# Tagged Questions

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

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 ...
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### 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 ...
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### 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 ...
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### 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?
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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.
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### 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
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### 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 ...
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### 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.
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### 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 ...
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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 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 ...
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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 ...
### 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 $y\ge 3$.
### 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 ...