Articles of prime numbers

If $n$ is an odd pseudoprime , then $2^n-1$ is also odd pseudoprime

I have some problems understanding the following proof: Definition: A composite number $n \in \mathbb{N}$ is called pseudo prime if $n \mid 2^{n-1}-1$ holds. Theorem: If n is a odd pseudo prime number, then $2^n-1$ is also an odd pseudo prime number, too. Proof: Let n be an odd pseudo prime number. Then we get […]

Sequence Of Primes

Hello I have a basic number theory question. I want to find a list of primes of the form a, a + d, a + 2d, … , a + 5d So a sequence of at least 6 or greater if I want to select a = 101 then what would I choose as my […]

A question about an asymptotic formula

I’ve been told that the asymptotic formula $\pi(x+y)-\pi(x)\sim y/\ln x$ holds for $y\ge x^{1/2+\varepsilon}$ if Riemann’s hypothesis is true, but I was unable to find a journal reference for this. Does anybody know of any journal reference or any other source where I can find this conditional result?

Prove that every number between two factors of primes is composite.

I am looking for some help with this problem: Let $p_1,p_2,\dots,p_{n+1}$ be the first $n+1$ primes in order. Prove that every number between $p_1p_2p_3\dots p_n+1$ and $p_1p_2p_3\dots p_n + p_{n+1}-1$ is composite (inclusive of the second term). How does this show that there are gaps of arbitrary length in the sequence of primes? I know […]

Can this approach to showing no positive integer solutions to $p^n = x^3 + y^3$ be generalized?

The following problem is a $2000$ Hungarian Olympiad question. Find all primes $p$ such that: $$p^n = x^3 + y^3$$ The answer is that there are only $2$ solutions: $2^1 = 1^3 + 1^3$ $3^2 = 2^3 + 1^3$ Here’s the argument: Assume $p \ge 5$ with $x,y,p,n$ positive integers and $p^n = x^3 + […]

Why do the first spikes in these plots point in opposite directions?

With the following Mathematica program: Clear[x, n, nn] nn = Prime[13] + 1; A = -Sum[Re[x^N[ZetaZero[n]]], {n, 1, 50}]; Plot[A, {x, 0, nn}, PlotRange -> {-80, 120}] I plotted the function: $$f(x)=-\sum _{n=1}^{50} \Re\left(x^{\rho _n}\right)$$ Where $\rho _n$ is the $n$-th Riemann zeta zero. With this second program: Clear[n, k, x, nn] nn = Prime[13] […]

Proof concerning Mersenne primes

Is this proof acceptable ? Lemma Let $M_p=2^p-1$ with $p$ prime and $p>2$ , thus If $M_p$ is prime then $3^{\frac{M_p-1}{2}} \equiv -1 \pmod {M_p}$ Proof Let $M_p$ be a prime , then according to Euler’s criterion : $3^{\frac{M_p-1}{2}} \equiv \left(\frac{3}{M_p}\right) \pmod {M_p}$ , where $\left(\frac{3}{M_p}\right)$ denotes Legendre symbol . If $M_p$ is prime then […]

3 primes conjecture

let be $ p,q,r $ prime numbers AND ‘n’ an integer is then true that we can always look for p,q,r and an integer n so $$ p^{n}+q=r $$ $ 5+2=7$ $ 2^{3}+3=11 $ $ 3^{4}+2=83 $ abnd so on

Pythagorean triples with additional parameters

I want to find solution in $\mathbb{Z}$ to the following quadratic Diophantene equation: $$na^2 + kb^2 = c^2$$ where $n,k,a,b,c \in \mathbb{Z}$, $n,k > 0$ and $(n,k) = 1$ I know that for some this won’t work for all values of $n$ and $k$ that satisfy the upper condition, but anyone know what will be […]

If $2$ divides $p^2$, how does it imply $2$ divides $p$?

I’m trying to understand a proof by contradiction. It’s proving by contradiction that $\sqrt2$ isn’t rational. (It’s a standard proof involving $\sqrt2=\frac{p}{q}$, where $p,q$ are already simplified integers) There’s a part of the proof that reads: Hence $2|p^2$ But then $2|p$ because $p$ is prime. Could someone explain how this is implied? Thanks Edit: Would […]