1
vote
2answers
197 views

Solutions to the Mordell Equation modulo $p$

It is well known that the Mordell Equation $x^2 = y^3 + k$ has finitely many solutions, but has solutions modulo $n$ for all $n$. One proof of this involves using the Weil Bound to show that $x^2 = ...
2
votes
2answers
86 views

A transcendental number from the diophantine equation $x+2y+3z=n$

Let $\displaystyle n=1,2,3,\cdots.$ We denote by $D_n$ the number of non-negative integer solutions of the diophantine equation $$x+2y+3z=n$$ Prove that $$ \sum_{n=0}^{\infty} ...
-2
votes
3answers
149 views

Equation $a^{n}+b^{n}=2008$ has no integers solutions. [on hold]

Prove that the equation $a^{n}+b^{n}=2008$ has no solutions for $a,b,n\in\mathbb{Z}, n\geq2.$
0
votes
0answers
21 views

No of unique values [on hold]

How to find the number of uniques values AX+BY can take under various scenarios Like (A,B) being co-prime or GCD(A,B)=C,with the imposed conditions like X,Y being integers from the set (E,F).
7
votes
2answers
256 views

Is $7^{8}+8^{9}+9^{7}+1$ a prime? (no computer usage allowed)

Prove or disprove that $$7^{8}+8^{9}+9^{7}+1$$ is a prime number, without using a computer. I tried to transform $n^{n+1}+(n+1)^{n+2}+(n+2)^{n}+1$, unsuccessfully, no useful conclusion.
1
vote
2answers
31 views

Rational Number of a given fraction

Find all rational numbers $\frac pq$ such that $\frac pq=\frac {p^2 +30}{q^2 +30}$. How can I go about it. If I substitute p and q by real values $\frac pq$ gets innumerable rational numbers
2
votes
1answer
90 views

Solutions to a diophantine equation

I tried to find integer solutions to the following diophantine equation $$x^3 - 3y^3 + 5z^3 - 3xy^2 + 3x^2y + 9xz^2 + 7x^2z + 3yz^2 - 3y^2z + xyz = 0$$ but was unable to do so. I suspect that there ...
0
votes
0answers
29 views

Differential Diophantine Equations?

So this is both a question on its own as well as a request for where I can find information on a given topic. Consider Differential Equations in two variables of the form: $$P(Z,Z', Z'' ... Z^{[n]}, ...
1
vote
3answers
90 views

Solving $a^2+3b^2=c^2$

I'm looking for how to solve the equation $a^2+3b^2=c^2$ where $a,b,c$ are integers and $b$ is even, I'm looking for the algorithm used to solve this kind of equations, not just the solution. Regards ...
5
votes
0answers
128 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 > ...
0
votes
2answers
41 views

$A^7 \not\equiv A(\mod 13) \Rightarrow A^{78} + 1 \equiv 0 (\mod 169)$

Let variable $A$ is integer and $A^7 \not\equiv A(\mod 13)$. Prove that $A^{78} + 1 \equiv 0 (\mod 169)$ Could someone explain, how to solve this type of problems? Any help would be greatly ...
13
votes
2answers
212 views

Integer values of $\frac{x}{y}+\frac{y}{z}+\frac{z}{x}$?

What are the possible integer values of $$\frac{x}{y}+\frac{y}{z}+\frac{z}{x}$$ where $x$, $y$, and $z$ are positive integers? My suspicion is the the only integer values are $3$ and $5$, the former ...
3
votes
0answers
70 views

Does the special Pell equation $X^2-dY^2=Z^2$ have a simple general parameterization?

In Carmichael's Diophantine Analysis ($\S8$), he notes that the equation $$X^2-dY^2=Z^2 \qquad(\dagger)$$ has a two-parameter solution $$x=m^2+dn^2, \quad y=2mn, \quad z=m^2-dn^2. \qquad(\star)$$ He ...
3
votes
1answer
55 views

For what positive integers $p$ and $q$: $(p+1)!+(q+1)!=(pq)^2$

I tried this problem using brute force and got the answers as $(3,4)$ and $(4,3)$,but is there a way to solve this question?
2
votes
2answers
49 views

Solving $a^2-24 a=k^2$ across the integers

$$a^2-24 a=k^2$$ I was trying to solve another problem that essentially boiled down to finding all possible integer values of a such that $a^2-24 a$ was a perfect square, and I thus came up with the ...
0
votes
3answers
59 views

Prove the solutions of following equation exists [closed]

$x^2+y^2=z^n$ has a solution in $\mathbb{N}$, for all $n \in \mathbb{N}$. The problem is to show that for every natural number n there exists 3 integers which show the above relation for ...
3
votes
3answers
89 views

Solve the following equation in positive integers $x$ and $y$

What are the solutions in positive integers of the equation: $${1+2^x+2^{2x+1}=y^2}$$ I tried to factorize the equation but it didn't help much. Clearly $y $ is an odd integer. Substituting $y ...
0
votes
3answers
38 views

