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Results tagged with prime-numbers
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user 127690
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.
30
votes
Why is integer factoring hard while determining whether an integer is prime easy?
In the particular case of primality testing, primality testing is easier than factoring mainly because $p$ is prime if and only if $\mathbb{Z}/p\mathbb{Z}$ forms a field. This has a lot of consequence …
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10
votes
Accepted
A weaker version of the Brocard's Conjecture
Theorem: For any constant $c$ there are infinitely many primes $p_k$ such that there are at least $c$ primes between $p_k^2$ and $p_{k+1}^2$.
Proof: Fix a $c$. Assume that for sufficiently large $k$ …
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10
votes
Are there infinitely many $n$ such that $n!-1$ and $n!+1$ are prime numbers?
Almost certainly not. But note that if it were true, proving it would be extremely tough. We can't even prove now that there are infinitely many primes of the form $n!+1$ or that there are infinitely …
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10
votes
Is every prime the largest prime factor in some prime gap?
Heuristically this should be the case. For any prime p greater than 5, consider the set of numbers of the form $2^a3^b5^c p \pm 1$. The "probability" that one of of these is prime should be about $$\ …
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10
votes
Accepted
Error term in Mertens' third theorem
There's been a lot of work on unconditional results of this sort.
Rosser and Schoenfeld showed in a 1962 paper that one can take
$$\dfrac{e^{-\gamma}}{\log x} \left(1- \frac{1}{2\log^2 x} \right) < …
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7
votes
Accepted
The equivalent proposition of Legendre's conjecture
Your conjecture for sufficiently large $n$ is implied by Cramer's conjecture. In general though, conjectures like this unless they are coming from some specific application aren't that interesting. It …
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6
votes
Is such a generalization of the twin prime conjecture known?
As written, this is hopeless false. $(2,3)$ is an obvious counterexample. Slightly less trivially, $(3,5,7)$ is a counterexample.
One can correct for these, and if one does so, one gets a version of t …
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5
votes
1
answer
344
views
Quadratic Diophantine equations with all values prime
Given a quadratic Diophantine equation over the integers in two variables, can we say much about when it has only finitely many solutions with the additional assumption that both variables are prime?
…
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5
votes
The smallest solution to $2^{2k}-1=\text{powerful}$
Note that if there are only finitely many non-Wieferich primes, then that would imply that there is such a $k$. If there were only finitely many non-Wieferich primes, one could make a sequence based o …
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4
votes
2
answers
360
views
A specific Diophantine equation restricted to prime values of variables.
Consider the following Diophantine equation $$x^2+x+1=(a^2+a+1)(b^2+b+1)(c^2+c+1).$$ Assume also that $x,a,b,c,a^2+a+1,b^2+b+1, c^2+c+1$ are all primes. We'll call such a quadruplet $(x,a,b,c)$ a trip …
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4
votes
Accepted
Existence of some prime $x_k | k > 2$ in $x_n = x_{n-1} + x_{n-2}$ whenever $x_1$ is coprime...
As Gerhard Paseman said in his comment, there are counterexamples. This is discussed in A3 of UPINT. According to the third edition, the smallest known counterexample as of the printing (in 2003) is $ …
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4
votes
Is every integer $\ge 312$ the sum of two integers with triangular divisors?
Not a complete answer, but it made sense I think to summarize the comment thread above:
Theorem: Every sufficiently large positive integer is expressible as a sum of three numbers which have triangul …
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4
votes
Accepted
New experiments involving Ramanujan primes: Benford's law
If I am following what is being asked, the answer is no.
Set $R$ to be the set of Ramanujan primes. Let $R_d$ be the set of Ramanujan primes with lead digit $d$. For a set of positive integers integer …
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3
votes
Accepted
A conjecture about an inequality that involve Ramanujan primes
Not a complete answer, but a bit too long for a comment: Conjecture 1 is very likely to be very difficult if true. The corresponding conjecture for general primes is open. Let $p_n$ be the $n$th prime …
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3
votes
Accepted
p2 - p1 = 2n for every 2n
The two relevant papers for Chen's original proof are Chen, J.R. (1973). "On the representation of a larger even integer as the sum of a prime and the product of at most two primes". Sci. Sinica. 16: …
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