Free Republic

I wondered if any FRiends were interested in prime numbers. You'll know that prime numbers are integers, and those which have only two factors, 1 and themselves. Except for 2, all other primes are odd numbers. By convention, the number 1 is not considered prime (but this is strictly a matter of definition because 1 does not entirely contradict the premise of being a multiple of 1 and itself). I cannot post this entire thread in one post so if you come across it in an early stage, be aware that a number of posts will follow rapidly until I...

A previously unnoticed property of prime numbers seems to violate a longstanding assumption about how they behave. o mathematicians have uncovered a simple, previously unnoticed property of prime numbers — those numbers that are divisible only by 1 and themselves. Prime numbers, it seems, have decided preferences about the final digits of the primes that immediately follow them. Among the first billion prime numbers, for instance, a prime ending in 9 is almost 65 percent more likely to be followed by a prime ending in 1 than another prime ending in 9. In a paper posted online today, Kannan Soundararajan...

The great mystery since the NSA and other intelligence agencies’ cyber-spying capabilities became watercooler fodder has not been the why of their actions, but the how? For example, how are they breaking crypto to decode secure Internet communication? A team of cryptographers and computer scientists from a handful of academic powerhouses is pretty confident they have the answer after having pieced together a number of clues from the Snowden documents that have been published so far, and giving the math around the Diffie-Hellman protocol a hard look. The answer is an implementation weakness in Diffie-Hellman key exchanges, specifically in the...

RSA-640 Factored By Eric W. Weisstein November 8, 2005--A team at the German Federal Agency for Information Technology Security (BSI) recently announced the factorization of the 193-digit number 310 7418240490 0437213507 5003588856 7930037346 0228427275 4572016194 8823206440 5180815045 5634682967 1723286782 4379162728 3803341547 1073108501 9195485290 0733772482 2783525742 3864540146 9173660247 7652346609 known as RSA-640. The team responsible for this factorization is the same one that previously factored the 174-digit number known as RSA-576 (MathWorld headline news, December 5, 2003) and the 200-digit number known as RSA-200 (MathWorld headline news, May 10, 2005). RSA numbers are composite numbers having exactly two prime factors (i.e.,...