Equation involving power of two

I want to show that the equation $2^x - 1 = 3^y$ does not have any positive integer solutions except for $ x = 2 , y = 1$ . Is it possible to prove the assertion using binary representation of powers ...
2
votes
1answer
45 views

Integer solutions to equations of the form $a^n+b^n+\cdots=c^n$

I shall refer to the number of terms on the left side of the equation as $m$. Suppose that all numbers in the equation are positive integers. I am wondering if anything is known about for which ...
0
votes
2answers
32 views

A diophantine question about squares

I have been trying to solve the following problem: Classify triples of integers $(m,n,k)$ satisfying the following equation $2mn+m+n=k^{2}$. It is very easy to obtain some solutions. However, I am ...
0
votes
2answers
33 views

Find all solutions in positive integers of the Diophantine equation (proof explanation)

An example problem in my textbook asks: Find all solutions in positive integers of the diophantine equation $x^2 + 2y^2 = z^2$. The provided proof appears as follows: $2y^2 = z^2 - x^2 = (z - ...
1
vote
3answers
54 views

Diophantine equations problem/exercise 3

Find all the pythagorean tripples (x,y,z) with x=40. Well I started with the known formulas for the pythagorean tripples but got me nowhere. Or I was not able continue the thought process required. I ...
0
votes
1answer
49 views

Diophantine equation exercise [duplicate]

Prove that the diophantine equation $x^4-2(y^2)=1$ has only 2 solutions. Any hint on how to start and what to do .. I do not have a lot of experience on non linear diophantine equations and do not ...
2
votes
1answer
68 views

Find all integers n such that n−2014 and n+ 2014 are both triangular numbers.

I came across this problem when searching for triangular numbers questions. I know that I need to use the equation, $$\frac {n(n+1)}{2} $$ but I don't know how to apply it to this problem.
2
votes
1answer
97 views

Number Theory Question: $x^2-33y^3=10$ no solutions

I've been struggling to get my head around this for a while! Show that: $x^2 - 33y^3 = 10$ has no integral solutions
1
vote
2answers
56 views

Solving a problem with a diophantine equation without trial and error.

I have the following problem: A teacher bought toys for the students of an academy, every toy for a boys costs $290$ and every toy for a girl costs $330$. If he spends $24300$, how many of each ...
3
votes
3answers
78 views

Divisor of $3^{2n+1}+61$

I have difficulty to show the following: If $p$ is a prime and $p^2$ divides $3^{2n+1}+61$, then $p$ must be $2$. I appreciate any help.
4
votes
1answer
109 views

Positive integer solutions to $x^2+y^2+x+y+1=xyz$

The question asks for positive integer solutions to $x^2+y^2+x+y+1=xyz$ . We at first note that $x|y^2+y+1$. Now,let there exist positive integers $x,y$ that satisfy the given equation.Then ...
1
vote
1answer
42 views

Solving diophatine equation of form $x^2+y^2=25$

How would you solve diophatine equations of the form $x^2+y^2=25$? I know how to solve linear diophatine equations but I have not done any of quadratic form before. I could use trial and error because ...
1
vote
1answer
59 views

Solve a system of diophantine equations

I have a problem with elemental number theory. I started with the expression $$ (a - \frac{1}{b})(b - \frac{1}{c})(c - \frac{1}{a}) $$ and task to find all natural $a,b,c$ so that the result of the ...
8
votes
2answers
284 views

diophantine equation $ |x^2-py^2|=\frac{p-1}{2} $

Prime $p\equiv3\pmod4$, then diophantine equation $$ |x^2-py^2|=\frac{p-1}{2} $$ has a solution in integers en, $x^2-py^2=-1$ has no solution in integers. I'd be grateful for any help you are ...
0
votes
1answer
41 views

Diophantine equation with condition

The question is to find the general solution in integers $x,y,z$ to $$2x+3y+5z=7$$ where none of $x,y$ or $z$ are divisible by $7$. Without the divisible by $7$ condition I found that the general ...
1
vote
1answer
21 views

General form of Bezout numbers

Bézout's lemma can be generalized to $n$ co-prime integers $a_1, \dots a_n$ : there exists integers $x_1, \dots, x_n$ such that $$a_1 x_1 + \dots + a_n x_n = 1$$ For the case $n = 2$, one can show ...
0
votes
5answers
63 views

Finding $7$ inverse modulo $11$

I'm trying to find the inverse of $7$ modulo $11$. From what I understand, the steps are: \begin{align} &11 = 1(7) + 3 \\ &7 = 2(3) + 1 \\ \end{align} From here, you work backwards ...
0
votes
3answers
66 views

Solving the congruence $7x \equiv 41 \mod{13}$

I have to solve the following linear congruence: $$7x \equiv 41 \mod{13}$$ The question where I got this from comes in two parts. The first is that it asks to find the set of the inverses of $7 ...
1
vote
1answer
43 views

Linear diophantine equation word problem

I have the following word problem: A small clothing manufacturer produces two styles of sweaters: cardigan and pullover. She sells cardigans for $\$31$ each and pullovers for $\$28$ each. If her ...
0
votes
0answers
22 views

Find the general solution to diophantine equation $-221x + 187y - 493 = 0$

I have to find the general solution to $$-221x + 187y - 493 = 0$$ The main issue, I'm figuring out if I have found the general solution or not. Below, are my steps: The $\gcd{(-221,187)} = 17$ and ...
0
votes
1answer
43 views

What is the best way of solving for $x,y\in\mathbb N$ given the conditions $\begin{cases}x\mid y+a\\y\mid x + b\end{cases}$?

