2-x)(y^2+x)=77$$ But I am really unsure of how to solve it from here. Thanks in ...
1
vote
1answer
76 views

how to solve $x^2-y^2=(2n-1)^2$

If $n$ is known,and $x,y,n$ belong to $\mathbb{N}^+$. What is $x$ and $y$? I know there exists a answer, for example,when $n=111$, $x=6161$, $y=6160$, but I do not know if the answer is unique.
2
votes
2answers
80 views

Multiple solutions to Diophantine Equation $X = \tfrac{1}{2} xy(x^2 - y^2) $

I am trying to figure out if the following is true or false: $X = \tfrac{1}{2} xy(x^2 - y^2) $ has three (or more) solutions of $x, y$ if and only if $X = 6561555*n^4 = 3*5*7*11*13*19*23*n^4 $ where ...
5
votes
1answer
160 views

Trying to solve the equation $\sum_{i=0}^{t}(-1)^i\binom{m}{i}\binom{n-m}{t-i}=0 $ for non-negative integers $m,n,t$

While considering a previous unanswered question, I started looking for the non-negative integer solutions $ m,n,t , (n\ge m)$ to the equation: $$ S(m,n,t)=\sum_{i=0}^{t}(-1)^i\binom{m}{i}\binom{...
0
votes
1answer
83 views

$x^3 + 5x + 6 = 3\cdot 2^{1+x-k} $

Does anyone know how to solve $$n^3 + 5n + 6 = 3\cdot 2^{1+n-k} $$ where n,k are natural numbers? I was told that there are prime number arguments that can be used but I am totally stuck. It is a ...
4
votes
3answers
151 views

proof - if $x^2 + y^2 + z^2 = 2xyz$ then $x = y = z = 0$ [duplicate]

Well, I have been trying to prove that: $$x^2 + y^2 + z^2 = 2xyz \implies x = y = z = 0$$ and have made little progress. Till now, I have only been able to prove that if this is to happen then $x$, $...
4
votes
1answer
67 views

Show that there exist no $a, b, c \in \mathbb Z^+$ such that $a^3 + 2b^3 = 4c^3$

Find all positive integer solutions of $a^3 + 2b^3 = 4c^3$. Proof: There don't exist any integer solutions for the give equation. Proof by the Well Ordering Principle. Let $d$ be the set of all ...
0
votes
1answer
28 views

Diophantine equation in the integers

Find all integers $a,b$ such that $6(a^2-ab+b^2) = 31(a+b)$ Ideas I have had so far: Move everything to the left side of the equation and try to solve a polynomial in $a$,whose discriminant in $b$ ...
1
vote
1answer
43 views

Looking for a general method for this type/class of Diophantine equation

I have the following Conjecture: If $w$ and $z$ are non-negative integers satisfying the equation $$ w(w+6) = z(16z^2+36z+27), \tag{$\star$} $$ then $w=z=0$. I believe it to be true for the ...
3
votes
2answers
67 views

Natural numbers $a,b,c$ satisfaying $abc=2(a+b+c)$

How can one find all natural numbers such that: $a≤b≤c$ $$abc=2(a+b+c)$$ I tried this : $abc-2c=2a+2b$ so $c=\frac{2(a+b)}{ab-2}$
3
votes
1answer
107 views

Equation $x^3+2x+1=2^n$ in positive integers

Determine all pairs of positive integers $(x,n)$ which satisfy the condition $$x^3+2x+1=2^n$$ My work so far: Obviously, $x$ is odd. We show that solutions exist only for $n \in\{1,2\}$. Suppose ...
0
votes
1answer
34 views

Solve the equation for $x,y\in\mathbb{Z}$: $x^4-2x^3+x=y^4+3y^2+y$.

As in the title. I have no idea how to deal with such equations, I'm completely new to this topic.