# Tagged Questions

Questions on more advanced topics of number theory, such as quadratic residues, primitive roots, prime numbers, non-linear Diophantine equations, etc. Consider first if (elementary-number-theory) might be a more appropriate tag before adding this tag.

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### Is the relationship between coprime integers and irreducible fractions biconditional?

Are coprime integers and irreducible fractions related biconditionally? That is, if two integers are coprime ($a$ and $b$ say) then the fractions $\frac{a}{b}$ and $\frac{b}{a}$ are both irreducible. ...
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### Maximum length of a string that has no substring divisible by a prime number $p$ is $p-1$?

What is the maximum length of a string of nonzero digits that has no substring that is divisible by a given prime number? I want to find a string of length n which has no substring divisible by the ...
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### multiples (of primes) coverage formula

I apologize in advance if my explanation is not clear. Please let me know if clarification is required and I will do my best to fix it! I am attempting to find an explicit formula (in terms of ...
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### Find the gcd of the following Gaussian integers

$\gcd(5 + 8i, 3 + 2i)$ in $Z[i]$. I found it and I got 1 then I look at the manual solution and it turns out it can be i or -i or -1 or 1. why?
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### Distribution of numbers of the form $p^{q-1}$, p and q prime

Prime numbers are exactly the integers having 2 divisors and of course, 2 is itself prime. If one considers the set of positive integers with a prime number of divisors, one can easily figure out that ...
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### Prove that the Area of triangle whose vertices are $(0,0)$, $(b,a)$ and $(x,y)$ is $|by-ax|/2$

Prove that the Are of triangle whose vertices are $(0,0)$, $(b,a)$ and $(x,y)$ is $\displaystyle \frac{|by-ax|}{2}$. I found this problem in Number theory by George Andrews, but I wonder how it ...
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Let's consider binary words of length $n$ with elements {-1,1}. There are $2^n$ binary words of length $n$. Now let's consider a subset of $n$ such binary words. All possible subsets are $\binom{2^n}{... 1answer 32 views ### Mapping finite discrete numbers to the infinite set This is an extension of my earlier question: Mapping discrete numbers Given that we can "map"$\mathbb{N}$to$\mathbb{Z}$via a bijection, I then wondered if it is possible to map a small subset of$\...
Let $E$ be an elliptic curve defined over $\mathbb{Q}$. Is there an example of such an $E$ such that the only rational point in $E(\mathbb{Q})$ is the point at infinity? In other words, consider the ...