Articles of elementary number theory

Primes $p$ such that $2$ is a primitive root modulo $p$ , where $p=4\cdot k^2+1$?

Consider the prime numbers of the form : $p=4 \cdot k^2+1~$ , where $~k~$ is an odd prime number . For the first $~1200000~$ primes of this form except when $~p=4 \cdot 193^2+1~$ $2~$ is a primitive root modulo $~p$ . Note that : Euler’s totient function is given by : $\phi(p)=4 \cdot k^2$ One […]

How to prove: $2^\frac{3}{2}<\pi$ without writing the explicit values of $\sqrt{2}$ and $\pi$

How to prove: $2^\frac{3}{2}<\pi$ without writing the explicit values of $\sqrt{2}$ and $\pi$. I am trying by calculus but don’t know how to use here in this problem. Any idea?

Show that $89|(2^{44})-1$

Show that $89|(2^{44})-1$ My teacher proved this problem using mod can someone explain the process step by step? Thank you so much!

Solve $3x^2-y^2=2$ for Integers

If $x$ and $y$ are integers, then solve (using elementary methods) $$3x^2-y^2=2$$ I tried the following If $y$ is even, then $4|y^2$ and hence $2|y^2+2$ (and $4$ doesn’t divide it), but $3x^2=y^2+2$ and since for R.H.S. to be even, $2|x^2 \implies 4|x^2$, and we get a contradiction. So $x$ and $y$ both are odd. If […]

How to solve $(2x^2-1)^2=2y^2 – 1$ in positive integers?

I encountered this question (posed by Fermat) in a letter from Fermat to Carcavi and was wondering what would be the best elementary way to solve it. Solve in positive integers$$(2x^2-1)^2=2y^2 – 1$$ Any help will be appreciated. Thanks in advance.

Show that $a – b \mid f(a) – f(b)$

I have seen this lemma elsewhere. Suppose $A$ is a domain, and $f \in A[X]$. Prove that $$a – b \mid f(a) – f(b)$$ I need to prove this. $$f(a) – f(b) \equiv 0 \pmod{a-b}$$ basically. Let, $a – b = c$ $$f(a) – f(b)/(a-b) = f'(\xi)$$ for Some $\xi \in (a, b)$. But I […]

Prove that : $(a-b)(a-c)(a-d)(b-c)(b-d)(c-d)$ divisible by $12$

Prove that : $(a-b)(a-c)(a-d)(b-c)(b-d)(c-d)$ divisible by $12$, with $a,b,c,d\in\mathbb{Z}$.

Show that $\sum\nolimits_{d|n} \frac{1}{d} = \frac{\sigma (n)}{n}$ for every positive integer $n$.

Show that $\sum\nolimits_{d|n} \frac{1}{d} = \frac{\sigma (n)}{n}$ for every positive integer $n$. where $\sigma (n)$ is the sum of all the divisors of $n$ and $\sum\nolimits_{d|n} f(d)$ is the summation of $f$ at each $d$ where $d$ is the divisor of $n$. I have written $n=p_1^{\alpha_1}p_2^{\alpha_2}p_3^{\alpha_3}…….p_k^{\alpha_k}$ then:- $$\begin {align*} \sum\nolimits_{d|n} \frac{1}{d}&=\frac{d_2.d_3……d_m+d_1.d_3……d_m+……..+d_1.d_2.d_3……d_{m-1}}{d_1.d_2.d_3……d_m} \\&=\frac{d_2.d_3……d_m+d_1.d_3……d_m+……..+d_1.d_2.d_3……d_{m-1}}{p_1^{1+2+…+\alpha_1}p_2^{1+2+…+\alpha_2}p_3^{1+2+….+\alpha_3}…….p_k^{1+2+….+\alpha_k}} \\ \end{align*}$$ where […]

Ternary Quadratic Forms

Let $Q(x,y,z) = ax^2 + by^2 + cz^2$ where $a,b,c \in \mathbb{Z}_{\neq 0}$. Suppose that the Diophantine equation $Q(x,y.z) = 0$ has a non-trivial integral solution. Show that for any rational number $g$, there exist $x,y,z \in \mathbb{Q}$ such that $Q(x,y,z) = g$ I have trouble starting, any help will be appreciated! I know Legendre’s […]

Permutation Partition Counting

Consider the number $n!$ for some integer $n$ In how many ways can $n!$ be expressed as $$a_1!a_2!\cdots a_n!$$ for a string of smaller integers $a_1 \cdots a_n$ Let us declare this function as $\Omega(n)$ Consider the value of $10!$ $$10! = 7!6!$$ $$10! = 7!5!3!$$ Thus we know that $\Omega(10)\ge 3$ We note that […]