Articles of number theory

Gauss sums and module endomorphisms

Let $p$ be an odd prime and $n \in \mathbb{N}$. Let $a,b,c$ be arbitrary integers such that $ab \neq 0$. We write $p^{\alpha}A = a$ and $p^{\beta}B = B$ for some $\alpha, \beta \in \mathbb{N}_0$ and $A, B \in \mathbb{Z}$ are such that $(AB,p)=1$. Further, we assume that $c$ is chosen such that $p^{\alpha} \mid […]

The equation $x^n+y^n+z^n=u^n+v^n+w^n=p$, where $p$ is prime.

I’m looking for positive integer solutions to $x^n+y^n+z^n=u^n+v^n+w^n=p$, where $p$ is prime. Background. I was looking at “Primes which are the sum of three nonzero 8th powers” and the like, and wondered if there are rules similar to those for a prime being the sum of two squares. My efforts. I’ve found primes of […]

Defining the integers and rationals

What axiomatic system is most commonly used by modern mathematicians to describe the properties of the integers and the rationals? Properties like $a+0=a$, $a*1=a$, $a+b=b+a$, Also given these axioms, can I show that a contradiction cant be derived from them, such as $0=1$.. etc

Prove that $x^2+1$ cannot be a perfect square for any positive integer x?

I started this problem by trying proof by contradiction. I first noted that the problem stated that $x$ had to be a positive integer, and thus $x=0$ could not be a solution. I then assumed that $x^2+1=n^2$ for some integer $n$ other than $1$. From here I have tried various methods, to no avail: Factoring: […]

Maximal gaps in prime factorizations (“wheel factorization”)

A wheel factorization is when you remove all the multiples of primes (up to a prime number P) from the product of all primes up to and including P. Examples: For P=5, you remove all the multiples of 2,3 and 5 from 1 to 2x3x5=30 You are then left with the set {1, 7, 11, […]

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 […]