Articles of number theory

Prime Number Theorem and the Riemann Zeta Function

Let $$\zeta(s) = \sum_{n=1}^\infty \frac{1}{n^s}$$ be the Riemann zeta function. The fact that we can analytically extend this to all of $\mathbb{C}$ and can find a zero free region to the left of the line $Re(s)=1$ shows that $$\pi(x) := |\{p\leq x: p \mbox{ prime }\}| \sim \frac{x}{\log(x)}.$$ Moreover, improving the zero free region improves […]

Find all $x,y\in \mathbb{N}$ such that: $2^x+17=y^2$.

Find all $x,y\in \mathbb{N}$ such that: $2^x+17=y^2$. Its easy to find that $x=6$ is the only even value for $x$, the others have to be odd. One more thing is that we get $y^2 \equiv 19 \pmod p$, for every prime factor of $x$. But I have no ides what next to do.

Find the positive $n$ such $2^{n-2i}|\sum_{k=i}^{\lfloor\frac{n-1}{2}\rfloor}\binom{k}{i}\binom{n}{2k+1}$

Question: if such $\forall i\in\{1,2,\cdots,\lfloor\dfrac{n-1}{2}\rfloor\}$,have $$2^{n-2i}|\sum_{k=i}^{\lfloor\frac{n-1}{2}\rfloor}\binom{k}{i}\binom{n}{2k+1}$$ Find all the positive intger $n$ I want to find this sum $\sum_{k=i}^{\lfloor\frac{n-1}{2}\rfloor}\binom{k}{i}\binom{n}{2k+1}$?this step is right?

Limiting value of $\lim \frac{1}{k}\sum_{n=1}^k \frac{p(n+1)-p(n)}{\log p(n)}$

Empirically it seems $$\lim_{k\to \infty} \frac{1}{k}\sum_{n=1}^k \frac{g(n)}{\log p(n)} = 1\tag{1} $$ in which p(n) is the nth prime and g(n) is the prime gap $p(n+1)-p(n).$ Cramer conjectured that $$g(n) = O\left(\log^2 p(n)\right). $$ My somewhat open-ended question is: Formally how does (1) relate to Cramer’s estimate (if at all)…and can we prove (1)? For instance, […]

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 […]

how to find integer solutions for $axy +bx + cy =d$?

How can I find the integer solutions for the diophantine equations $axy +bx + cy =d$ ? the smallest particular solution ($x_0$,$y_0$) and a way to generate the rest.

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 […]

When is $991n^2 +1$ a perfect square?

What should be the value of $n$ so that the number obtained after adding $1$ to $991$ times its square is itself a perfect square? Can you please give me a few hints on this topic with a few specific reasons?

How often does $D(n^2) = m^2$ happen, where $D(x)$ is the deficiency of $x$?

Let $\sigma=\sigma_{1}$ denote the classical sum of divisors. For instance, $\sigma(12) = 1 + 2 + 3 + 4 + 6 + 12 = 28$. Let $x \in \mathbb{N}$ ($\mathbb{N}$ is the set of natural numbers/positive integers). Denote the number $2x – \sigma(x)$ by $D(x)$. We call $D(x)$ the deficiency of $x$. Now, let $m, […]