Questions on finding integer/rational solutions of polynomial equations.

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1
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3answers
196 views

Number of solutions for $\frac{1}{X} + \frac{1}{Y} = \frac{1}{N!}$ where $1 \leq N \leq 10^6$

Note: this is a programming challenge at this site For this equation $$\frac{1}{X} + \frac{1}{Y} = \frac{1}{N!}\quad ( N \text{ factorial} ),$$ find the number of positive integral solutions for ...
2
votes
1answer
136 views

Prove that there are no natural numbers, $i, j$ such that $ 3i^2+3i+7=j^3$

I'm not sure if this is true but, I've tried with many different values of $i, j$ and didn't get any contradictions. The question again, here Prove that there are no natural numbers, $i, j$ such ...
2
votes
1answer
299 views

Diophantine equations in positive integer solutions

I want to know the solution of the equation $x^3$ + $y^3$ = $31z^3$ in integers. I know the fundamental solution ($137, -65$, $42$), but want to have all the values positive. I know also that there is ...
2
votes
2answers
4k views

How many integer solutions to a linear combination, with restrictions?

I've already done a few problems such as this, other problems where I'm supposed to find the number of combinations or permutations, subject to certain restrictions. Here's been my basic strategy: ...
1
vote
4answers
182 views

Solve $2a + 5b = 20$

Is this equation solvable? It seems like you should be able to get a right number! If this is solvable can you tell me step by step on how you solved it. $$\begin{align} {2a + 5b} & = {20} ...
3
votes
2answers
128 views

solutions of $a^2+b^2=c^2$

I am trying to figure the following out. If you have $a^2+b^2=c^2$ and let $x=a/c$ and $y=b/c$ how can you show that $x=\frac{m^2-n^2}{m^2+n^2}$ and $y=\frac{2mn}{m^2+n^2}$ for some relatively ...
13
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5answers
879 views

How to find solutions of $x^2-3y^2=-2$?

According to MathWorld, Pentagonal Triangular Number: A number which is simultaneously a pentagonal number $P_n$ and triangular number $T_m$. Such numbers exist when ...
1
vote
1answer
266 views

integer solutions of an equations

Now I came with an equation to find the solutions in integers. Not aonly that, I would like to know other types of solutions (if exists). Find the solutions and method of solving the equation $p^3 - ...
6
votes
3answers
445 views

The positive integer solutions for $2^a+3^b=5^c$

What are the positive integer solutions to the equation $$2^a + 3^b = 5^c$$ Of course $(1,\space 1, \space 1)$ is a solution.
8
votes
3answers
762 views

Integer Solutions to $x^2+y^2=5z^2$

I'm looking for a formula to generate all solutions $x$, $y$, $z$ for $x^2 + y^2 = 5z^2$. Any advice?
3
votes
1answer
173 views

Diophantine equation: fermat numbers and fibonacci numbers

My question is how to find all solutions $(m,n)\in\mathbb N^2$ for $F_n=f_m$, where $F_n=2^{2^n}+1$ and $f_m$ is the $m$th fibonacci number: $f_0=0$, $f_1=1$ and $f_n+f_{n+1}=f_{n+2}$ for each ...
4
votes
4answers
200 views

a theorem of Fermat

While I was surfing the web, searching things about math, I read something about a particular theorem of Fermat. It said: let $a$ and $b$ be rational. Then $a^4-b^4$ cannot equal the square of a ...
21
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6answers
3k views

Integral solution for $|x | + | y | + | z | = 10$

How can I find the number of integral solution to the equation $|x | + | y | + | z | = 10.$ I am using the formula, Number of integral solutions for $|x| +|y| +|z| = p$ is $(4P^2) +2 $, So the ...
5
votes
3answers
1k views

Show that the equation $y^2 = x^3 + 7$ has no integral solutions.

Show that the equation $y^2 = x^3 + 7$ has no integral solutions.
5
votes
2answers
220 views

Are Euclid numbers squarefree?

Are Euclid numbers squarefree ? An Euclid number is by definition a Primorial number + 1. See http://mathworld.wolfram.com/Primorial.html. In notation the $n$ th Euclid number is written as $E_n = ...
8
votes
3answers
229 views

$a+b=c \times d$ and $a\times b = c + d$

There is a 'nice' relationship between the integers (1,5) and (2,3) as $$1+5=2 \times 3;$$ $$1\times 5 = 2 + 3.$$ So I tried to find all positive integers pairs $(a, b)$ and $(c, d)$ such that ...
1
vote
1answer
122 views

Count the number of integer solution to $\sum_{i=1}^ {2}{a_i\times b_i} \geq 6 $ [closed]

Count the number of integer solution to $\sum_{i=1}^ {2}{a_i\times b_i} \geq 6 $ such that condition 1: $1 \leq a_i \leq 7$ condition 2: $1 \leq b_i \leq 4$ condition 3: $\sum_{i=1}^{2} {a_i} = 8$ ...
7
votes
2answers
309 views

Find the possible values of $a$, $b$ and $c$?

