# Tagged Questions

Questions on finding integer/rational solutions of equations.

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### If $a$ and $b$ are whole numbers from $1$ to $100$, how many pairs of numbers $(a,b)$ are there which satisfy $a^{\sqrt{b}}=\sqrt{a^b}$

If $a$ and $b$ are whole numbers from $1$ to $100$, how many pairs of numbers $(a,b)$ are there which satisfy $a^{\sqrt{b}}=\sqrt{a^b}$ This was from a math contest I did earlier today and I was ...
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### How can you solve this type of (not quite linear) diophantine equation in 2 variables?

Is there a general technique to find solutions of this type of equation? 555555=t+2rt+r I'll provide the only answer I know in the comments. Thanks.
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### How to find the first integer making two progressions have gcd $> 1$

Is there a technique to efficiently find the first positive integer, $r$, that makes: $$\gcd(97+r, 106-r) > 1\text{?}$$
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### Perfect powers by Oblath's result [duplicate]

What do you mean by this statement? Obl\'ath proved that the only perfect powers all of whose digits are equal to a fixed one $a \neq 1$ in decimal representation are 4, 8 and 9. This is equivalent ...
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### Application of the theorem about diophantine equations having either infinite or finite solution.

How can i apply the theorem below in an equation like $$\label{eq:(4)} 10^{n+3} a - 10^3 a + 999b = (3y)^2.$$ that equation is actually from letting $m = 3$, from the ...
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### Right angled triangle and Pythagorean triplet

Show that there exists a right angled triangle with rational sides and area $d$ if and only if $x^2,y^2$ and $z^2$ are squares of rational numbers and are in arithmetic progression with common ...
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### Why does the modulo affect other terms in the equation? [closed]

i just want to ask if why does the modulo affect the other terms in an eqution? Why does the 4th equation has to be multiplied by $a^2$? Then as the modulo becomes $n≡1(mod3)$ in the 5th eq. then the ...
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### Probability of solution amongst a set of Diophantine equations

I have a set of Diophantine equations which I know only one equation has a single solution. I am trying to find a way to give probabilities to which equation contains the solution. For example, ...
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### Find a positive integer solution to $xyzw=504(x^2+y^2+z^2+w^2)$

Find positive integer values of $x,y,z,w$, such that $$xyzw=504(x^2+y^2+z^2+w^2)$$ I found it at some point and now I am unable to find the solution anymore, maybe this equation isn't satisfiable? ...
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### Need help with proof about Diophantine equations

The way I am planning to arrange this is by providing fragments of the proof, so I can understand what's going on before forging ahead, so if you are going to help me, keep in mind that I am going to ...
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### How does $x^4+y^4=z^2 \implies x^4+y^4=z^4$?

Why is the statement "the following cannot be satisfied" for $x^4+y^4=z^2$ more strong than for $x^4+y^4=z^4?$ More specifically, how does $x^4+y^4=z^2 \implies x^4+y^4=z^4?$ This statement was ...