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Results tagged with prime-numbers
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user 3206
Questions where prime numbers play a key-role, such as: questions on the distribution of prime numbers (twin primes, gaps between primes, Hardy–Littlewood conjectures, etc); questions on prime numbers with special properties (Wieferich prime, Wolstenholme prime, etc.). This tag is often used as a specialized tag in combination with the top-level tag nt.number-theory and (if applicable) analytic-number-theory.
4
votes
Accepted
Conjecture:if $i<j$,then $\pi(p[i]+i)-i<=\pi(p[j]+j)-j+1$
I believe your inequality $p(i+1) \leq p(i) + i$ is true, but that there is no short and elementary proof. It follows from inspection and some known results on the length of gaps between primes, cf. …
- 3,209
6
votes
Accepted
A conjecture on the prime counting function
Look for a large gap in the distribution of primes. For this conjecture, the gap between $n!+2$ and $n!+n$ will suffice. Set $y = n!+2$ (which is composite) and set $m$ (which will be $\frac{x+y}{2} …
- 3,209
1
vote
The number of totatives to the nth primorial, in an interval shorter than the nth primorial
Since I haven't posted this construction on MathOverflow
(only referred to an ArXiv posting at Erik Westzynthius's cool upper bound argument: update? ),
let me show that one can have intervals of th …
- 3,209
3
votes
Accepted
Relative-totient function (2nd attempt)
I will not comment on the soundness of the approach, but I will render a subjective opinion: I don't like it. One of the reasons is that I have found most people do not have a good understanding of …
- 3,209
2
votes
Smallest constant so that there are at least $n/\log_2{n}$ primes between $n$ and a constant...
Indeed, the suggestion given in the other thread is quite appropriate. Use a lower bound from Dusart for $\pi(cn)$, and an upper bound for $\pi(n)$, and you want the difference between these bounds t …
- 3,209
2
votes
Approximating a real by a ratio of primes
Here is an idea which might help show that the smallest prime is not much larger than the smallest integer needed.
One of the processes I like to use is the mediant $\frac{a+c}{b+d}$ of two positive …
- 3,209
1
vote
Small quotients of smooth numbers
Here is some (unverified) computational data, which I encourage others to extend.
I look at the squarefree p-smooth numbers (being $2^k$ in number for $p$ being the $k$th
prime) and compared successi …
- 3,209
2
votes
Accepted
Distribution of composite numbers
I would like to maintain my basic position on the other post: that this forum does not do well with questions that frequently change. Since the last version has hit rather close to home, I will remar …
- 3,209
2
votes
Distribution of composite numbers
I am afraid the weak version (at this writing) involving density being less than $1/(x-2)$ doesn't work either. Simply pick $d_i$ and $x$ near but less than $\sqrt{N}$, and
arrange $K$ and $L$ so tha …
- 3,209
8
votes
How did Cole factor $2^{67}-1$ in 1903?
I imagine it took Cole longer than he said. If I were to undertake the project, here is how I would proceed:
I would start sifting the set of numbers {134k + 1} for primes. One can modify the Sieve …
- 3,209