What is the best way of solving for $x,y\in\mathbb N$ given the conditions $\begin{cases}x\mid y+a\\y\mid x + b\end{cases}$? The letters $a,b\in\mathbb N$ denote constant known numbers. The ...
0
votes
0answers
25 views

prove that $ax+by=b+c$ has solution in $\mathbb Z$ if and only if $ax+by=c$ has solution in $\mathbb Z$

Let $a,b,c,d\in \mathbb Z$ prove that: a)$ax+by=b+c$ has solution in $\mathbb Z$ if and only if $ax+by=c$ has solution in $\mathbb Z$ b)$ax+by=c$ has solution in $\mathbb Z$ if and only if ...
0
votes
2answers
78 views

Solve and explain diophantine equation

A Diophantine equation ax+by = c always has a solution whenever a and b are relatively prime. Find x ,y such that $$93x-81y=3 $$
1
vote
1answer
34 views

Is every unit vector in $\mathbb{Q}^n$ the first column of a rational orthogonal matrix?

Equivalently, does every unit vector in $\mathbb{Q}^n$ belong to some orthonormal basis for $\mathbb{Q}^n$? This is clearly true for $\mathbb{Q}^2$, and for $\mathbb{Q}^3$ it seems to be true for ...
1
vote
2answers
99 views

Integer solutions of $800000007 = x^2+y^2+z^2$

Prove that the equation, $800000007 = x^2+y^2+z^2$ has no solutions in integers.(That is $8$ followed by $7$ zeroes, with a $7$ at the end). I tried checking modulo $3$, $5$, $7$, and $10$, but ...
1
vote
1answer
35 views

On representing the general solution for the diophantine equation $a_1x_1+\dotsb+a_nx_n=c$

On representing the general solution with the special solutions for the diophantine equation $$a_1x_1+a_2x_2+\dotsb+a_nx_n=c$$ here $a_1 ,a_2, \dotsb,a_n,c\in\Bbb Z,(a_1 ,a_2, \dotsb,a_n)=1$. Can ...
2
votes
3answers
103 views

Show that $0 = 2a^3-5ab^2+25b^3$ has no other integer solutions than $a = b = 0$.

I am trying to solve the following problem: I have the equation $0 = 2a^3-5ab^2+25b^3$, where $a,b \in \mathbb Z$. Obviously, $a = b = 0$ is a solution of this equation. But how can I show that there ...
5
votes
2answers
132 views

Integers can be expressed as $a^3+b^3+c^3-3abc$

$$S=\{a^3+b^3+c^3-3abc|a,b,c\in\Bbb Z\}$$ Can we decide $S$? that is, we want to find all integers of the form $a^3+b^3+c^3-3abc$. obviously, if $m,n\in S$, then $mn\in S$, so we only need to ...
3
votes
3answers
67 views

Solving for $3^x - 1 = 2^y$

Besides $x=2, y=3$, are there any other solutions? I know that if there is another solution: $y$ is odd since $2^y \equiv -1 \pmod 3$ $x$ is even since $3^x - 1 \equiv 0 \pmod 8$ $3 | y$ since $-1 ...
3
votes
1answer
43 views

Finding all possible values

we have to find all possible prime values $(p,q,r)$ such that $ pq = r + 1 $ $ 2(p^2+q^2) = r^2 + 1 $ I do not know how to start looking for an answer.
1
vote
3answers
70 views

Can we find positive integers $a$ and $k \geq 2$ with $2^n - 1 = a^k$?

I would appreciate if somebody could help me with the following problem: For a given positive integer $n$, can we find positive integers $a$ and $k$ ($k\geq 2$) such that $2^n-1=a^k$? The ...
0
votes
1answer
96 views

Prove that $2^x \cdot 3^y - 5^z \cdot 7^w = 1$ has no solutions

Prove that $2^x \cdot 3^y - 5^z \cdot 7^w = 1$ has no solutions in $\mathbb{Z}^+$, if $y\ge 3$.
4
votes
3answers
81 views

Four integers that satisfy $a+b+c+d\; =\; -3$ and $a^{3}+b^{3}+c^{3}+d^{3}\; =\; 3$

Find a set of 4 integers that satisfy $$a+b+c+d\; =\; -3$$ and $$a^{3}+b^{3}+c^{3}+d^{3}\; =\; 3$$ I am really not sure how to proceed. I tried letting $d = -c$ to see if that would yield a ...