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Results tagged with prime-numbers
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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.
3
votes
Accepted
Formula with prime-density 1 in the integers
It depends a bit on what you accept as "explicit". E.g., there is a positive real number $A$ such that the integer part of $A^{3^n}$ is prime for all positive integers $n$. See Wikipedia on Mills' con …
- 39.9k
1
vote
Accepted
Extended Euclid proof and primes in form $|\prod\limits_{n \neq m} p_n -\prod\limits_{m \neq...
See Guy, Lacampagne, and Selfridge, Primes at a Glance, Mathematics of Computation, volume 48, number 177, January 1987, pages 183-202.
Abstract. Let $N = B - L$, $B \ge L$, $\gcd(B,L) = 1$, $p \mid B …
- 39.9k
3
votes
Accepted
On the set of divisors of $q-1$ and $q'-1$
$q=27=3^3$, $q'=13=13^1$. $3\mid13-1$, $13\mid27-1$. $$\pi(26)\cup3=\pi(12)\cup13=\{2,3,13\}$$
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6
votes
Accepted
Is $2^{p}-1$ prime iff for $\frac{p-1}{2}$ odd positive integers $n$ below $p$, $(n+2)\vert ...
$2^{19}-1$ is prime, but neither $(2^{19}+3)/5$ nor $(2^{19}+9)/11$ is an integer.
- 39.9k
3
votes
Accepted
Giuga and Carmichael numbers
If $n$ is Carmichael, then $a^{n-1}\equiv1\pmod n$ for all $a$ with $\gcd(a,n)=1$. If $\gcd(a,n)\ne1$, then it is clearly impossible to have $a^{n-1}\equiv1\pmod n$. So, let $n$ be Carmichael, let $r$ …
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2
votes
6
votes
Accepted
A question about primes as an additive basis
I think the heuristic evidence suggests quite the opposite, that $r_2(N)$ increases without bound. See http://www.ieeta.pt/~tos/goldbach.html for both theoretical background and computational results. …
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5
votes
Accepted
An estimate for 'almost primes'?
The number of almost-primes up to $x$ is asymptotic to $x\log\log x/\log x$. It doesn't matter whether you include the primes or not, as there are only $x/\log x$ of them. I think this gives $n\log n/ …
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4
votes
Lower bound of the number of relatively primes(each-other) in an interval
Let's turn the question around and let $f(n)$ be the number of consecutive integers required to guarantee that $n$ of them are pairwise relatively prime. E.g., $f(4)=6$ because you can find a set of 5 …
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8
votes
Prime Number Theorem w/o Complex Analysis
Another exposition of an elementary proof (that is, a proof not using complex analysis) is in Gerald Tenenbaum and Michel Mendes France, The Prime Numbers and Their Distribution, which is Volume 6 of …
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3
votes
Accepted
How does one prove that the density of unusual numbers is $\ln 2$?
A reference is Greene and Knuth, Mathematics for the Analysis of Algorithms, 3rd ed., pages 95-98. The section of the book containing those pages may be found at https://link.springer.com/content/pdf/ …
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9
votes
Accepted
Is there a 2-power-twinless prime?
Erdos proved that there is an arithmetic progression of odd numbers, none of which can be expressed as a sum of a power of two and a prime. I believe one can arrange for such an arithmetic progression …
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32
votes
111...11 base p = 111...11 base q
Richard Guy, Unsolved Problems in Number Theory, 3rd Ed., section D10, writes,
The conjecture of Goormaghtigh, that the only solutions of
$$
{x^m-1\over x-1}={y^n-1\over y-1}
$$
with $x,y>1$ and $n>m> …
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22
votes
Accepted
Prime factorization of n+1
Check out the literature on Fermat numbers, $2^{2^n}+1$. If factoring $m$ helped you factor $m+1$, these numbers would be a cinch, but they're not.
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14
votes
Floors of rationals to powers: Infinite number of primes?
The question is much too hard. Forman and Shapiro proved that $[r^n]$ is composite infinitely often for $r=3/2$ and for $r=4/3$. Dubickas and his students have found a few more results along these lin …
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