Questions on finding integer/rational solutions of equations.

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30
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455 views

On Ramanujan's curious equality for $\sqrt{2\,(1-3^{-2})(1-7^{-2})(1-11^{-2})\cdots} $

In Ramanujan's Notebooks, Vol IV, p.20, there is the rather curious, $$\sqrt{2\,\Big(1-\frac{1}{3^2}\Big) \Big(1-\frac{1}{7^2}\Big)\Big(1-\frac{1}{11^2}\Big)\Big(1-\frac{1}{19^2}\Big)} = ...
27
votes
0answers
669 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 ...
18
votes
0answers
753 views

Are there unique solutions for $n=\sum_{j=1}^{g(k)} a_j^k$?

Edward Waring, asks whether for every natural number $n$ there exists an associated positive integer s such that every natural number is the sum of at most $s$ $k$th powers of natural numbers ...
12
votes
0answers
125 views

Congruence properties of $x_1^6+x_2^6+x_3^6+x_4^6+x_5^6 = z^6$?

It is known that given primitive (co-prime) integer solutions to, $$x_1^4+x_2^4+x_3^4+x_4^4 = z^4$$ then there is one $x_i$ such that $z^4-x_i^4$ is divisible by $d_4=5^4$. Additionally, Ward ...
11
votes
0answers
484 views

Find All $x$ values where $f(x)$ is Perfect Square

Is there a formula, method or anyway to find all positive $x$ integer values (if exists) such that $f(x)$ is Perfect square where $f(x)$ is a quadratic equation? For example if I have the following ...
11
votes
0answers
387 views

If $p$, $q$ are naturals, solve $p^3-q^5=(p+q)^2$.

In If $p,q$ are prime, solve $p^3-q^5=(p+q)^2$., the author asks to solve the equation $p^3-q^5=(p+q)^2$ for primes $p$ and $q$. A proof is given that $p=7, q=3$ is the only solution. In this ...
10
votes
0answers
192 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} ...
10
votes
0answers
855 views

How many integer solutions to a diophantine equation

Starting with the equation: $\frac{1}{a}+\frac{1}{b}=\frac{p}{10^n}$, I reached the equation: $10^{n-log(p)} = \frac{ab}{a+b}$. Now given the positive integer $n$, for what integer values of $p$ ...
9
votes
0answers
70 views

$2n^2-\lfloor m^b\rfloor=k$ has only finitely many integer solutions

Let $k$ be an nonzero integer and $b>2$ a real. Is it true that there exist only finitely many positive integer pairs $(n,m)$ for which $$ 2n^2-\lfloor m^b\rfloor=k? $$ I don't know the answer, ...
9
votes
0answers
217 views

Number of ways to express a binary number in a certain way

So I'm working on a problem where I get to a point where I have to count the number of solutions to an equation or at least find a decent upper bound to be used in an estimate I need later. The ...
9
votes
0answers
266 views

Coefficients in expansion of $(\sqrt[3]{2} - 1)^m$

In trying to solve $a^3 - 2b^3 = 1$ over the integers I came across the need to answer the question: when does $(1+ \sqrt[3]{2} + \sqrt[3]{2}^2)^n$ have no $\sqrt[3]{2}^2$ term in it's expansion (in ...
7
votes
0answers
134 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 ...
7
votes
0answers
102 views

About solutions to $x^2+y^2+z^2=8n+3,n\in \mathbb N$

As we know that $x^2+y^2+z^2=8n+3,(n\in \mathbb N)\tag{1}$ has integer solutions $x,y,z\in \mathbb N.$ If $k\in \mathbb N$ has at least one prime factor which is $\equiv 3 \mod 4,$ then we call $k$ ...
6
votes
0answers
136 views

Seemingly easy Diophantine equation $a^3+a+1=3^b$

How to prove that $a=b=1$ is the only positive integer solution to the following Diophantine equation?$$a^3+a+1=3^b$$
6
votes
0answers
113 views

Solving (n+1)(n+2)…(n+k)−k = x^2

Let $n$ and $k$ be positive integers. Need to find all pairs of $(n,k)$ such that $$(n+1)(n+2) \cdots (n+k)−k = x^2,$$ where $x^2$ is a perfect square.
6
votes
0answers
137 views

Can a sum of three fifth power of integers be 8?

By congruence computation we get that $n= a^5+b^5+c^5$ implies $n \not \equiv 4,5,6,7 \pmod{11} $ (with $a,b,c \in \mathbb{Z}$) For $a,b,c \in \{-100,-99, \dots , 99, 100\}$, the set of integers ...
6
votes
0answers
204 views

What is stopping every Mordell equation from having a [truly] elementary proof?

