0
votes
1answer
35 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
46 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 ...
3
votes
0answers
62 views

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

Prime $p\equiv3\pmod4$, then the equation $$ |x^2-py^2|=\frac{p-1}{2} $$ has a solution in integers obviuosly, en, $x^2-py^2=-1$ has no solution in integers. Thanks a lot!
0
votes
1answer
37 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
20 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
58 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
48 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
23 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
19 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
40 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
24 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
67 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
29 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
96 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
33 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
87 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
120 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
61 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
32 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
65 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
94 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
79 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 ...
1
vote
1answer
59 views

What is quadratic equations in Algebra?

Yesterday someone asked a question in SE about indeterminate quadratic equations(of the form $x^2−ny^2=1$ which got me really interested in them and I thought I would try to learn something related to ...
4
votes
2answers
102 views

There is no Pythagorean triple in which the hypotenuse and one leg are the legs of another Pythagorean triple.

According to Wikipedia, There are no Pythagorean triples in which the hypotenuse and one leg are the legs of another Pythagorean triple. I cannot find the proof in the citation provided. I am ...
2
votes
4answers
97 views

Solving $6ab+a+b-36pq-19p-13q=7$ where $a,b,p,q \in \mathbb{N}$, $a,b,p,q \neq 0$

Is there an efficient way to find solutions to the equation: $6ab+a+b-36pq-19p-13q=7$ where $a,b,p,q \in \mathbb{N}$ and $a,b,p,q \neq 0$ If the equation has no solutions, how could you prove that, ...
1
vote
4answers
43 views

how do i prove $ab|n$ if $\gcd(a, b) = 1$ and $a|n $ and $b|n$?

Suppose that, for integers $a, b,$ and $n,$ $$\gcd(a, b) = 1\text{ and }a|n\text{ and }b|n.$$ How do I prove that $ab|n$ using linear Diophantine equations? Can I extend the above result to the ...
3
votes
1answer
76 views

Solving $x^p + y^p = p^z$ in positive integers $x,y,z$ and a prime $p$

The question is from Zeitz's ''The Art and Craft of Problem Solving:" Find all positive integer solutions $x,y,z,p$, with $p$ a prime, of the equation $x^p + y^p = p^z$. One thing I noticed is ...
1
vote
1answer
38 views

Co-primality of coefficients of coprime integers

Given that $a,b$ are co-prime, we have infinitely many solutions for $x,y$ to the equation $$ax+by=c.$$ Furthermore, solutions have the form: $x=ca^{-1}+tb,y=cb^{-1}-ta$. Given that $c$ ...
0
votes
1answer
47 views

Diophantine Equation with 15th power

So I'm working on the Diophantine equation $2x^2-1=y^{15}$ (1) with x,y>1 In particular I want to show that x must be a multiple of 5. I have found that it suffices to show that for $y=1 mod 10$ ...
4
votes
1answer
70 views

Integer solutions of $ x^3+y^3+z^3=(x+y+z)^3 $

Consider the equation $$ x^3+y^3+z^3=(x+y+z)^3 $$ for triples of integers $(x, y, z) $. I noticed that this has infinitely many solutions: $ x, y $ arbitrary and $ z=-y $. Are there more solutions? ...
0
votes
2answers
61 views

Proving that if $a^2+b^2=c^2$ for $a,b,c \in \Bbb Z^+$, then either $a$ or $b$ is even. [closed]

Prove that if $a, b, c \in \Bbb Z^+$, and $a^2+b^2=c^2$, then either $a$ or $b$ is even. It seems like a proof by contradiction can be used here. I have my own proof below but it may need some ...
5
votes
3answers
112 views

If $(m,n)\in\mathbb Z_+^2$ satisfies $3m^2+m = 4n^2+n$ then $(m-n)$ is a perfect square.

I came across this question on another forum. The question is: $$ \text{If $m,n\in \mathbb{Z}_+$ such that $3m^2+m=4n^2+n$, then $(m-n)$ is a perfect square.}$$ I have managed to partially prove ...
0
votes
0answers
44 views

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

just so this doesn't get deleted, I want to make it clear that I already know how to solve this using the UFD $\mathbb{Z}[\sqrt{-2}]$, and am in search for the infinite descent proof that Fermat ...
1
vote
3answers
92 views

Find all $n\in\mathbb N$, $n>3$ such that $p(n)=2n+16$, where $p(n)$ is the product of all the prime numbers less than $n$.

Find all $n\in\mathbb N$, $n>3$ such that $p(n)=2n+16$, where $p(n)$ is the product of all the prime numbers less than $n$. E.g. $p(7)=2\cdot3\cdot5$ (and $n=7$ is a solution). Let ...
4
votes
1answer
123 views

Do there exist 4 rationals satisfying $a^2+b^2+c^2+d^2=1$ and $2a+b+c+d=0$?

