Articles of number theory

Factorization of $p\mathcal{O}_K$ in $\mathcal{O}_K$

Let $K=\mathbb{Q}(\alpha)$, where $\alpha$ is a root of $f(x)=x^3+x+1$. If $p$ is a rational prime. What can you say about factorization of $p\mathcal{O}_K$ in $\mathcal{O}_K$? I have this: The discriminant is -31 and the minkowski bound, $M_K \leq 1.57$. Then $N(\mathcal{A})=1$, $\mathcal{A}$ a ideal class. I can say anything about the question?

$X^n + X + 1$ reducible in $\mathbb{F}_2$

I was told that sometimes in characteristic 2 that $X^n + X + 1$ is reducible mod 2. What is the smallest $n$ where that is true?

integer solutions to $x^2+y^2+z^2+t^2 = w^2$

Is there a way to find all integer primitive solutions to the equation $x^2+y^2+z^2+t^2 = w^2$? i.e., is there a parametrization which covers all the possible solutions?

unique factorization of matrices

If I have a set of matrices, call this set U, how can I make this a UFD (unique factorization domain)? In other words, given any matrix $X \in U$, I would be able to factorize X as $X_1 X_2 … X_n$ where $X_i \in U$ and this factorization is unique? We may assume the […]

Describe all integers a for which the following system of congruences (with one unknown x) has integer solutions:

$$x\equiv a \pmod {100}$$ $$x\equiv a^2 \pmod {35}$$ $$x\equiv 3a-2 \pmod {49}$$ I’m trying to solve this system of congruences, but I’m only familiar with a method for solving when the mods are pairwise coprime. I feel like there ought to be some simplification I can make to reduce it to a system where they […]

Largest idempotent

Given the prime power factors of $N$, is there a non-quadratic algorithm for finding the largest idempotent of ring $\mathbb{Z}/N\mathbb{Z}$? (That is, the largest number $A \lt N$ such that $A^{2} \equiv A \text{mod} N$.) I know that there are at most $8$ prime power factors ($=K$) for $N$ in the range of interest, thus […]

In how many ways I can write a number $n$ as sum of $4$ numbers?

The precise problem is in how many ways I can write a number $n$ as sum of $4$ numbers say $a,b,c,d$ where $a \leq b \leq c \leq d$. I know about Jacobi’s $4$ square problem which is number of ways to write a number in form of sum of $4$ squares number. There is […]

Equation over the integers

Find all quartuples $(a,b,c,d)$ with non-negative integers $a,b,c,d$ satisfying $$2^a3^b-5^c7^d=1$$ One solution is $$(2,2,1,1)$$

Given a fixed prime number $p$ and fixed positive integers $a$ and $k$; Find all positive integers $n$ such that $p^k \mid a^n-1$

The question is the natural generalization of the Find all positive integers solutions such that $3^k$ divides $2^n-1$ . Let $p$ to be a prime number and let $k$ & $a$ be fixed natural numbers. Find all natural numbers $n$, such that $p^k \mid a^n-1$?

Number of decimal places to be considered in division

This must be a basic question. But i need some help. What is the number of decimal places that needs to be considered normally in division operations in order to represent the dividend value as a multiple of divisor and quotient by rounding off. Example: Say i want to divide 3475934 and 3475935 by 65536. […]