The Mordell equation is the Diophantine equation $$Y^2 = X^3-k \tag{1}$$ where $k$ is a given integer. There is no known single method — elementary or otherwise — to solve equation $(1)$ for all $k$, ...
6
votes
0answers
77 views

To solve for $x,y,n$ in non-negative integers , $\dfrac{x!+y!}{n!}=p^n$ , $p$ a given prime

Let $p$ be a given prime , then how do we find non-negative integers $(x,y,n)$ $\space$ , such that $\dfrac{x!+y!}{n!}=p^n$ ?
6
votes
0answers
159 views

Twin Prime Powers

What are all the possible triplets of numbers $a$, $b$, $c$ such that $a+2=b$, $a+4=c$, and all $3$ are prime powers (where one must be a power of $3$)? I'm aware of the cases for when they are ...
5
votes
0answers
95 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 ...
5
votes
0answers
96 views

Describe the integral solutions to this cubic equation.

Consider the following cubic equation in $c$: $c^3 - 3c^2(a+b) + 3c(a+b) -3ab(a+b)-3=0$ Does this equation have infinitely many integer solutions $(a,b,c)$ ? EDIT: My attempt was rerwriting it as a ...
5
votes
0answers
98 views

On the fourth power $2^4 + 15045^4 + 26870^4 + 34090^4 = 37239^4$

The Diophantine equation, $$x_1^4+x_2^4+x_3^4+x_4^4 = z^4\tag1$$ can be solved with $x_1 = 0$ (Elkies), or as the title shows, with $x_1 = 2$ (Wroblewski). See link here. Q: Can $(1)$ be solved in ...
5
votes
0answers
123 views

The smallest non-zero integer $c$ such that $\sum\limits_{n=1}^m\pm(x+n)^6 = c$?

We have the neat equalities, I. Group 1 For $k=2,3,4,5,\dots$ $$\sum_{n=1}^{2^k}\epsilon_n(x+n)^k = 2^{\frac{k(k-1)}{2}}k! = 4,\;48,\;1536,\;\color{brown}{122880},\dots$$ for appropriate ...
5
votes
0answers
81 views

Probability of another 3 integers with same sum and product as the first 3 integers

Let us suppose $3$ integers are selected at random from a large range, say $$-1000\leq x\leq y\leq z\leq 1000$$ Now, we define the sum and product: $$\begin{align*}s&=x+y+z ...
5
votes
0answers
175 views

Diophantine: $x^3+5=y^5$

Find all integers $x$ and $y$ such that $x^3+5=y^5$. I found this after solving the equation $3^a+5=2^b$. For this equation, since $(a,b)=(3,5)$ is a solution, it is possible to write it as ...
5
votes
0answers
133 views

Binomial triplets

Solutions to the equation $$\dbinom{a}{n}+\dbinom{b}{n}=\dbinom{c}{n}$$ I will refer to as 'Binomial triplets of order $n$'. These triplets describe simplicial $n$-polytopic numbers that can be ...
5
votes
0answers
219 views

$(b-a)^2-2ab$ is a perfect square.

I'm in need of some help if possible, about a formula, theorems, old works, ideas, or even an existing solution are welcome. The problem is that i have two distinct natural numbers as $b > a > ...
5
votes
0answers
159 views

Generalizing Ramanujan's 6-10-8 Identity

Let $ad=bc$. Then Ramanujan's 6-10-8 Identity is the bizarre, $$\small 64[(a+b+c)^6+(b+c+d)^6-(c+d+a)^6-(d+a+b)^6+(a-d)^6-(b-c)^6]\\ \small ...
5
votes
0answers
132 views

Is there always a telescopic series associated with a rational number?

Here is something I thought up while I was bored and my, erm, fish were busy: Given a rational number $p\in(0,1)$, are there always positive integers $n$, $c_m$, and $w_m$ such that ...
5
votes
0answers
116 views

Approach to elliptic curve $y^2=x^3+1/4+p/a^2$

While taking a brute-force look at this question I discovered that it seems that almost every prime (I'll conjecture every prime larger than 20627) can be written as $p=w^2+wc+d$ for $w,c,d\in ...
4
votes
0answers
61 views

Original proof of Ljunngren's equation

The equation $$x^2=2y^4-1$$ was studied and solved by Ljunngren, who showed that 1,1 and 293,13 are the only integer solutions.However, his proof was very difficult and L.J.Mordell thought there must ...
4
votes
0answers
63 views

Classifying Diophantine Equations

Take a given Diophantine equation. Chances are, we can't find any solutions. But if it's an equation of a certain form, we may get lucky and may be able not only to find a solution, but be able to ...
4
votes
0answers
99 views

Find all $x,y\in \mathbb{N}$ such that: $2^x+17=y^2$.