Does there exist 4 rationals $(a,b,c,d)$ which satisfy the following two relations?, $a^2+b^2+c^2+d^2=1$ and $2a+b+c+d=0$? I spend a lot o' time with it and tried the criteria of quadratic ...
2
votes
3answers
78 views

Does $b^2 = 4a + 2$ have integer solutions?

I got the following question on an exam I had today: $$b^2 = 4a+2$$ I said that it didn't since $b^2 -2$ was not a multiple of $4a$ or in other words, was not divisible by $4$. Is this correct?
0
votes
2answers
41 views

The number of ordered triples $(a, b, c)$ of positive integers which satisfy the simultaneous equations $ab + bc = 44$, $ac + bc = 33$

My try: Subtracting the eqns: $a(b-c) = 11$ $a=1,b-c=11$ OR $a=11,b-c=1$ Substituting these values back int the original eqn. does not give an integral answer. Thus number of ordered ...
0
votes
2answers
32 views

For each of the following values of ($a,b$), find the largest number that is not of the form $ax+by$ with $x\geq 0$ and $y \geq 0$.

For each of the following values of ($a,b$), find the largest number that is not of the form $ax+by$ with $x\geq 0$ and $y \geq 0$. $(i) (a,b) = (3,7)$ $(ii) (a,b) = (5,7)$ $(iii) (a,b) = (4,11)$ ...
0
votes
0answers
10 views

Deciding a linear diophantine equation with real coefficients

Fix $F,G,a,b\in\mathbb{R}$. What is a fast algorithm to determine whether there exist integers $k, l$ such that $Ft+a=\pi k$ and $Gt+b=\pi l$? I'm having difficulty with my computations because ...
0
votes
1answer
23 views

$x \equiv b^u\pmod m$ is a solution of given congruent relation

$b,k \in {\Bbb Z};$ $\operatorname{gcd}(b,m) = 1;$ $\operatorname{gcd}(k,φ(m)) = 1; $ $x^k \equiv b\pmod m$ Show that $x \equiv b^u\pmod m$ is a solution of above congruent relation. $u$ is the ...
17
votes
2answers
655 views

Prove that $x^3 + y^3 = z^3$ has no integer solutions as simply as possible

Can someone prove the special case of Fermat's Last Theorem for $n=3$, i.e., that $$x^3 + y^3 = z^3,$$ has no positive integer solutions, as simply as possible? I have seen some good proofs, but ...
1
vote
2answers
84 views

parametric solution for the sum of three square

Is there a parametric integer solution for $x,y,z,t$ when the sum of three square is equal to a square, i.e, $$x^2+y^2+z^2=t^2$$?
6
votes
1answer
151 views

Finding integer cubes that are $2$ greater than a square, $x^3 = y^2 + 2$ [duplicate]

I was given an example of a cube that is $2$ greater than a square number. The pair: $27$ and $25$. What's the best way to find further pairs ?
0
votes
1answer
40 views

Solving Diophantine Equation

$x$,$y$ are integers such that $~x^{2}+1=y^{x}$. Find all pairs of $(x,y)$. I know that it's a diophantine equation but don't have any idea. Also I can't find anything related to it by search ...
1
vote
1answer
70 views

Proving that diophantine equation has no solutions

I am trying to show that the equation $x^5y + 5x^3 - xy^5 = 1$ has no solutions. Anyone has an idea on this?
6
votes
2answers
103 views

Equation: $(x^2-9y^2)^2=33y+16$

I want to know the solution of the equation $(x^2-9y^2)^2=33y+16$ in positive integers. I know it has solution $(\pm2;0)$ but I can't prove that it doesn't have other solutions. Please help.
0
votes
1answer
45 views

How to solve a system of equations with only integers

$a+2y = 320$ $2b+3y = 320$ $3c+4y = 320$ And a, b, c, and y are all integers. How do I find all possible solutions, if any?
0
votes
2answers
71 views

If $n\in\mathbb N$ and $k\in\mathbb Z$, solve $n^3-32n^2+n=k^2$.

If $n\in\mathbb N$ and $k\in\mathbb Z$, solve $$n^3-32n^2+n=k^2$$ I've tried checking congruence modulo $4$. I have that n is divisible by 4. Let $n=4u$, where $u\in\mathbb N$. Then the equation ...
4
votes
0answers
83 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 $x^3-y^5=(x+y)^2$ for primes $p$ and $q$. A proof is given that $p=7, q=3$ is the only solution. In this ...
6
votes
1answer
112 views

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

If $p,q$ are prime, solve $$p^3-q^5=(p+q)^2$$ I can't think of a nice idea for the solution. Since there's a solution $(7;3)$, consisting of two distinct numbers, I really doubt modular arithmetic ...