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Results tagged with prime-numbers
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user 3684
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.
1
vote
Why is integer factoring hard while determining whether an integer is prime easy?
Determining whether the multiplicative group of ${\bf Z}/n{\bf Z}$ can be generated by a single element is easy (pretty much as easy as determining whether $n$ is prime), finding a single element whic …
- 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
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 …
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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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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 …
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14
votes
12
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1
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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 …
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3
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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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4
votes
What did Yu Jianchun discover about Carmichael numbers?
On page 136 of Cai Tianxin, The Book of Numbers, it says,
In 2015 the Chinese amateur, Jianchun Yu, who was then a packer in a bookstore warehouse in Hangzhou found that all the numbers like $(6k+1 …
- 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 …
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2
votes
Have you seen this prime distribution before?
The numbers you give match "Second-order Eulerian triangle" at OEIS.
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9
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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 …
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6
votes
What are conjectures that are true for primes but then turned out to be false for some compo...
An aliquot sequence is a recursive sequence in which each term is the sum of the proper divisors of the previous term. Catalan conjectured that no aliquot sequence is unbounded. The conjecture is triv …
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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 …
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