Find all $x,y\in \mathbb{N}$ such that: $2^x+17=y^2$. Its easy to find that $x=6$ is the only even value for $x$, the others have to be odd. One more thing is that we get $y^2 \equiv 19 \pmod ...
4
votes
0answers
64 views

Systems of linear modular equations with unknowns in the moduli

I am interested in systems of linear modular equations, where the unknowns also appear in the moduli. The general form would be: $A \vec{x}= \vec{b} \;\textrm{mod} \; (C \vec{x}+\vec{d})$ where A ...
4
votes
0answers
131 views

Diophantine equation: $13^x+3=y^2$

$x,y$ are positive integers. $$13^x+3=y^2\iff \left(4+\sqrt{3}\right)^x\left(4-\sqrt{3}\right)^x=\left(y+\sqrt3\right)\left(y-\sqrt3\right)$$ $\gcd\left(y+\sqrt3, y-\sqrt3\right)=1$, therefore ...
4
votes
0answers
68 views

Structure of first-coordinate-projection of set of solutions of “elliptic” diophantine equation $xy(6-(x+y))=6$

Say that a rational number $a$ is good iff there is a rational number $b$ with $ab(6-a-b)=6$, or equivalently iff $a^4 - 12a^3 + 36a^2 - 24a$ is the square of a rational number. Denote by $G$ the set ...
4
votes
0answers
121 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 ...
4
votes
0answers
72 views

A question on the Pell equation $x^2-pqy^2 = -1$, with prime $p,q$.

We know that a necessary but not sufficient condition such that, $$x^2-dy^2 = -1\tag1$$ is solvable is that $d$ is not divisible by a prime of form $4m+3$. It is not sufficient because the prime ...
4
votes
0answers
82 views

Solving quadratic diophantine equations in two variables

I've looked at the recommended questions, but none of them seem to match my question. Consider the equation $2015 = \frac{(x+y)(x+y-1)}{2} - y + 1$. This can trivially be simplified to $4030 = x^2 + ...
4
votes
0answers
162 views

Diophantine equation: $7^x=3^y-2$

I've tried using mods but nothing is working on this one: solve in positive integers $x,y$ the diophantine equation $7^x=3^y-2$.
4
votes
0answers
120 views

Number of Solutions to a Diophantine Equation

I am asked the following: Show that the number of integer solutions to $y^p=x^2+2$ for any odd prime $p$ is at most $p-1$. I checked that for $y^p=x^2+2$, the same method for $y^3=x^2+2$ works ...
4
votes
0answers
161 views

Modified Pell equation: $x^2-D y^2 = m$, $m\neq1$.

How does one solve the Diophantine equation $$ x^2-Dy^2=m, $$ where $m$ is some fixed arbitrary integer? I understand that given the fundamental solution to $r^2-D s^2=1$, and any solution to the ...
4
votes
0answers
185 views

How many natural numbers $x, y$ are possible if $(x - y)^2 = \frac{4xy}{(x + y - 1)}$.

How many natural numbers $x$, $y$ are possible if $(x - y)^2 = \frac{4xy}{x + y - 1}$. Does this system has infinite solutions which can be generalized for some integer $k \geq 2?$ $(x - y)^2(x + y) ...
3
votes
0answers
16 views

Diophantine equation $10^x=yzwt-3$

I have resolved, brute force, the following problem someone asked me: Solve the Diophantine equation $$10^x=yzwt-3\space \text{where}\space \space y,z,w,t \space \text {are distinct primes}$$ $$ ...
3
votes
0answers
55 views

Diophantine equation involving factorials

$$x!+y=y^3$$ $$y=\sqrt[3]{x!+\sqrt[3]{x!+\sqrt[3]{x!+\cdots}}}$$ The only integer solutions to these identities that I have found are: $$3!+2=2^3$$ $$4!+3=3^3$$ $$5!+5=5^3$$ $$6!+9=9^3$$ I ...
3
votes
0answers
121 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 ...
3
votes
0answers
87 views

Symmetry and trivial solutions to Pell equations

Below is a representation of the solutions to the equation $x^2-Dy^2=1$ for $6(6-1)\leq D \leq 6(6+1)$: \begin{array}{c} & 30 & 31 & 32 & 33 & 34 & 35 & 36 & 37 & ...
3
votes
0answers
24 views

How many quadruples are there?

How many quadruples are there $(x_1,x_2,x_3,x_4) \in \mathbb{Z}^{+}_{0}$ such that $(x_1+x_2)(2x_2+2x_3+x_4)=95$? My attempt. We have that $x_1+x_2 = 19$ and $2x_2+2x_3+x_4 = 5$ or $x_1+x_2 = 5$ and ...
3
votes
0answers
88 views

How can I find values for which a given expression gives a perfect square?

There have been several posts on this topic on math.se, such as this one with the same title. However all the posts I found contained coefficients to $x^2$, that were perfect squares. I am looking for ...
3
votes
0answers
51 views

Can the determinant of an integer matrix with $k$ given rows be the gcd of the determinants of the $k\times k$ minors of those rows?

I'm interested if the following is true: Let $n\geq k\geq1$ be integers, let $A\in\mathbb Z^{k\times n}$ and denote the $\binom nk$ $k\times k$ minors of $A$ by $A_1,\ldots,A_N$. Then the ...