Questions on finding integer/rational solutions of polynomial equations.

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0answers
34 views

How to find whole number answers in systems of square root equations

Given the following 4 equations, can you find 4 whole number answers using whole number variable inputs? $x,y,z$ where $x>y>z$ $Eq 1 = (x^2-2xy+y^2-2xz+z^2)^{\frac{1}{2}} $ $Eq 2 = ...
1
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0answers
19 views

The Nagell-Ljunngren Equation

I have been trying to find the papers of Nagell and Ljunngren, which deal with the equation $$\frac{x^n - 1}{x - 1} = y^2$$ and solve it completely. Many papers cite these papers, but I haven't found ...
1
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1answer
62 views

Diophantine equation $\frac{1}{x}+\frac{1}{y}+\frac{1}{z}=\frac{3}{5}$

I'm trying to solve the diophantine equation $\frac{1}{x}+\frac{1}{y}+\frac{1}{z}=\frac{3}{5}$ for all $x,y,z \in \mathbb{Z}$. I did this. Since that the diophantine equation is symmetric, we can ...
1
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3answers
61 views

Solve in non-negative integers: $m^2+n^2=1997 (m-n)$

Solve in $\mathbb{N}$:$$m^2+n^2=1997(m-n)$$ I try with quadratic equation or with factorising, but I have no idea what to do after that.
3
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0answers
144 views

An argument for “Brocard's problem has finite solution”

Brocard's problem is a problem in mathematics that asks to find integer values of n for which $$x^{2}-1=n!$$ http://en.wikipedia.org/wiki/Brocard%27s_problem. According to Brocard's problem ...
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0answers
22 views

Argument for finite solution of power Diophantione Equation.

Assume the equation $4x^3=y^2+3$ has infinite positive integer solution. If $x,y$ has general solution then it is clear that for any $x$(rational, integer), there is a $y$. It can be said there is a ...
6
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2answers
174 views
+100

diophantine equation $x^3+x^2-16=2^y$

Solve in integers: $x^3+x^2-16=2^y$. my attempt: of course $y\ge 0$, then $2^y\ge 1$, so $x\ge 1$. for $y=0,1,2,3$ there is no good $x$. so $y\ge 4$ and we have equation $x^2(x+1)=16(2^z+1)$, ...
0
votes
1answer
54 views

Integer solution to the equation

Does there exists an integer solution (for every integer $m\geq 1$) for the following equation? $$x_1x_2...x_n+(2y+1)z+y=4m+3$$ where, $1\leq x_1\leq x_2\leq...\leq x_n\leq l$,$0\leq y \leq ...
7
votes
2answers
152 views

Solve $x^3=y^2-y+1$ in positive integers.

I recently started doing number theory and have finished with all the basic, intermediate and some of the advanced stuff with ease. However, I encountered this question and have been stuck for about a ...
1
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1answer
39 views

Diophantine Equations with Factor Exponents

I'm trying to prove that the following equations have no solutions to finish a problem. They're intuitively impossible but I'm looking for rigorous arguments (if they are actually possible, then prove ...
12
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2answers
575 views

diophantine equation in positive integers

would you please help me solve this? solve this equation in positive integers: $x^2+y^2+z^2=3xyz$ I could prove that it's solutions are infinite, for if $(x,y,z)$ is a solution, with $x\le y \le ...
1
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1answer
38 views

How do I solve these questions using Diophantine equations?

