# Tagged Questions

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### 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 ...
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### 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 ...
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### 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!
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### 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 ...
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### 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 ...
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### 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 ...
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### 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.
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### 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 ...
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### 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 $y\ge 3$.
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### 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 ...
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### 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 ...
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### 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 ...
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### 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, ...
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### 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 ...
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### 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 ...
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### 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 $c$ ...
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### 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 $y=1 mod 10$ ...
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### 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? ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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?
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### 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 ...
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### 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 $y \geq 0$. $(i) (a,b) = (3,7)$ $(ii) (a,b) = (5,7)$ $(iii) (a,b) = (4,11)$ ...
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### 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 ...
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### $x \equiv b^u\pmod m$ 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 ...
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### 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 ...
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### 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, $$x^2+y^2+z^2=t^2$$?
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### 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 ?
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### 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 ...
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### 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?
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### 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.
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### 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?
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### 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 ...
### 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 ...
### 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 ...