Articles of number theory

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. […]

How do I get a sequence from a generating function?

For example if I have the generating function $\frac{1}{1-2x}$ then it corresponds to the sequence $1 + 2x + 4x^2 + 8x^3 +~…$. I know how to start from the sequence and get the generating function, but I don’t know how to start from the generating function and get the sequence. Similarly, what if I […]

Does Fermat's Little Theorem work on polynomials?

Let $p$ be a prime number. Then if $ f(x) = (1+x)^p$ and $g(x) = (1+x)$, then is $f \equiv g \mod p$? I’m trying to prove that for integers $a > b > 0$ and a prime integer $p$, ${pa\choose b} \equiv {a \choose b}.$ To do this I use FLT to show that […]

Prime number theory and primes in a specific interval

The prime number theorem states that $$\pi(x) \sim \frac{x}{\log x}$$ as $x \to \infty$. How can we prove the existance of a prime number in the interval $[54n, 55n]$ for $n > 30$ ($n$ is a natural number) using the prime number theorem?

Number Theory: Ramification

The question is as follows. Let $K = \mathbb{Q}(\sqrt[m]{a},\sqrt[n]{b}) $, where $m,n,a,b$ are positive integers such that they are pairwise coprime. Assume that $[K:\mathbb{Q}]=mn$/ Prove that no prime numbers can totally ramify in $K/\mathbb{Q}$. I assume we would need to find such prime numbers that are ramified in $\mathbb{Q}(\sqrt[m]{a})/\mathbb{Q}$, $\mathbb{Q}(\sqrt[n]{b})/\mathbb{Q}$ respectively. I know if $p|\text{disc}(L)$, […]

Computing $\text{Gal}(\mathbb{Q}^{ab}/\mathbb{Q})$

Algebraic class field theory tells us that $\text{Gal}(\mathbb{Q}^{ab}/\mathbb{Q})$ is isomorphic to the group of connected components of the quotient $\mathbb{Q}^{\times}\backslash \mathbb{A}_{\mathbb{Q}}^{\times}\cong \prod_p \mathbb{Z_p}^{\times}\times \mathbb{R}_{>0}$, where $\mathbb{A}_{\mathbb{Q}}$ is the ring of adèles of $\mathbb{Q}$. It’s then said that the group of connected components is given by $\prod_p \mathbb{Z_p}^{\times}$, how can I see this? Thank you very […]