Given $(a,\space b,\space c)\in \mathbb Z^3$ and that $$\sqrt[3]{\sqrt{a}+\sqrt{b}} + \sqrt[3]{\sqrt{a}-\sqrt{b}} = c$$ Find the possible values of $a $, $b$, and $c$.
1
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1answer
286 views

Recursive solution to a Diophantine equation

I'd like to find a recursive formula giving positive integer solutions to this Diophantine equation $$5L^2 - a^2 - 1 =0$$ It can be seen that I need $5L^2 - 1$ to be a square of a number $\in \mathbb ...
0
votes
1answer
106 views

Solutions of $a^2 = b^d -3^c$

The solutions of $a^2 = b^d -3^c$ are in the form $(a, b, c, d) = ((46)27^t, (13)9^t, 6t+4, 3)$. This is done by using calculator. As per my calculator, I have checked some terms, which are satisfied ...
26
votes
5answers
668 views

When is $1^5 + 2^5 + \ldots + n^5$ a square?

When is $1^5 + 2^5 + \ldots + n^5$ a square? I found that this happens sometimes: $n=13$ gives $1001^2$, $n=133$ gives $9712992^2$ and $n=1321$ gives $942162299^2$. I feel that the ...
0
votes
2answers
117 views

Sylvester Theorem

Bonjour, The equation $\binom{n}{k}=m^l$ has no entire solution for l$\ge$2 and 4$\le$k$\le$n-4. Suppose that n$\ge$2k (since $\binom{n}{k}=\binom{n}{n-k}$). According to the Sylvester theorem, the ...
8
votes
2answers
166 views

Solve $x^{y^2}=y^x$ which $x,y\in\mathbb{N}$.

Solve $x^{y^2}=y^x$ which $x,y\in\mathbb{N}$. I can observe that $(x,y)$ can be $(1,1)$, then I don't know how to carry on. Please help. Thank you. p.s. I wonder if there's solution without ...
0
votes
1answer
94 views

Show the number of solutions

Show that the number of solutions of $x^2+y^2=m$, where $m=2^{\alpha}r$ and $r$ is odd, is given by $U(m)=4\sum_{u|r}(-1)^{\frac{u-1}{2}}=4\gamma(m)$, where $\gamma(m)$ denotes the number of positive ...
13
votes
4answers
601 views

When is $\left\lfloor \frac {7^n}{2^n} \right\rfloor \bmod {2^n} \ne 0\;$?

Is $$\left\lfloor \frac {7^n}{2^n} \right\rfloor \bmod{2^n} \ne 0\;$$ always true when $n \ge 3$. Baker's theorem on transcendental numbers that provide bounds for diophantine equations may be ...
6
votes
2answers
474 views

Diophantine equation (use class ideal group to solve)

Use ideal class group to find all integer solutions to the equation $$x^3=y^2+200$$ My approach: Observe that $\mathbb{Z}[\sqrt-2]$ is the field of integers in the ring $\mathbb{Q}(\sqrt -2).$ ...
1
vote
2answers
111 views

number of integral solutions for $x^2+y^2=5^k$

Prove that the equation $x^2+y^2=5^k$ has $4k+4$ integral solution. Any ideas would be appreciated. Thanks
0
votes
0answers
123 views

Pell type equation cum elliptic curve equation

I have seen this equation $y^3 - 3x^2 = p^m$ to determine the solutions. I know this is elliptic curve. I had some knowledge of elliptic curve. But, I was totally upset to determine the solutions of ...
1
vote
1answer
186 views

Pell equation of the special form

I have tried several times to solve the following equation and finally, I was failed to complete. Help me to find the solutions of Pell equation $y^2-2x^2 = p^m$, where $p$ is prime and $8|(p-1)$ or ...
0
votes
0answers
109 views

The equation $( m^2 + n^2 + q^2 )^2 = 36 ( u^2 + s^2 + t^2 )$

What is known about solutions in integers of the following equation ? $$( m^2 + n^2 + q^2 )^2 = 36 ( u^2 + s^2 + t^2 )$$ I am asking this because I just recently have got these: $$(a-b)^2 + ...
0
votes
1answer
32 views

Simultaneously smooth

I came across a problem recently which can be reduced to finding numbers $m$ such that $m$ and either $5m+1$ or $5m-1$ are $\{2,3,5\}$-smooth, i.e., of the form $2^a3^b5^c$ for nonnegative integers ...
0
votes
3answers
318 views

Are there any integer solutions to $a^3=b^2$?

