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66 votes
3 answers
6k views

Chebyshev polynomials of the first kind and primality testing

Can you provide a proof or a counterexample for the claim given below ? Inspired by Agrawal's conjecture in this paper and by Theorem 4 in this paper I have formulated the following claim : Let $...
Pedja's user avatar
  • 2,661
65 votes
6 answers
14k views

What is the simplest proof that the density of primes goes to zero?

By density of primes, I mean the proportion of integers between $1$ and $x$ which are prime. The prime number theorem says that this is asymptotically $1/\log(x)$. I want something much weaker, namely ...
Kim's user avatar
  • 4,164
32 votes
3 answers
8k views

Ideas in the elementary proof of the prime number theorem (Selberg / Erdős)

I'm reading the elementary proof of prime number theorem (Selberg / Erdős, around 1949). One key step is to prove that, with $\vartheta(x) = \sum_{p\leq x} \log p$, $$(1) \qquad\qquad \vartheta(x) \...
Basj's user avatar
  • 587
20 votes
3 answers
3k views

What is the simplest proof that the density of coprime pairs does not go to zero?

By density of coprime pairs, I mean the proportion of pairs integers between $1$ and $x$ which are coprime. This is known to be asymptotically $1/\zeta(2)$. I want something much weaker, namely that ...
domotorp's user avatar
  • 19k
17 votes
0 answers
891 views

An elementary proof that, for every fixed $n \in \mathbf N^+$, there are infinitely many primes $\equiv -1 \bmod n$

This morning, I made a comment to a comment to a question of Ayman Moussa, only to point out that, among many others, there is an elementary proof of Dirichlet's theorem on the existence of infinitely ...
Salvo Tringali's user avatar
14 votes
1 answer
1k views

Nonstandard proofs of the fundamental theorem of arithmetic

Thirty or so years ago, someone showed me a clever proof of the Fundamental Theorem of Arithmetic that did not make use of the lemma "If $p\mid ab$ then $p\mid a$ or $p\mid b$". I'm unable ...
James Propp's user avatar
  • 19.7k
13 votes
3 answers
1k views

At what point would an elementary generalization of Bertrand's Postulate be interesting?

I know that in 1952 Jitsuro Nagura was able to show that there is always a prime between $k$ and $\frac{6k}{5}$ for $k > 24$. At what point would an improvement on Nagura's result be interesting? ...
Larry Freeman's user avatar
7 votes
1 answer
462 views

Primality test for $N=2^a3^b+1$

Can you prove or disprove the following claim: Let $N=2^a3^b+1$ , $a>0 , b>0$ . If there exists an integer $c$ such that $$c^{(N-1)/3}-c^{(N-1)/6} \equiv -1 \pmod{N}$$ then $N$ is a prime. You ...
Pedja's user avatar
  • 2,661
6 votes
1 answer
826 views

Going beyond the Sylvester and Schur theorem with regard to $x,x+1,\dots,x+n-1$

I was recent reading through Paul Erdos's classic elementary proof of Sylvester-Schur. It occurred me that there is a simple argument that when $x$ is sufficiently large and if $p_i$ represents the $...
Larry Freeman's user avatar
5 votes
0 answers
586 views

Primality test for specific class of generalized Fermat numbers

Can you provide a proof or a counterexample for the following claim : Let $P_m(x)=2^{-m}\cdot((x-\sqrt{x^2-4})^m+(x+\sqrt{x^2-4})^m)$ . Let $F_{p,n}= (2p)^{2^n}+1 $ where $p$ is a prime number ...
Pedja's user avatar
  • 2,661
3 votes
1 answer
383 views

Primality test for specific class of $N=k \cdot b^n-1$

This question is successor of Compositeness test for specific class of $N=k \cdot b^n-1$ . Can you provide a proof or a counterexample to the following claim : Let $P_m(x)=2^{-m}\cdot \left(\left(...
Pedja's user avatar
  • 2,661
3 votes
0 answers
1k views

Formula for $\pi$ involving exponents of Mersenne primes

Can someone provide a proof for the following claim? $$\pi=\dfrac{S_0S_2}{M_3M_5} \cdot\left(\displaystyle\prod_{p \equiv 1 \pmod{4} } \frac{p}{p-1}\right) \cdot \left(\displaystyle\prod_{p \equiv 3 \...
Pedja's user avatar
  • 2,661
2 votes
1 answer
365 views

Primality test for specific class of $N=8kp^n-1$

My following question is related to my question here Can you provide a proof or a counterexample for the following claim : Let $P_m(x)=2^{-m}\cdot \left(\left(x-\sqrt{x^2-4}\right)^{m}+\left(x+\...
Pedja's user avatar
  • 2,661
1 vote
0 answers
93 views

Primality test for specific class of $N=12k \cdot 5^n-1$

Can you provide a proof for the following claim: Let $P_m(x)=2^{-m}\cdot \left(\left(x-\sqrt{x^2-4})^m+(x+\sqrt{x^2-4}\right)^m\right)$ . Let $N= 12k \cdot 5^{n} - 1 $ where $n\ge3$ , $12k <5^n$ ...
Pedja's user avatar
  • 2,661