# 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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### When does -1 have a squareroot in a finite field? (-1 as a quadratic residue)

For example in $\mathbb{F}_5$, $2^2=3^2=-1$. However, in $\mathbb{F}_3$, there is no solution to $x^2=-1$. When do the squareroot(s) exist, and if they do, can we say anything about their ...
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### Let f(x,y) be a positive semidefinite quadratic form with discriminant 0. Show that f is equivalent to the form h(x,y) = $gx^{^{2}}$.

Let f(x,y) = $ax^{^{2}} + bxy + cy^{^{2}}$ be a positive semidefinite quadratic form of discriminant 0. Put g = gcd(a,b,c). Show that f is equivalent to the form h(x,y) = $gx^{^{2}}$. I know that if ...
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### Sums involving square of Moebius function

I try to estimate the following sum: $$\sum_{n \leq x}\mu(n)^2 f(n)$$ where $\mu(n)$ is a Moebius function and $f(n)$ is some multiplicative arithmetic function. If I understand it correctly it is ...
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### Last 7 digits of 7th powers

Alice and Bob play the following game. They alternately select distinct nonzero digits from $1$ to $9$, until they have chosen seven such digits. Consider the resulting seven-digit number by joining ...
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### What is $\zeta(n)$ as $n$ tends to $\infty$? How fast it goes to the limit?

What is $\zeta(n)$ as $n\to\infty$? How fast it goes to the limit?
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### How to have this equation $s^2-2(p+q+r+2pqr)s+(p^2+q^2+r^2-2(pq+qr+rp)-4)=0$？

Old Question: For $x,y,z\in N^{+}$, if such $(xy+1)(yz+1)(zx+1)$ is a perfect square ,show that $$(xy+1),(yz+1),(xz+1)$$ are all perfect square . and I konw this PDF have solution, http://math....
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### Help with proof regarding degrees of polynomials

How do you prove that if $f(x)\mid g(x)$ in $F[x]$, then either $g(x) = 0$ or $\deg(g(x)) \geq \deg(f(x))$ I'm not really sure how to prove these types of statements
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### Möbius function verification

I am looking to verify my answer to the question $$F(n)=\sum_{d|n}{\mu(d)\sigma(d)}=(-1)^{\omega(n)}\prod_{j=1}^{\omega(n)}{p_j}$$ Where $\mu$ is the Möbius function, $\sigma$ is the sum of divisors ...
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### Diophantine solution to a fraction

How can we find solutions to the following equation: $$y=\dfrac{x^2-1085}{14718-2x}$$ where $x,\ y$ are integers.
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### Sizes of Blocks of Consecutive Integers Divisible by at Least One Prime Less than or Equal to $r$.

Let $f(r)$ be the largest integer such that there exists a block of $f(r)$ consecutive integers each divisible by some prime that is less than or equal to $r$. For example, $f(2)=1$ because it is ...
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### regarding pseudo-prime numbers.

If $W$ is an odd composite number and $-1+2^{W-1}$ is divisible by $W$ yet not by $W^2$, then $W^2$ does not divide $-1+ 2^{W(W-1)}$. Is this true? (forgive my use of symbols,I have no good math ...
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### An Inequality with Prime Numbers

Let $m\geq 8$ be an integer, and let $q$ be the smallest prime that is greater than $m$. Let $p$ be a prime that satisfies $q\leq p<q^2$, and let $p_0$ be the smallest prime such that $p_0q>p^2$....
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### Finding the number of elements in $\left(ℤ[i]\right)_m$

If $m$ = $m_1$ + $m_1i$ ∈ $ℤ[i]$, is there a general formula to find the number of elements in $\left(ℤ[i]\right)_m$?
### Is there a polynomial equation for $f(n) = n!$ and if so what is it?
And I am not necessarily talking about $f(n) = n(n-1)(n-2)...(3)(2)(1)$ in its factored form; Well it could be that but then I would like a general way of expansion. Thanks in advance!
### Proof that $26$ is the one and only number between square and cube
$x^2 + 1 = z = y^3 - 1$ Why $z = 26$ and only $26$ ? Is there an elementary proof of that ?