All Questions
7 questions
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 $\vartheta(x) = \sum_{p\leq x} \log p$,
$$(1) \qquad\qquad \vartheta(x) \...
- 587
13
votes
3
answers
1k
views
At what point would an elementary generalization of Bertrand's Postulate be interesting?
I know that in 1952 Jitsuro Nagura was able to show that there is always a prime between $k$ and $\frac{6k}{5}$ for $k > 24$.
At what point would an improvement on Nagura's result be interesting? ...
- 1,049
7
votes
2
answers
636
views
How to use the Prime Number Theorem in order to prove Selberg's Formula?
I`m reading Melvin B. Nathanson's "Elementary Methods in Number Theory"
and I can't think of a way of deducing Selberg's formula (9.3) from the prime number theorem.
This is one of the tasks ...
- 129
6
votes
1
answer
826
views
Going beyond the Sylvester and Schur theorem with regard to $x,x+1,\dots,x+n-1$
I was recent reading through Paul Erdos's classic elementary proof of Sylvester-Schur. It occurred me that there is a simple argument that when $x$ is sufficiently large and if $p_i$ represents the $...
- 1,049
6
votes
0
answers
381
views
A possible variant of Zagier's one-sentence proof for Fermat's sum of two squares theorem?
Is it possible to modify Zagier's one-sentence proof of Fermat's sum of two squares theorem (see here) to prove certain non-trivial cases of Jacobi's four-square theorem (see here)?
Let $p$ be a prime ...
- 81
3
votes
0
answers
1k
views
Formula for $\pi$ involving exponents of Mersenne primes
Can someone provide a proof for the following claim?
$$\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 \...
- 2,661
1
vote
0
answers
127
views
Some property of the greatest prime factor
Let $n$ be a positive integer $\geq 2$ et denote by $ P^{+}(n)$ the greatest prime factor of $n$ my question is as follows:
If $a$ and $b$ are two numbers, is there any method to express or to bound $...
- 1,257