a := sum ({n,1,k}, prime_n^(1/n) – prime_(n+1)^(1/(n+1))) ,  {n,1,k} means n=1 to k. For k <= 1,000,000, observe a < 1, and a-->1.  What is the largest...
24023
John
reddwarf2956
Feb 2, 2012 4:04 pm
I do not know why the extra characters were place in, but here it is again. a := sum ({n,1,k}, prime_n^(1/n) – prime_(n+1)^(1/(n+1))) , {n,1,k} means n=1 to...
24024
djbroadhurst
Feb 2, 2012 5:45 pm
... which was an unhelpful way of defining a(k) = 2 - prime(k+1)^(1/(k+1)) At large k, the PNT gives (1 - a(k))*log(k)/k = 1 + O(log(log(k))/log(k)) David...
24025
djbroadhurst
Feb 2, 2012 6:05 pm
... but meant to write (1 - a(k)) = (log(k)/k)*(1 + O(log(log(k))/log(k)))...
24026
Chris Caldwell
primemogul
Feb 2, 2012 7:42 pm
I have a couple undergraduate students researching the history of the primality of one. For example, most of the early Greeks did not consider one to be a...
24027
Walter Nissen
nissen@...
Feb 2, 2012 8:41 pm
A most interesting subject . I think the most accessible definition of prime is geometrical , rectangularization forces linearization , ...
24028
Jack Brennen
jbrennen
Feb 2, 2012 9:45 pm
phi(1) has to equal 1 in order to preserve the multiplicative nature of phi(). In other words, If gcd(a,b) == 1, then phi(a)*phi(b)==phi(a*b). 1 is the only...
24029
Paul Leyland
paul@...
Feb 3, 2012 12:22 pm
... To the Pythagoreans, primality and irrationality were closely related ... Although odd by modern standards, it arose from their fundamentally geometric...
24030
djbroadhurst
Feb 3, 2012 5:27 pm
... But then some Germans disagreed, circa 1900: http://www.peterhug.ch/lexikon/primzahl/13_0390 http://de.academic.ru/dic.nsf/meyers/111736 Des goûts et des...
24031
djbroadhurst
Feb 3, 2012 9:28 pm
... V. A. Lebesgue designated 1 as prime on page 5 of his 1859 textbook: http://books.google.co.uk/books?id=ea8WAAAAQAAJ He is not to be confused with Henri...
24032
djbroadhurst
Feb 3, 2012 9:38 pm
... Here is a thumbnail biography: http://www.les-mathematiques.net/phorum/read.php?17,323622 ... David...
24033
Phil Carmody
thefatphil
Feb 4, 2012 1:24 am
... Fundamental, pervasive, and perverted. It is, after all, a language in which the singular 'thou' has been jetisoned for the plural 'you', and similarly the...
24034
djbroadhurst
Feb 4, 2012 1:48 am
... Is/are Manchester ******* United singular/plural? Meanwhile, back in the archives: on page 252 of http://www.gutenberg.org/ebooks/31246.html Rouse Ball...
24035
Samir Musalı
samir.musali
Feb 4, 2012 2:04 pm
We say that primes must have only 2 divisors. In other hand, if number has 2 divisors, it's prime. 2 is also prime and the first element and only even element...
24036
djbroadhurst
Feb 5, 2012 1:47 am
... Then "a" is either a product of distinct primes or the cube of a prime. David...
24037
paulunderwooduk
Feb 6, 2012 4:19 am
Hi, the conjecture now has gcd(30,N)==1 dropped. 4.X Selfridge Conjecture Non-square N>1 is prime if and only if for any integer x such that ...
24038
paulunderwooduk
Feb 6, 2012 4:32 am
... I should use Kronecker! The revision is: 4.X Selfridge Conjecture Non-square N>1 is prime if and only if for any integer x such that ...
24039
Juan Ignacio Casaubon
jicasaubon
Feb 9, 2012 11:24 am
Mensaje marcado sábado, 4 de febrero de 2012 10:23 La unidad no es primo porque tiene dos divisores en vez de cuatro; ademas, si lo fuera, seria invalido EL...
24040
Alan Eliasen
aeliasen
Feb 9, 2012 7:14 pm
... Dr. Casaubon, I hope you see that your definition as you state it doesn't clarify things. Of which primes is 1 the product? Either your statement of the ...
24041
WarrenS
warren_d_smi...
Feb 15, 2012 12:39 am
front page of NY Times just featured this paper http://eprint.iacr.org/2012/064.pdf which is hilarious! Just when you thought you'd seen the ultimate in human...
24042
djbroadhurst
Feb 15, 2012 2:26 am
... "among the 4.7 million distinct 1024-bit RSA moduli that we had originally collected, more than 12500 have a single prime factor in common" Hint to...
24043
Paul Leyland
paul@...
Feb 15, 2012 10:15 am
... Easier said than done. First, you have to get Arjen's database. Once you have it, finding factors through GCDs is a non-trivial task. That is the limit...
24044
Phil Carmody
thefatphil
Feb 15, 2012 3:09 pm
... Presumably you get it from the same places that Arjen got it? ... I think I can do it in about the cost of about a dozen half-gigabyte GCDs. Looking at...
24045
Paul Leyland
paul@...
Feb 15, 2012 5:06 pm
... That's one way. It's also consistent with my statement that it's easier said than done. ... If you think you can do it, you can easily verify whether you...
24046
djbroadhurst
Feb 15, 2012 5:20 pm
... I suggest that Ron = R.L. Rivest Whit = W. Diffie but I am not sure if they had a direct argument? David...
24047
djbroadhurst
Feb 15, 2012 5:40 pm
... R[SA] is not keen on D[H], but more relaxed about D[VW]: http://www.rsa.com/rsalabs/node.asp?id=2248 ... or Station-to-Station (STS) protocol, was...
24048
djbroadhurst
Feb 15, 2012 6:16 pm
... https://listserv.nodak.edu/cgi-bin/wa.exe?A2=ind0503&L=nmbrthry&P=R684 give a lot a help, by publicly certifying semi-primality. Might Paul care to try to...
24049
Phil Carmody
thefatphil
Feb 15, 2012 6:26 pm
... All hail the great godess Obscurity, saving the day once again! ... Well, I don't have a machine with multiple gigabytes of RAM, which would be necessary...
24050
djbroadhurst
Feb 15, 2012 6:43 pm
... Paul Leyland quite correctly remarked that Arjen's database is private. However, the GCD idea seems to work OK: ...
24051
Paul Leyland
paul@...
Feb 15, 2012 7:02 pm
... Two things. 1) Never ascribe to malice that which is adequately explained by incompetence. 2) I never said your algorithm couldn't work, only that it might...
