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Mathematicians Discover Prime Conspiracy (quantamagazine.org)
An anonymous reader writes with an intriguing story at Quanta Magazine, which begins: Two 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 and Robert Lemke Oliver of Stanford University present both numerical and theoretical evidence that prime numbers repel other would-be primes that end in the same digit, and have varied predilections for being followed by primes ending in the other possible final digits. "We've been studying primes for a long time, and no one spotted this before," said Andrew Granville, a number theorist at the University of Montreal and University College London. "It's crazy."
Only 65%? Pft. In base 2, every prime number is 100% likely to be followed by a prime ending in 1.
"We've been studying primes for a long time, and no one spotted this before," said Andrew Granville, a number theorist at the University of Montreal and University College London. "It's crazy."
I can tell you that it's not crazy, the information has simply been occulted ("occult" means to hide). Why do you think RSA selects two consecutive prime numbers? The answer is known to the NSA, and now you do too.
I bet you think that we actually thought there would be WMDs in Iraq, that we couldn't have stopped 9/11, and that we didn't know it was strategically folly to deploy such a fleet in close quarters in Perl Harbor, meanwhile embargoing Japan...
Besides, "it's crazy" to think otherwise, eh?
I wonder if this has anything to do with Twin primes. If a prime ends in 9, then its twin will end in 1, and so we should expect primes ending in 9 to more often be followed by primes ending in 1. The number of twin primes is believed to be infinite, but they get more sparse as you go towards infinity (proportional to 1/(ln(n)^2)), even faster than primes (proportional to 1/ln(n)), so if they are responsible for the bias, then the bias should diminish as you go up.
... or base 16, or base 7, or base 64, for that matter?
Will they still exhibit the 'twin prime' / 'prime number conspiracy' phenomena?
This seems wrong to me. Both Alice and Bob have equal chance of rolling a head, hence on average they will need the same number of tries to arrive at a head toss; and since coins dont have memory, the next toss has equal chance of being head or tail. So I do expect the chances of head head and head tail to be the same.
Yes, it's called a veridical paradox. That's something that seems impossible but is nonetheless true. You can verify it by flipping a coin, or running a computer simulation using a good random number generator.
If I can be modded down for being a troll, can I be modded up for being an orc, or a balrog?
What's this base 1010 drama? Everyone knows in binary ALL primes end in "1".
Jokes aside, the fact that there's plenty of bases to choose from means that what they are really talking about is the modulo remainders of primes having a pattern- and modulo division and primes have had a pretty flirty relationship. Unquestionably interesting. The thing with the prime number set is that it's immutable- a set of fixed numeric stars shining the same light since before time began, and yet even with that constancy, many functions involving the prime number web have proven frustrating to calculate for large values- there's hardly any shortcuts compared to the integer math you run into on a daily basis.