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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.
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> …
- 39.9k
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.
- 39.9k
16
votes
A conjecture by Euler about $8n+3$
Barry Mazur discussed this problem in his presentation of January 2012, Why is it plausible? Mazur says there that it is still unsettled.
- 39.9k
15
votes
Do consecutive integers have a big prime factor?
Shorey and Tijdeman, "On the greatest prime factors of polynomials at integer points", Compositio Mathematica, tome 33, no 2 (1976), p. 187-195, MR424681, Zbl 0338.10040 note that if $f$ is a polynomi …
- 39.9k
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 …
- 39.9k
14
votes
Priming for the primes
The nonzero characteristics of fields are precisely the prime numbers.
14
votes
Accepted
What are examples of problems we know how to solve for primes (or prime powers), but not for...
There is a projective plane of order $N$ for every prime power $N$. The existence of projective planes of other orders is an open question; in particular, it is not known whether there is a projective …
12
votes
Priming for the primes
The abelian simple groups are precisely the groups of prime order.
12
votes
Asymptotic density of k-almost primes
According to Dickson's History, Gauss, in a manuscript of 1796, stated empirically that the number $\pi_2(x)$ of integers $\le x$ which are products of two distinct primes, is approximately $x\log\log …
- 39.9k
11
votes
What are the connections between pi and prime numbers?
There are a few formulas relating $\pi$ to arithmetic functions. For example, if $\sigma(n)$ is the sum of the divisors of $n$, then $\sum_1^n\sigma(n)=\pi^2n^2/12+O(n\log n)$. If $d(n)$ is the number …
9
votes
Accepted
On the distribution of roots modulo primes of an integral polynomial
In my paper, Polynomial congruences and density, Mathematics Magazine 80 (2007) 299-302, I cite the result of Christopher Hooley, On the distribution of roots of polynomial congruences, Mathematika 11 …
- 39.9k
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 …
- 39.9k
9
votes
Can exist a positive integer number $x$ such that $a_1=x$ and $a_n=2a_{n-1}+1$ are not prime...
From https://en.wikipedia.org/wiki/Riesel_number "a Riesel number is an odd natural number $k$ for which ${\displaystyle k\times 2^{n}-1}$ is composite for all natural numbers $n$."
"In 1956, Hans Rie …
- 39.9k
8
votes
Accepted
Optical methods for number theory?
There's a way to use physics to calculate the digits of $\pi$. I quote from https://math.stackexchange.com/questions/138289/intuitive-reasoning-behind-pis-appearance-in-bouncing-balls
Let the mass of …
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 …
- 39.9k