but has since been put to practical use in cryptography
So, how long before the FBI asks to ban prime numbers?
The Great Internet Mersenne Prime Search (GIMPS) has announced the discovery of a new largest Mersenne prime number, 277,232,917 -1. The figure, viewable here (.zip file), was found by GIMPS' network of volunteer prime hunters, and is the 50th Mersenne prime discovered. It comprises 23,249,425 digits, and is 910,807 digits …
I suggest that mere mortals are required to use a non-decimal numbering system, possibly based on irrationals that would make finding primes much more interesting.
Or maybe requiring all CPUs to only deal with floating-point numbers (no integer calculations) which would kill the processing time and eradicate any exactitude. Of course, the powers-that-be would be allowed to do integer arithmetic.
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"Whenever we had to find prime numbers at school the ones I 'found' were usually divisible by 3."
yeah too much busywork, doing all of those divisions. Imagine doing it WITHOUT an electronic calculator. That would be when _I_ was in school... through Jr. High anyway.
thinking of high school, I had a friend who came up with a really interesting way of calculating prime numbers. He proposed prime numbers "by addition", basically a set of 'for' loops that marked an array (you could use a bit array) for every value divisible by 'n' and then you just examine the array afterwards and print out anything with a zero in it. It would be significantly faster than dividing by every odd integer <= sqrt(number), but maybe not faster than dividing by "discovered prime numbers" <= sqrt(number). Anyway, for a value of this magnitude (re: article's number), I think you'd run out of RAM...
(then again it's only 2 ^ 77 million, so perhaps not?)
I had to change a password as it contained an asterisk. It was created on a Mac, but my Android phone must use a different ASCII code for asterisks or something and it simply would not work on that. Which was a pain. Here was me under the impression that ASCII was set in stone and should be the same across systems.
Well, IIRC some infinities are "even", and infinity can be defined as a fraction of -1/12 (note that is *minus* one).
So finding the "last" four digits might not be impossible in some aspects of mathematical analyses. It's the infinite set between two points that is more difficult. ;)
I'd be interested to know what the actual odds are of any specific ten consecutive digit number occurring in a truly random long digit sequence like this monster prime, I would hope its around 10^10 or one in ten billion? I searched for the same string sequence (9999999999) in the decimal digits of pi and couldn't find it in the first 200 million or so, perhaps finding it so much sooner in this sequence has a very low probability?
I think the ten digit statement above is only true for infinitely long random sequences and not for a specific sequence of a given length? If the probability of finding ten recurring digits in any randomly generated number sequence are less than one in the FIRST few billion digits then the odds of a specific ten digit sequence turning up far sooner (in say tens of millions of digits rather than billions) that should be much lower but perfectly acceptable, just less than 1% maybe? Isn't this how bitcoin miners use sha256 hash collisions for proof-of-work, as the odds of finding a really large (2^256) number with lots of recurring leading zeroes is extremely low for randomly generated inputs but every now and then someone in the pool gets lucky and guesses sooner than expected? You wouldn't expect to find the complete works of Shakespeare (or even the first page of Hamlet!) to ever be generated using monkeys on typewriters during the expected age of the universe due to the probability being 1 in 26 to the power thousands of characters?
https://en.wikipedia.org/wiki/Infinite_monkey_theorem#Probabilities
Probabilities - wiki? load of bollocks. A truly random number has no probable outcome nor odds. Any sequence can appear in an infinite number: if it is probable then is is random? yes and no - the cat is dead or alive.
The probability of me winning the lottery is 50% - I either win it or don't.
There is structure in primes:
Mathematicians Discover Prime Conspiracy: A previously unnoticed property of prime numbers seems to violate a long-standing assumption about how they behave.
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.
Anyone know of a really, really good random number generator?
Despite frequentists' conniptions, probabilities ARE ontological and Nature has very good generators with fixed values built-in: HotBits: Genuine random numbers, generated by radioactive decay.
(Beware line breaks every 100 characters)
I was disappointed to find that 0118999 wasn't there - let alone the rest of it.
For Pi fans: 3141592 is there, but not 31415926.
There are 23,714,413 decimal digits; so an arbitrary sequence of 7 digits (of which there are 10 million combinations) has a reasonable chance of being there.
Looking at sequences of repeated digits, the largest ones are:
zeros - 7
ones - 7
twos - 8
threes - 8
fours - 7
fives - 7
sixes - 8
sevens - 6
eights - 7
nines - 10
So nine does seem to be a special case.
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Technically, a prime is an integer p such that 1) p is not a unit (i.e., 1 or -1), and 2) whenever p divides ab, then p divides a or p divides b (or both). What is described here are irreducibles, not primes. That primes and irreducibles are the same thing is the definition of a unique factorization domain. (All primes are irreducible, but not all irreducibles are prime.)
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"69" occurs 229,286 times (first instance: digit 255)
"420" occurs 22,948 times (first instance: digit 613)
"80085" occurs 217 times (first instance: digit 145943)
Interesting pattern to note: The frequency of each number decreases proportionally to the length of that number, almost exactly to the rate of one order of magnitude per additional digit. It's almost like... base 10 or something.
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...but I do giggle to myself when I can read the intelligent, thought-inspiring comments on a Reg thread, swipe straight past anything that is headed with “bombastic bob” and see just a few milliseconds glimpse of his/hers downvotes before I read then next post.
Happy New Beers :)
Look, tie Mersenne Prime search to a block chain, Einsteinium for example. Payout in EMC2 when a prime is found. You'll have so much computing power, new Primes will be popping like corn.
Although the finders likely won't be boffins, so it will deprive El Reg of its customary reporting style...
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