Articles of chinese remainder theorem

modules with several powers $x^{y^z}$

I have an assignment(which gives no credit) which I cant solve so I would like to get some help $3^{{1001}^{1002}}\pmod {43}$ (All original numbers were replaced) Our hint was to use the Chinese-remainder-theorem but I don’t see how when $43$ is a prim number.

Solvability of $a \equiv x^2 \mod b$

Suppose you want to prove that $\exists x \in \mathbb{Z}$ with $a \equiv x^2 \mod b$. Write $b = \prod_{i = 1}^{k} p_i^{e_i}$, the prime factorisation of $b$. Why is the equivalent with finding solutions to $a \equiv x_i^2 \mod p_i$? How does one apply the Chinese Remainder Theorem here? Thanks in advance!

Sheafyness and relative chinese remainder theorem

The relative chinese remainder theorem says that for any ring $R$ with two ideals $I,J$ we have an iso $R/(I\cap J)\cong R/I\times_{R/(I+J)}R/J$. Let’s take $R=\Bbbk [x_1,\dots ,x_n]$ for $\Bbbk $ algebraically closed. If $I,J$ are radical, the standard dictionary tells us $R(I\cap J)$ is the coordinate ring of the variety $\mathbf V(I)\cup \mathbf V(J)$. Furthermore, […]

Doubts about a nested exponents modulo n (homework)

As part of my homework I am supposed to find the remainder of the division of $2^{{14}^{45231}}$ by $31$. Using the ideas explained in calculating nested exponents modulo n I tried the following: $\phi(31)=30$ since $31$ is prime. Then: $2^{{14}^{45231}}$ (mod $31$) = $2^{{14}^{45231} (\textrm{mod}30)}$ (mod $31$) Since $30 = 2 \cdot 3 \cdot 5 […]

Flaw or no flaw in MS Excel's RNG?

I have a question about my understanding of an article of B.D. McCullough (2008) about Excel’s implementation of the Wichmann-Hill random number generator (1982). First, a bit of context The Wichmann-Hill algorithm is given in AS 183 here. As one can see in the program’s comments, it’s merely three linear congruential generators (LCG) combined to […]

Solving a congruence without Fermat's little theorem

Given $n\in\Bbb N$, what is the least $a>1$ with $a^{2^n}\equiv1\bmod2015$? Is there a solution not using Fermat’s little theorem or the Chinese remainder theorem, any ideia?

Why is this congruence true?

$$\eqalign{ x &\equiv 5 \mod 15\cr x &\equiv 8 \mod 21\cr}$$ The extended Euclidean algorithm gives $x≡50 \bmod 105$. How/why? I am trying to understand how this is true when the Euclidean algorithm typically needs only two inputs $a,b$. I see four numbers in these two congruences.

Show that $R/(I \cap J) \cong (R/I) \times (R/J) $

My question actually follows from this one: Chinese Remainder Theorem for Rings What I don’t understand is why is it necessary for $I+J=R$, in order for $$ \varphi\colon R\to R/I\times R/J:a\mapsto (a+I,a+J) $$ to be surjective. This map seems surjective by construction (I worked through the same question myself and came up with the same […]

Show that $15\mid 21n^5 + 10n^3 + 14n$ for all integers $n$.

I’m not sure if it’s correct, but what I have so far is; $$21n^5 + 10n^3 + 14n ≡ (1 + 0 – 1) ≡ 0 \mod 5$$ but I’m having trouble solving it in $\bmod 3$. I have: $$21n^5 + 10n^3 + 14n ≡ (0 + (?) + 2),$$ I’m not sure how to […]

Multi-pullbacks and the relative chinese remainder theorem

Let $I,J$ be two-sided ideals of a ring $R$. In this question I asked for an “automatic” proof of the fact the natural map $R/(I\cap J)\rightarrow R/I\times _{R/(I+J)} R/J$ is an isomorphism (a direct proof is not hard). At any rate, if $I,J$ are comaximal, this immediately yields an isomorphism $R/(I\cap J)\cong R/I\times R/J$. Now […]