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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."
https://www.quantamagazine.org...
7 did it.
Stop anthropomorphizing prime numbers. They hate that!
Only 65%? Pft. In base 2, every prime number is 100% likely to be followed by a prime ending in 1.
>Two mathematicians have uncovered a simple, previously unnoticed property of prime numbers — those numbers that are divisible only by 1 and themselves.
Did anyone else LOL when they read the first sentence. My first thought was who wouldn't notice primes are only divisible by 1 and themselves it's the definition, duh.
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.
Hi all, your input please on this paragraph in the linked article:
"Soundararajan was drawn to study consecutive primes after hearing a lecture at Stanford by the mathematician Tadashi Tokieda, of the University of Cambridge, in which he mentioned a counterintuitive property of coin-tossing: If Alice tosses a coin until she sees a head followed by a tail, and Bob tosses a coin until he sees two heads in a row, then on average, Alice will require four tosses while Bob will require six tosses (try this at home!), even though head-tail and head-head have an equal chance of appearing after two coin tosses."
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.
This is how I see it, the statement above came from a mathematician so I am probably making a mistake in my reasoning but I don't see where.
Any input?
After a prime ending in 2 or 5, there has to be at least a billion primes before another one can end in 2 or 5.
Shachar
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.
Great, now he has to come up with a new one.
Private keys are supposed to be kept secret, dammit!
We used to have a Bill of Rights. Now, with the rights gone, all we have left is the bill.
Everyone with at least a passing interest in cryptography and computer security does. Primes is basically what we rely on in these fields.
Quite seriously, every time someone comes up with a claim that something can be done "more easily", "more efficiently" or generally "faster" in a field that remotely touches on prime numbers, you can see the ripples in the fabric of spacetime from cryptographers shaking in their boots.
We used to have a Bill of Rights. Now, with the rights gone, all we have left is the bill.
Oh for FFS. The US did not bait Japan with the fleet at Pearl Harbor. That was the home port long before they embargoed Japan and if they had move the fleet to the west coast then Japan would have just attacked the Philippines and you would be talking about how suspicious it was that US moved their fleet away from the Philippines after they embargoed Japan. If you're trying to start a war you don't make your opening move laying your most important assets out for the enemy to wipe out, you stage something that you control. Like Poland "attacking" Germany or having WMD's.
The fleet was lined up in Pearl Harbor because the US military had been stripped down after WWI and was run by politically motivated officers who often didn't know their arse from a hole in the ground (the prima donna General MacArthur being the prime example). Once war broke out those officers were quickly squirreled away and better officers brought up from the ranks who actually knew what they were doing. Most people don't realize that Eisenhower was a Major with no combat experience who was quickly promoted up to General because Marshall knew he had a sharp strategic mind (and could put up with politics, which he demonstrated as a member of MacArthur's staff). The German general staff couldn't (and for a time didn't) believe he was made Allied Supreme commander instead of Patton (who was tactically brilliant but questionable strategically and certifiably insane to boot).
Last time I looked at primes in binary I noticed a 100% chance that the next one ended in 1. No I am not trolling you I'm just making a point, go look at the primes in different bases and see what you notice.
Intuitively it makes sense. Assume the first H has been tossed. For Alice, she fails by tossing another H. However, this second H can be the first H of a successful HT sequence, so in failure there is a silver lining - she's halfway to success and can stop after tossing a single T. Full sequence: HHT.
For Bob, after tossing the first H, tossing a T means he has to start over. He needs to toss another H first, followed by yet another H to succeed. His task is harder. Full sequence: HTHH.
Except, there are no numbers ending with the digit of '9' in base 7. Quite a conspiracy!
the fact that it's extremely difficult to determine the factors of large prime numbers is the basis for a lot of cryptography
I think you might have jumbled your words.
It's exceptionally easy to determine the factors of any large prime number because there are only two; the number one the number itself. Determining the prime factors of a large, non-prime number, on the other hand, is a challenge.
The twin prime conjecture is independent of the base, so the base doesn't matter for it to be true or false.
I would find this surprising, since in a base 2 system every prime number ending in 1 is followed by a prime number ending in 1.
You missed the absolutely critical corollary that restores balance to the force: after Bob succeeds, he's already halfway to his next success where after Alice succeeds, she needs to snooze for one toss before she's back in the game, where apparently the game involves some gender-swap role play.
It's so totally male to cease thinking the problem through after attaining the initial success condition.
I think I could teach a very interesting grade XI math class.
Corollary: I would end up behind bars.
The twin prime conjecture is independent of the base, so the base doesn't matter for it to be true or false.
I would find this surprising, since in a base 2 system every prime number ending in 1 is followed by a prime number ending in 1.
In base 2, every prime ending in 0 is also followed by a prime ending in 1.
Not necessarily. Even though PI isn't a rational number, it doesn't mean that it necessarily contains any given subsequence. Let me take another non-rational number: 0.12112111211112... It won't ever contain the prime number 3, nor will it contain the complete works of Shakespeare. PI is similarly constrained to a specific pattern. I agree that an infinite un-patterned sequence will contain such sequences, but whether you include or exclude the axiom of choice will determine whether such numbers exist or not. Whether or not PI actually contains all primes as subsequences, I don't know. I'd suggest you present a proof one way or the other.
10 is divisible by 1,2, 5, and 10, so how is it prime?
I've fallen off your lawn, and I can't get up.