Articles of elementary number theory

Solve $2b(b-1) = t(t-1)$ as Pell's equation

I know the method of continued fractions to solve the Pell’s equation. I need help turning $2b(b-1) = t(t-1)$, with $b, t$ as integers, into the form $x^2 – ny^2 = 1$, if possible. My attempt is $(t-1/2)^2 – 2(b – 1/2)^2 = -1/4$, but those aren’t integer solutions anymore.

Prove that if perfect squares divide each other, then so do the originals – without primes

I want to prove: $$\text{If }a^2|z^2,\text{ then }a|z.$$ Assume everyone is a positive integer, etc. Unless I’m deluding myself, this is pretty easy to show using unique prime factorization. But I want to do it without using primes or the (usual statement of the) FTA. That is, using coprime is fine, using the so-called Bezout […]

Prove that there are exactly $k$ pairs $(x,y)$ of rational numbers with $0\leq x,y<1$ for which both $ax+by,cx+dy$ are integers.

Let $a,b,c,d$ are integers such that $(a,b)=(c,d)=1$ and $ad-bc=k>0$. Prove that there are exactly $k$ pairs $(x,y)$ of rational numbers with $0\leq x,y<1$ for which both $ax+by,cx+dy$ are integers. What I did Let $ax+by=m$ and $cx+dy=n$, solving them we get $y=\frac{na-mc}{k}$ since $0\leq y<1\Rightarrow na-mc<k$ So $na-mc$ can take the values $0,1,2,…,k-1$ and so $y$ […]

Proof by induction: Show that $9^n-2^n$ for any $n$ natural number is divisible by $7$.

Can someone please solve following problem. Show that $9^n-2^n$ for any $n$ natural number is divisible by $7$. ($9 ^ n$ = $9$ to the power of $n$). I know the principle of induction but am stuck with setting up a formula for this.

The necessary and sufficient condition for a regular n-gon to be constructible by ruler and compass.

I have a problem concerning the necessary and sufficient condition for a regular n-gon to be constructible by ruler and compass. $\bf My$ $\bf question:$ For a given positive integer $n$, how can we prove that a regular $n$-gon is constructible by ruler and compass if and only if the number $cos(2\pi/n)$ (that is, the […]

Proof of for all integer $n \ge 2$, $n^3-n$ is divisible by 6 by mathematical induction.

Prove the following statement by mathematical induction: For all integer $n \ge 2$, $n^3-n$ is divisible by 6 My attempt: [Proof] Let the given sentence p(n) (1) $2^3-2$=6 is divisible by 6. p(2) is true. (2) Suppose for all integer $k \ge 2$, p(k) is true. That is, mathematical hypothesis is $k^3-k$ is divisible by […]

How many zeros are there at $1000!$ in the base $24$

I know $1000!$ has $\frac{1000}{5}+\frac{1000}{25}+\frac{1000}{125}+\frac{1000}{625}=249$ terminal zeros in decimal notation, but what if we write $1000!$ in base $24$, how many terminal zeros would it have? Is there a way to compute it? Regards

Can you make sense of roots of unity in modular arithmetic?

I tried to solve this question, but realized I have no idea how to justify the intermingling of complex numbers and modular arithmetic. I know how to justify rational numbers, it is easy to realize you just treat the denominator as inverse. Is there any way to justify it? Here is my (possibly flawed) solution: […]

If $a+b\equiv0\pmod m$ and $c+d\equiv 0\pmod m$ then $ac\equiv bd\pmod m$. Demonstration.

If $a+b\equiv0\pmod m$ and $c+d\equiv 0\pmod m$ then $ac\equiv bd\pmod m$ How to show?$$$$ I tried $$a+b\equiv0\pmod m\Longrightarrow m\mid a+b\\c+d\equiv0\pmod m\Longrightarrow m\mid c+d\\a+b=mk\;\;\text{and}\;\;c+d=mj\\(a+b)(c+d)=m(mjk)\\m\mid ab+ad+bc+bd$$ But from here could solve anything.

If $w^2 + x^2 + y^2 = z^2$, then $z$ is even if and only if $w$, $x$, and $y$ are even

I’m trying to go through the MIT opencourseware Mathematics for Computer Science (6.042J). I’ve been stumped for half a day trying to figure it out. Something isn’t clicking, and I could use some help. Here is the problem: Suppose that $w^2 + x^2 + y^2 = z^2$, where $w$, $x$, $y$, and $z$ always denote […]