Articles of elementary number theory

Is it trying to say that $(\operatorname{mod} 7)$ is neither associated with $29$ nor $15$?

I am reading Notes of Mathematics for Computer Science(MIT 6.042J). And I’m stuck in the Modular Arithmetic Section which I mark with a RED LINE. I want to know what is trying to say. Is it trying to say that $(\operatorname{mod} 7)$ is neither associated with $29$ nor $15$?

Related to the partition of $n$ where $p(n,2)$ denotes the number of partitions $\geq 2$

I was trying the following question from the Number theory book of Zuckerman. If $p(n,2)$ denotes the number of partition of n with parts $\geq2$, prove that $p(n,2)>p(n-1,2)$ for all $n\geq 8$, that $p(n)=p(n-1)+p(n,2)$ for all $n \geq 1$, and that $p(n+1)+p(n-1)>2p(n)$ for all $n \geq 7$. Need some clue to solve it.

Irreducible 3rd degree polynomials over $\mathbb{Z}_3$ field?

I want to find all irreducible polynomials over $\mathbb{Z}_3$ field which have the form $x^3 + a_2 x^2 +a_1 x + a_0$. My thought process: Third degree polynomial is irreducible if it has no roots within $\mathbb{Z}_3$. $a_0 \neq 0$, otherwise $0 \in \mathbb{Z}_3$ is a root of the polynomial in question. However this leaves […]

Prove that $9\mid (4^n+15n-1)$ for all $n\in\mathbb N$

First of all I would like to thank you for all the help you’ve given me so far. Once again, I’m having some issues with a typical exam problem about divisibility. The problem says that: Prove that $\forall n \in \mathbb{N}, \ 9\mid4^n + 15n -1$ I’ve tried using induction, but that didnt work. I’ve […]

Is always $\small {rq-1 \over 2^B} +1 \le \min(q,r) $ with equality iff $\small q$ or $\small r$ is a divisor…

I had a simpler question before such that I could even answer it myself. For the next step I seem again to be too dense today. (Remark several days later: it’s not only being dense… I still don’t find the first step for the solution) Recall: I discuss q,r as residues to a modulus of […]

Trying to understand why a set of residues modulo a primorial $p_k\#$ has a range of values smaller than $2p_{k+1}$

I’ve been reviewing the following: $$v_i = \left\lfloor\frac{ip_k\#}{p_{k+1}}\right\rfloor + c_i$$ where: $c_i \in \left\{1,2\right\}$ so that $v_i$ is odd and $v_ip_{k+1} > ip_k\# > (v_i-c)p_{k+1}$ $i$ is an integer such that $1 \le i \le p_{k+1}-1$ $\left\lfloor\frac{a}{b}\right\rfloor$ is a floor function I hit a result that surprises me. Let $[v_i]$ be a residue modulo $p_k\#$ […]

The number $25!$ has exactly 7 trailing zeros, true or false?

I don’t know how to determine it… any hints?

What does $p^\alpha\| n$ mean?

What does $p^\alpha\| n$ mean ? I saw this in Euler totient function, $$\varphi(n)=\prod_{p^\alpha\| n}p^\alpha(p-1).$$

$a, b \in\Bbb N$, $A=\sqrt{a^2+2b+1}+\sqrt{b^2+2a+1}\in \Bbb N $, to show that $a = b$.

If $a$ and $b$ are positive integers such that $A=\sqrt{a^2+2b+1}+\sqrt{b^2+2a+1}\in \Bbb N $, to show that $a = b$. By contradiction assuming that $a <b$ then it follows that $2 (a +1) <A <2 (b +1)$. This means that $A = 2a + r,$ $r$ taking values $3, 4, …, 2b-2a +1$. From the equality […]

On a criterion for almost perfect numbers using the abundancy index

Let $\sigma(x)$ denote the sum of the divisors of $x$, and let $I(x) = \sigma(x)/x$ be the abundancy index of $x$. A number $y$ is said to be almost perfect if $\sigma(y) = 2y – 1$. In a preprint titled A Criterion For Almost Perfect Numbers Using The Abundancy Index, Dagal and Dris show that […]