I have been told that it is easier to solve the below 2 questions using Diophantine equations instead of simply trial and error. 1) Find the smallest positive integer which, when divided by 6, ...
0
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0answers
13 views

minimising multivariate quadratic function over integer variables

I have a quadratic function $x_1^2+x_2^2-(u_1x_1+u_2x_2)^2$ which I need to minimise over integer $x_1$ and $x_2$; also, the coefficients $u_1,u_2<1$. In other word, assuming coefficients ...
9
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1answer
76 views

Solving a Diophantine equation: $p^n+144=m^2$

I found this Diophantine equation: $$p^n+144=m^2$$ where $m$ and $n$ are integers and $p$ is a prime number. I solved it but I want to know if there exist other proofs through the use of rules of ...
4
votes
0answers
77 views

Solve $(x+1)^n-x^n=p^m$ in positive integers

Solve in positive integers: $$(x+1)^n-x^n=p^m$$ $p$ is prime, $n\ge 2$. Seemingly Zsigmondy's Theorem and LTE won't work here. Though you can tell (as suggested by user barto), using ...
5
votes
3answers
325 views

Whenever Pell's equation proof is solvable, it has infinitely many solutions

Prove that whenever the equation $x^2 - dy^2 = c$ is solvable, then it has infinitely many solutions. I consider that, if $u$ and $v$ satisfy $x^2 -dy^2 = c$ and then $r$ and $s$ satisfy $x^2 ...
0
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0answers
13 views

Diophantine equation involving three variavles

I wonder if it is possible to enumerated all the integer solutions of the Diophantine equation $ax+bxy^2=xz^2+4z$, with $a,b\in\mathbb{N}$. Of course we may solve it for $x$ and obtain rational ...
1
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2answers
57 views

Diophantine equation resembling FLT

I was wondering if the equation $x^p+y^p=2z^p$ has been studied. For small cases it is seen that the only solutions are trivial: $x=y=z$. There are probably methods to solve this for regular ...
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4answers
2k views

$n!+1$ being a perfect square

One observes that $4!+1 =25=5^{2}$, $5!+1=121=11^{2}$ is a perfect square. Similarly for $n=7$ also we see that $n!+1$ is a perfect square. So one can ask the truth of this question: Is $n!+1$ a ...
2
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0answers
47 views

Sums of three cubes in arithmetic progression equal to a cube $x^3+(x+y)^3+(x+2y)^3 = z^3$

Using exhaustive search, small positive and primitive integer solutions to, $$x^3+(x+y)^3+(x+2y)^3 = 3 x^3 + 9 x^2 y + 15 x y^2 + 9 y^3= z^3\tag1$$ are, $$x,y = 3,1$$ $$x,y = 149,107$$ $$x,y = ...
3
votes
5answers
84 views

Can every perfect square exist as the sum or difference of two perfect squares?

I believe this is trivial and I'm over-complicating it. But can every squared integer be expressed as the sum of two squared integers OR the difference of two squared integers? And is there a proof ...
2
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5answers
53 views

Possible solutions of a diophantine equation:$p^2+pq+275p+10q=2008$

What are couples of prime integers that verify this diophantine equation:$$p^2+pq+275p+10q=2008?$$ I tried to solve this equation trough the rules of modular-arithmetic. I rewrite the equation as: ...
0
votes
1answer
29 views

Solving a Diophantine Equation of the form $N(N-1) = 2X(X-1)$ for $N, X > 0$

When working on a problem on Project Euler I came up with a formula I need to solve: $N(N-1) = 2X(X-1)$ for $N > 10^{12}, X > 0$ with $N$ and $X$ being integer numbers. After some ...
2
votes
1answer
371 views

What are the necessary and sufficient conditions for a cubic equation to have integers roots

Let's start with Fermat equation with the lowest power, $x^3 + y^3 = z^3$. Now let's set $y = x + a, z = x + b$ with $b > a$ and $a,b$ integers. then the equation becomes $$x^3 + (3a-3b)x^2 + ...
1
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1answer
80 views

Diophantine equation $a^3 + b^3 + c^3 = 2$

I have a pretty difficult math question that I have no idea even how to begin. Here it goes: Find the nonzero integers $a$, $b$, $c$ such that $a^3 + b^3 + c^3 = 2$? I would assume that at ...
3
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2answers
62 views

Faster Sage Code for Diophantine Equation? [closed]

I'm having trouble with the computation time. Does anyone have any ideas for faster code? ...
4
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2answers
91 views