I was wondering if there were any two integers $a$ and $b$ where $a^3=b^2$.
1
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1answer
158 views

Two Diophantine equations

What is known about solutions in integers of the equations; $x^4 - y^2 = z^6$ I got $x=4st(s^4 - t^4)$ , $z=4st(s^2 - t^2)$ , $y=(4st(s^2 - t^2))^2 (s^4 + t^4 - 6 (st)^2) $ and, the equation $x^2 - ...
20
votes
1answer
519 views

Integer solutions of $x! = y! + z!$

There was an interesting problem asked about triples $(x,y,z)$ which are solutions of $$x! = y! + z!.$$ Here $(2,1,1)$ is a solution because $2! = 1! + 1!$, as are $(2,1,0)$ and $(2,0,1)$. Now I ...
3
votes
1answer
72 views

Unique solution of a diophantine equation

Suppose $m_{1}^{h}+\cdots m_{k}^{h}=n_{1}^{h}+\cdots n_{k}^{h}$ for $h=1,\dots ,k$, where $0<m_{v}\leq q, 0<n_{v}\leq q, q$ positive integer. How do one show that the natural number $n_{v}$ must ...
2
votes
1answer
312 views

GCD to Linear Diophantine Equation without Euclid Algorithm

Is there a technique other than performing Euclid's algorithm in reverse that can elegantly show that if GCD$(a,b) = d$ then there exist integers $x$ and $y$ such that $ax + by = d$?
0
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2answers
198 views

Quadratic equations

Does anyone know how to find integer solutions of the quadratic equation $$y^2+y+z=f$$ where $z$ is a fixed odd prime or $1$ and $f$ is a fixed odd prime greater than $3$? This problem arose from ...
2
votes
1answer
331 views

How to find all solutions for Pell's equation $x^2 - Dy^2 = -1$ after the first $x_0$ and $y_0$?

How to find all solutions for Pell's equation $x^2 - Dy^2 = -1$ after the first $x_0$ and $y_0$? for example if we have $x^2 - 2 y^2 = -1$ then the smallest integer solution for $(x,y) = (1,1)$ How ...
0
votes
0answers
118 views

integer solutions to $a^m+nx^2 = y^n$ with various conditions

I consider the following equation with conditions of obtaining solutions $$a^m+nx^2 = y^n$$ This equation has solution when $a$ is an even prime and $x, y, m$ are positive integers with $(nx, y) = ...
26
votes
1answer
2k views

How to compute rational or integer points on elliptic curves

This is an attempt to get someone to write a canonical answer, as discussed in this meta thread. We often have people come to us asking for solutions to a diophantine equation which, after some clever ...
3
votes
3answers
151 views

What is the number of combinations of the solutions to $a+b+c=7$ in $\mathbb{N}$?

My professor gave me this problem: Find the number of combinations of the integer solutions to the equation $a+b+c=7$ using combinatorics. Thank you. UPDATE Positive solutions
7
votes
3answers
303 views

Finding the all integer solutions

How to Find the all integer solutions for: $$x+y+z=3$$ $$x^3+y^3+z^3=3$$
0
votes
1answer
194 views

What are all the solutions to ax + by = 0 with nonzero integer coefficients?

Suppose we have an equation of the form $ax+by=0$ with $a,b,c \in \mathbb{Z}$. For simplicity, $a \neq 0, b \neq 0$. Then, a single solution to this equation is $(x_0, y_0)=(-a, b)$. My book states ...
3
votes
0answers
101 views

how many natural numbers on a sphere

how many natural solutions are there to the following equation: $$ \sum_{i=0}^k x_i^2 = n$$ where $n,k \in\ \Bbb{N}$ i well like to get a answer for every n and k, but could do with just $k=2,3$.
3
votes
2answers
301 views

Modification of 5th question from BMO'81

First of all I will introduce original problem (Question 5 from British Mathematical Olympiad). You can find complete list of BMO'81 there BMO'81. Find, with proof, the smallest possible value ...
2
votes
2answers
496 views

How to find integer solutions for $x^3 - y^2 = 0 $?

How can I find integer solutions for $x^3 - y^2 = 0 $ ? In case that there are infinite number of solutions .How can we prove that ? and how to generate first few solutions ?
0
votes
1answer
396 views

How to find integer solutions for an ellipse equation?

How can I find the positive integer solutions to $x$ and $y$, given the integers $a$, $b$ and $c$ in the following ellipse equation in the form: $\frac{x^2}{a^2} + \frac{y^2}{b^2}=c$ For example, ...
4
votes
1answer
190 views

How to find odd solutions only for Pell's equation $x^2 - Dy^2 = 1$?

How can I find only the odd solutions for Pell's equation: $$x^2 - Dy^2 = 1$$ Specifically where $x$ is odd (but $y$ may be even or odd). Is there a way to generate the odd solutions to $x$, and can ...
3
votes
1answer
213 views

What is the sixth Martin quadruple $\sqrt[n]{x_1^k+x_2^k+x_3^k+x_4^k} =\text{Integer}$ for $k=1,2,3$?

Define a Martin quadruple {a,b,c,d} as a solution in non-zero integers to the system, $a+b+c+d = x^2$ $a^2+b^2+c^2+d^2 = y^2$ $a^3+b^3+c^3+d^3 = z^3$ It can be shown that there are an infinite ...
1
vote
0answers
65 views

Relation between b and c such that $ b^2 + c^2 + b^2 c^2$ is a perfect square [duplicate]

Possible Duplicate: On the equation $(a^2+1)(b^2+1)=c^2+1$ I came across a problem: What is relation between $b$ and $c$ such that $b^2 + c^2 + b^2 c^2$ is a perfect square? After trying ...