Articles of number theory

Are $4ab\pm 1 $ and $(4a^2\pm 1)^2$ coprime?

Let $a\ne b$ be two positive integers. Are $4ab+1$ and $(4a^2+1)^2$ coprime always? Can you find $a$ and $b$ such that they are not coprime? Edit: It has been proved that $4ab-1$ is not a divisor of $(4a^2-1)^2$. Are $4ab-1$ and $(4a^2-1)^2$ always coprime?

An identity involving the Pochhammer symbol

I need help proving the following identity: $$\frac{(6n)!}{(3n)!} = 1728^n \left(\frac{1}{6}\right)_n \left(\frac{1}{2}\right)_n \left(\frac{5}{6}\right)_n.$$ Here, $$(a)_n = a(a + 1)(a + 2) \cdots (a + n – 1), \quad n > 1, \quad (a)_0 = 1,$$ is the Pochhammer symbol. I do not really know how one converts expressions involving factorials to products of the Pochhammer […]

Show $\mathrm{gcd}(7a+5,4a+3)=1$.

I have been trying to do this problem for a couple of days for better or worse. I suppose that $d = \mathrm{gcd}(7a+5,4a+3)$. Since $4a+3=2(2a+1)+1$ it must be that $d$ is odd. I know that $d|(7a+5)$ and $d|(4a+3)$ so $d|(11a+8)$. I also know that $\mathrm{gcd}(a,b) \leq \mathrm{gcd}(a+b,a-b)$ but I haven’t been able to get anything […]

regularization of sum $n \ln(n)$

I was testing out a few summation using my previous descriped methodes when i found an error in my reasoning. I’m really hoping someone could help me out. The function which i was evaluating was $\sum_{n=1}^{\infty} n\ln(n)$ which turns out to be $-\zeta'(-1)$. This made me hope i could confirm my previous summation methode for […]

Squarefree binomial coefficients.

At $n=23$, all binomial coefficients are squarefree. Is this the largest value for $n$ where this is the case? Edit A plot up to $n=50$: A plot up to $n=500$: plotted against $n+1$ and $\frac{112}{\sqrt{239}}\sqrt{n}$ and the same up to $n=2000$: Do these bounds hold for all $n$? (Clearly, the bound $n+1$ holds for all […]

Where does googolplum lie in the fast growing hierarchy?

Here : Saibian presents the largest number coined by Andre Joyce, googolplum. It should lie at the $f_{\omega+2}$-level in the fast growing hierarchy, but where exactly ? In other words : Between which tight bounds does googolplum lie ? Is it really at level $f_{\omega+2}$ ? If yes, what is the smallest number $n$, […]

Solving Pell's equation(or any other diophantine equation) through modular arithmetic.

Let us take a solution of Pell’s equation ($x^2 – my^2 = 1$) and take any prime $p$. Then we have found a solution of the Pell’s equation mod $p$. Now, conversely, for any prime $p$, we can find a solution of Pell’s equation. My question is whether we can use these solutions mod $p$ […]

Sum of greatest common divisors

As usually, let $\gcd(a,b)$ be the greatest common divisor of integer numbers $a$ and $b$. What is the asymptotics of $$\frac{1}{n^2} \sum_{i=1}^{i=n} \sum_{j=1}^{j=n} \gcd(i,j)$$ as $n \to \infty?$

Checking a fundamental unit of a real quadratic field

I just want to check whether I have got the fundamental unit of a certain real quadratic field, but I can’t find how. For instance, if I am working in $\mathbb{Q}(\sqrt{2})$ then $\mathcal{O}_K=\mathbb{Z}[\sqrt{2}]$ and so the fundamental unit is of the form $a+b\sqrt{2}$. I suppose the fundamental unit is $1+\sqrt{2}$, ie. $a=1, b=1$. I know […]

Liouville's number revisited

Liouville’s Number is defined as $L = \sum_{n=1}^{\infty}(10^{-n!})$. Does it have other applications than just constructing a transcendental number? (Personally, I would have defined it (as “Steven’s Number” :-)) as binary: $S = \sum_{n=1}^{\infty}(2^{-n!})$, since each digit can only be “0” or “1”: the corresponding power of 2 (instead of 10) included or not. Since […]