Solving $x^2 - 11y^2 = 3$ using congruences

I'm looking to find solutions to $x^2 - 11y^2 = 3$ using congruences. The question specifically asks "Can this equation be solved by congruences (mod 3)? If so, what is the solution? (mod 4) ? (mod ...
2
votes
3answers
123 views

Integer solutions to $x^{x-2}=y^{x-1}$

Find all $x,y \in \mathbb{Z}^+ $ such that $$x^{x-2}=y^{x-1}.$$ I can only find the following solutions: $x=1,2$. Are there any other solutions?
2
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2answers
122 views

p odd prime. Prove that if $a\equiv b\pmod p$ then $a^p\equiv b^p\pmod p^2$. Then show $x^5+y^5=z^5$ has no integer solutions with $5\not\mid xyz$

Question: Let $p$ be an odd prime. Prove that if $a\equiv b \pmod p$ then $a^p\equiv b^p \pmod p^2$. Then show the Diophantine equation $x^5+y^5=z^5$ has no integer solutions with $5\not\mid xyz$. My ...
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1answer
31 views

Solutions $3 p\sin x - (p+\sin x)(p^2-p \sin x +\ sin ^{2} x) =1$

$3 p \sin x - (p+\sin x)(p^2-p \sin x + \sin ^{2} x) =1$ has a solution for $x$. Then number of integral solutions of $p$ are ?
1
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1answer
41 views

How many non-negative integral solutions?

How many non-negative integral solutions does this equation have? $$17x_{17}+16x_{16}+ \ldots +2x_{2}+x_1=18^2$$ I add some conditions that bring more limitations: $$\sum_{i=1}^{17}x_{i}=20 \quad 0 ...
2
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0answers
96 views

How to solve $x^2+11=y^3$?

I've been trying to solve the diophantine $$x^2+11=y^3$$ recently but to no avail. I tried the "UFD trick", re-writing as $(x-i\sqrt{11})(x+i\sqrt{11})=y^3$, but it didn't give me all the solutions. I ...
0
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0answers
28 views

Diophantine eqution with a parameter

My question is about the problem when is the number $$\frac{m^3 + n^3}{n^2+m^2+m+n+c}$$ a natural number. Here $c\in \mathbb{N}$ is a constant and $m, n \in \mathbb{N}$ are the variables. This ...
0
votes
2answers
50 views

The diophantine equation $z^2=a^2+bx^2+cy^2$

Is there a way to obtain (enumerate) the integer solutions $(x,y,z)$ of the following quadratic Diophantine equation $z^2=a^2+bx^2+cy^2$ where $a$ is an integer and $b, c$ are positive integers? I ...
3
votes
2answers
51 views

Solving the diophantine equation $p^2+n-3=6^n+n^6$

What are the pairs ($p,n$) of non-negative integers where $p$ is a prime number, such that $$p^2+n-3=6^n+n^6$$ How can I solve this diophantine equation?
3
votes
1answer
35 views

Bounding $x^2+6x$ between consecutive cubes when solving $y^3=x^2+6x$

I am familiar with the method of bounding a polynomial between consecutive squares to prove it is not a square. For example, this method can prove $y^2=x^2+x+1$ has no solutions since ...
7
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0answers
91 views

Looking for all sequences such that $a_i^2+a_j^2=a_k^2+a_l^2$ whenever $i^2+j^2=k^2+l^2$

I'm working in a difficult functional equation, and I have reduced the problem to the following question ($\mathbb{N}$ denotes the set of non negative integers $0,1,2,3,4\cdots$) Question: Can we ...
2
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4answers
308 views

Sum of square patterns

Can anyone give the name of this pattern $$136^2+137^2+138^2+139^2+140^2+141^2+142^2+143^2+144^2 =\\ 145^2+146^2+147^2+148^2+149^2+150^2+151^2+152^2$$
2
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2answers
46 views

