A few weeks ago I attempted to predict the size of the newly reported 45th Mersenne prime by extrapolating from the trends on previous Mersenne primes; I liked 14.5 million digits.
Brent Yorgey reports that it's actually about 11.2 million digits; the 46th prime (which hadn't been discovered when I made that post, but was reported to exist a few days later) has just under 13 million digits.
Oh well, I was wrong. Good thing I didn't put any money on it.
Showing posts with label Mersenne primes. Show all posts
Showing posts with label Mersenne primes. Show all posts
17 September 2008
26 August 2008
Predicting the next Mersenne prime
Robert at Casting Out Nines reports the new that the Great Internet Mersenne Prime Search claims to have found the 45th Mersenne prime, but they're waiting for verification before they report its identity. He invites guesses on the number of digits.
Here's my guess. A remark in Sloane's encyclopedia says the number of Mersenne primes up to exponent N is, conjecturally, about K log N for some constant K. (Here's a plot illustrating that.)
Let Nr denote the exponent of the rth Mersenne prime. The 44th Mersenne prime has exponent N44 = 32582657, so we have approximately
44 ~ K log 32582657
from which we get K ~ 2.5434. Thus the 45th Mersenne prime has exponent N45 satisfying approximately 45 ~ K log N45; thus N45 ~ 48276546. (Alternatively, we could have raised 32582657 to the 45/44 power, but it's not obvious why!) So the number of digits is approximately N45 log10 2 ~ 14532688; I'll take 14.5 million digits.
This method, it turns out, might show some sort of systematic bias. If I'd predicted Nr = (Nr-1)(r/(r-1)) with r = 2, 3, ..., 44, I would have had 16 underpredictions (the predicted exponent less than the true exponent) and 27 overpredictions. That's not quite statistically significant but it's not something you'd just neglect either. But this is just for fun -- if I were seriously trying to make a prediction I would have defined things more precisely, for one. (Oddly enough, since Mersenne primes get sparser as exponents get larger, the "most likely" single exponent is probably the single prime after N44 -- unless there's something that keeps them from clustering!) But if I were going to bet on this I'd also take into account the history of GIMPS; how long has it been since they've reported a prime, and how quickly does their search appear to move?
If you want to take a shot at that, be my guest. (I may regret what I've just unleashed.)
Here's my guess. A remark in Sloane's encyclopedia says the number of Mersenne primes up to exponent N is, conjecturally, about K log N for some constant K. (Here's a plot illustrating that.)
Let Nr denote the exponent of the rth Mersenne prime. The 44th Mersenne prime has exponent N44 = 32582657, so we have approximately
44 ~ K log 32582657
from which we get K ~ 2.5434. Thus the 45th Mersenne prime has exponent N45 satisfying approximately 45 ~ K log N45; thus N45 ~ 48276546. (Alternatively, we could have raised 32582657 to the 45/44 power, but it's not obvious why!) So the number of digits is approximately N45 log10 2 ~ 14532688; I'll take 14.5 million digits.
This method, it turns out, might show some sort of systematic bias. If I'd predicted Nr = (Nr-1)(r/(r-1)) with r = 2, 3, ..., 44, I would have had 16 underpredictions (the predicted exponent less than the true exponent) and 27 overpredictions. That's not quite statistically significant but it's not something you'd just neglect either. But this is just for fun -- if I were seriously trying to make a prediction I would have defined things more precisely, for one. (Oddly enough, since Mersenne primes get sparser as exponents get larger, the "most likely" single exponent is probably the single prime after N44 -- unless there's something that keeps them from clustering!) But if I were going to bet on this I'd also take into account the history of GIMPS; how long has it been since they've reported a prime, and how quickly does their search appear to move?
If you want to take a shot at that, be my guest. (I may regret what I've just unleashed.)
25 June 2007
The Simpsons use decimal numbers
In The Simpsons, people have four fingers on each hand. Eight fingers in total. Therefore, shouldn't they use numbers in base 8?
(The reason that they have four fingers is the same reason that most animated characters have four fingers -- it's easier to draw. In at least one episode, God appears; God has five fingers.)
This occurred to me while watching the episode The Canine Mutiny, in which "After using his credit card to buy another dog, Bart must choose between his new wonder-pooch and the bumbling but loyal Santa's Little Helper." Bart gets the credit card in the name of his old dog, Santa's Little Helper; to order the new dog from a catalog, he has to dial an 800 number. He says "I don't think our phone goes up to 800", which got me thinking about what kind of numbers they use in Simpsons-world.
simpsonsmath.com, by Sarah Greenwald and Andrew Nestler, has a list of mathematical references on the Simpsons. This is not one of them.
It's actually possible to prove that the Simpsons universe has numbers in base 10. The baseball attendance figures in Marge and Homer turn a Couple Play are 8191, 8128, and 8208. The use of 9 indicates that we're in base at least 10. If we assume this is supposed to be a mathematical joke, 8191 and 8128 are immediately recognizable as 213-1 (Mersenne prime) and (27-1)26 (a perfect number.) 8208 is also 213 + 24, but more importantly it's the sum of the fourth powers of its digits. This would only be true in base 10. (Incidentally, most mathematicians regard properties of numbers that are based on their digits as not worthy of investigation, because they are basically accidents of the fact that we have ten fingers.)
(The reason that they have four fingers is the same reason that most animated characters have four fingers -- it's easier to draw. In at least one episode, God appears; God has five fingers.)
This occurred to me while watching the episode The Canine Mutiny, in which "After using his credit card to buy another dog, Bart must choose between his new wonder-pooch and the bumbling but loyal Santa's Little Helper." Bart gets the credit card in the name of his old dog, Santa's Little Helper; to order the new dog from a catalog, he has to dial an 800 number. He says "I don't think our phone goes up to 800", which got me thinking about what kind of numbers they use in Simpsons-world.
simpsonsmath.com, by Sarah Greenwald and Andrew Nestler, has a list of mathematical references on the Simpsons. This is not one of them.
It's actually possible to prove that the Simpsons universe has numbers in base 10. The baseball attendance figures in Marge and Homer turn a Couple Play are 8191, 8128, and 8208. The use of 9 indicates that we're in base at least 10. If we assume this is supposed to be a mathematical joke, 8191 and 8128 are immediately recognizable as 213-1 (Mersenne prime) and (27-1)26 (a perfect number.) 8208 is also 213 + 24, but more importantly it's the sum of the fourth powers of its digits. This would only be true in base 10. (Incidentally, most mathematicians regard properties of numbers that are based on their digits as not worthy of investigation, because they are basically accidents of the fact that we have ten fingers.)
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