Which K will be the next to yield a prime?
It's been 7 months since we found a prime. I think that we are due to find one soon. Which k do you think it will be?
Code:
K Probability
4847 0.98860
5359 0.99469 Prime Found
10223 0.99539
19249 0.85992
21181 0.98736
22699 0.84058
24737 0.98830
27653 0.94479
28433 0.91430
33661 0.98703
44131 0.99661 Prime Found
46157 0.87826 Prime Found
54767 0.99464 Prime Found
55459 0.99832
65567 0.83838 Prime Found
67607 0.79880
69109 0.97235 Prime Found
How were the probabilities calculated?
Quote:
Originally posted by Joe O
It's been 7 months since we found a prime. I think that we are due to find one soon. Which k do you think it will be?
Code:
K Probability
4847 0.98860
5359 0.99469 Prime Found
10223 0.99539
19249 0.85992
21181 0.98736
22699 0.84058
24737 0.98830
27653 0.94479
28433 0.91430
33661 0.98703
44131 0.99661 Prime Found
46157 0.87826 Prime Found
54767 0.99464 Prime Found
55459 0.99832
65567 0.83838 Prime Found
67607 0.79880
69109 0.97235 Prime Found
How did you come up with this? The most probable still has no prime and the least and the 4th least probables have primes. I guess getting more accurate probabilities will be the next step in defining Sierpinski numbers.
How many Sierpinski numbers have been proved?
Re: How were the probabilities calculated?
Quote:
How did you come up with this? The most probable still has no prime and the least and the 4th least probables have primes. I guess getting more accurate probabilities will be the next step in defining Sierpinski numbers.
There's a Java applet online that will compute the "probabilities." (They're actually densities, not probabilities, where a density of 1.000 is equal to the density of primes in a random sequence of integers... see Mathworld's page on the prime counting function for more on that.) Without getting into heavy math, the density is estimated basically by picking a k and then counting the number of n candidates remaining after a sieve to a certain threshold.
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How many Sierpinski numbers have been proved?
Infinitely many, if you ignore the requirement that k be odd :) If k is a Sierpinski number then 2k is also a Sierpinski number, since (2k) * 2^n + 1 = k * 2^(n + 1) + 1.
Even when considering only odd k there are still an infinitude of Sierpinski numbers (Sierpinski himself proved this although it wasn't until 1960 that the first actual Sierpinski number was identified). Of course only a finite number are and will ever be known.
Re: Which K will be the next to yield a prime?
Code:
K Density N Probability
4847 0.09877 6578031 0.98860
5359 0.11761 5054502 0.99469 Prime Found
10223 0.11875 6578105 0.99539
19249 0.04339 6578078 0.85992
21181 0.09649 6578060 0.98736
22699 0.04054 6577318 0.84058
24737 0.09820 6578191 0.98830
27653 0.06394 6578097 0.94479
28433 0.05424 6578017 0.91430
33661 0.09592 6578208 0.98703
44131 0.14273 995972 0.99661 Prime Found
46157 0.05424 698207 0.87826 Prime Found
54767 0.12846 1337287 0.99464 Prime Found
55459 0.14102 6578218 0.99832
65567 0.04567 1013803 0.83838 Prime Found
67607 0.03540 6577227 0.79880
69109 0.08906 1157446 0.97235 Prime Found
These are the densities I used. They should correspond to those obtained from the Java applet.
1-EXP(-2*Density*LOG(N)/LOG(2)) Is the formula I used.
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The most probable still has no prime and the least and the 4th least probables have primes.
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In theory, there is no difference between theory and practice. In practice, there is.