A simple question:

let $a,b,c,d$ be all positive integers such that $a-bc \neq 0$,and $\gcd(a,b)=1$. Under what conditions, $(a-bc)$|$(a-b^d)$? In other words, does it exist any integer $k \neq 1 $ such ...
0
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1answer
98 views

On a theorem of Kronecker! [closed]

Let $\alpha$ be an irrational number and $\beta$ be an arbitrary real number, Prove that there are infinitely many pair of integers $(x,y)$ with $x\in\mathbb{N}$ such that: ...
3
votes
1answer
75 views

For which primes $p$ does $px^2-2y^2=1$ have a solution?

Let $p$ be an odd prime. If $px^2-2y^2=1$ is solvable, we can get Jacobi symbol $(\frac{-2}{p})=1$, so $p=8k+1,8k+3$. But when $k=12$, $p=97$, the Pell equation $97x^2-2y^2=1$ is unsolvable. I think ...
5
votes
1answer
187 views

To find all $n$ such that $(n-1)!+1$ is a perfect power of $n$

How to find all positive integers $n$ such that $(n-1)!+1$ can be written as $n^k , k\in \mathbb Z^+$ ?
6
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2answers
125 views

$1+n!=m^{2}$ for some n,m$\in\mathbb{N}$ [duplicate]

I have no idea whether this is known or not and I couldn't find anything related on Google. While I was studying , I come up with this idea $1+n!=m^{2} $ for some $n,m\in\mathbb{N}$ $1+4!=5^{2}$ ...
1
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3answers
47 views

How can i solve this diophantine equation:$x^2-(6p-4q)x+3pq=0$?

I found this diophantine equation $$x^2-(6p-4q)x+3pq=0$$ (p and q both prime numbers) and i posted my answer but i want to know if there are other methods to find the solutions of this equation. What ...
17
votes
1answer
349 views

Are there $a,b>1$ with $a^4\equiv 1 \pmod{b^2}$ and $b^4\equiv1 \pmod{a^2}$?

Are there solutions in integers $a,b>1$ to the following simultaneous congruences? $$ a^4\equiv 1 \pmod{b^2} \quad \mathrm{and} \quad b^4\equiv1 \pmod{a^2} $$ A brute-force search didn't turn up ...
7
votes
1answer
367 views

Ramanujan-Nagell Theorem Proof Question

I'm currently working through Stewart and Tall's Algebraic Number Theory. In particular, section 4.9 of this book provides a proof of the Ramanujan-Nagell Theorem, which states that the only integer ...
1
vote
1answer
18 views

Number of coins using Diophantine equation

I'd like to solve this question using Diophantine equations: We have an unknown number of coins. If you make 77 strings of them, you are 50 coins short; but if you make 78 strings, it is exact. ...
3
votes
5answers
69 views

If $a,b,c$ are integers such that $4a^3+2b^3+c^3=6abc$, is $a=b=c=0$?

If $a,b,c$ are integers such that $4a^3+2b^3+c^3=6abc$ , then is it true that $a=b=c=0$ ? I was thinking of infinite descent but can't actually proceed , please help. Thanks in advance
11
votes
1answer
280 views

How to get Fermat's descent working on the conic $x^4+y^4=2z^2$?

Fermat solved the Diophantine equation $(x^2)^2 + (y^2)^2 = z^2$ using descent, the key step was using the Pythagorean triples: $x^2 = u^2 - v^2$ $y^2 = 2 u v$ $z = u^2 + v^2$ but then it is seen ...
1
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0answers
69 views

Can Fermat's descent for $x^4+y^4=z^2$ be interpreted on a conic?

Fermat proved the Diophantine equation $$(x^2)^2 + (y^2)^2 = z^2$$ has only solutions $(0,0,0)$, $(0,\pm 1,\pm 1)$ and $(\pm 1,0,\pm 1)$ using "infinite descent". The conic $C : X^2 + Y^2 - 1$ has a ...