# Tag Info

1 vote

### How to check for solutions of non-linear systems over the integers modulo m?

Let's see, lemma, given a prime $p$ and a target $t,$ there always exist $x,y$ such that $x^2 + y^2 \equiv t \pmod p.$ This one uses just a counting/pigeonhole argument This can always be done for ...
• 141k
1 vote
Accepted

### A question about integrality in polynomial rings

Pick an $\alpha \in B$, and consider $B'$ the subring of $B$ generated by the $x_i$'s and the coefficients of $\alpha$ (denoted $c_1,...,c_m \in \overline{k}$ in no particular order). Then $\alpha$ is ...
Accepted

### Approximating the Prime Counting Function as $\pi(x) \approx \frac{x^2}{\ln\left(\Gamma(x+1)\right)}$

Since $\log \Gamma(x+1) = x\log x - x + O(\log x)$, \begin{align*} \frac{x^2}{\log \Gamma(x+1)} &= \frac x{\log x - 1 + O(x^{-1}\log x)} \\ &= \frac x{\log x} \biggl(1 - \frac1{\log x} + O(x^{-...
• 81.9k

• 7,103
1 vote

### Fermat's last Theorem and elliptic curve cryptography

I don't think there is any connection and the temporal link is weak. First, Fermat's last theorem was proven in 1994. Second, elliptic curve cryptography was proposed in 1985, so about 10 years before ...
• 6,083
Accepted

### 2-adic valuations of $k\cdot 3^n-1$

The best way to understand the situation, for odd $k$ (even $k$ is trivial) is to decompose your expression, to wit: $(k\cdot3^n)-1=k[(3^n-1)-(k^{-1}-1)]$ where $k^{-1}$ is the reciprocal of $k$ in $2$...
• 41.1k

### A simple question about bounding a sum

First, the condition $\|x\|_2>\gamma^{-1}$ leads to $\frac{1}{\|\gamma x\|_2} <1$, which implies that $\frac{\gamma^2}{\|\gamma x\|_2^2}$ dominates the other terms in the sum. Hence, we only ...
• 531
Accepted

• 181k
### Lemma about $n$-adic numbers
If the element $x \in \mathbb{Z}_{mn}$ is given by the numbers $x_k$, then those same numbers $x_k$ also define an element $x' \in \mathbb{Z}_n$ and similarly an element $x'' \in \mathbb{Z}_m$. Try to ...