So as I rewrite my Balanced Ternary (ī,0,1) library once again, something occurs to me...
While it is not visually obvious which numbers are multiples of two, we do know that all multiples of 3 end in 0. Since no prime (except 3) can be a multiple of 3, that means that all primes are going to end in ī or 1. Well, since we know all primes are odd, that also means that the specific number with a 0 ending would have also been a multiple of 2. IE: The number ending in 0 would have been a multiple of 6.
Where am I going with this? Well, let's look at the first few multiples of 2*3
6 - 5 and 7 are twin primes
12 - 11 and 13 are twin primes
18 - 17 and 19 are twin primes
24 - 23 and 25... nope, not that one (23 is prime though, other is 52)
30 - 29 and 31 are twin primes
36 - 35 and 37... nope, not that one (37 is prime though, other is 5*7)
42 - 41 and 43 are twin primes
48 - 47 and 49... nope, not that one (47 is prime though, 49 is 72)
54 - 53 and 55... nope, not that one (53 is prime though, other is 5*11)
60 - 59 and 61 are twin primes
66 - 65 and 67... nope, not that one (67 is prime though, other is 5*13)
72 - 71 and 73 are twin primes
80 - 79 and 81... nope, not that one (79 is prime though, 81 is 92)
well obviously that wasn't the "correct solution"...
let's start with a simple question - did we miss any twin primes?
well we miss (3,5) - but that could just be because we were specifically looking for multiples of 6.
other than that, we did not skip any twin primes
what about those 'extra' entries?
well some are when we have squares (52, 72, 92)
the others are 5*{7,11,13}
could be a pattern there... partial one anyways...
let's look at it from another aspect... what primes did we miss?
obviously anything below 6 since we started with multiples of 6
other than that, we got all primes...
and 6 extra values below 82... so about 7% too many
it has to become less usable as we get higher, doesn't it?
600 - 599 and 601 are twin primes
606 - 607 is prime
612 - 613 is prime
618 - 617 and 619 are twin primes
624 - (neither 623 or 625 are prime)
630 - 631 is prime
636 - (neither 635 or 637 are prime)
642 - 641 and 643 are twin primes
648 - 647 is prime
652 - 653 is prime
660 - 659 and 661 are twin primes
and no primes are missing (though a couple extras)
26 January 2007
18 January 2007
Prime Numbers, Assumptions and Thoughts
Prime numbers have been a challenge to people for a very long time. If they are something that we still can't write an equation for, perhaps it is time to challenge our assumptions about these fascinating and extremely important numbers.
First, it is commonly believed that there is no pattern to prime numbers. Research studied under this belief might very well be simply falling prey to self-fullfilling prophecy. We know that prime numbers do indeed have a pattern. How? Because color-coding a Pascal Triangle to look like a Sierpinski Gasket shows that prime numbers form perfect fractals, while composite numbers show overlapping fractcals. I discussed this a little while ago. Using this method, with just a couple sentences of explanation, ANYONE can look at a picture (mod some number) and tell you if it is a prime number or not (primes look clean, composites look cluttered). The pattern is visually obvious.
Second, let's tackle the concept that '1' is not a prime number... Some people will tell you that '1' isn't prime because it doesn't fit the rules. Others because it isn't convenient. Still others say it was due to the Greeks thinking number games weren't as challenging with it. Hell, you are probably wondering why we even care... Simple: If '1' IS a prime number, then all the equations that don't quite work change... the math changes... We get rid of "except 1" or ">=2" type of exceptions... but most importantly, because it would force us to look at things differently -- which is what we desperately need if we are going to 'solve' primes.
So, what about the argument that it doesn't fit the rules? What is the definition of a prime number? Generally, people assume it is "any number greater than 1 that has only 1 and itself as factors".. So 1*3 is prime because no other number is a factor. 1*4 is not prime because 2 is a factor. 1*1 has no other factors, therefore it should be considered prime. But, you also see why people have a problem with it. Because it is also a square ;) They always say "1 and itself"... Well, under that definition '1' should be prime. The only reason they say it isn't is because they add the clause "greater than 1" or "except 1" to the rule. Occam's razor fans should be well aware of the problem here. You don't add a single exception (for a number that would otherwise work) just because you don't like or understand it.
You are probably still wondering why we care... Let's say you are writing an equation... f(n)=nth prime. Well, if '1' is prime, then all the current attempts are actually doing f(n)=(n+1)st prime... thus patterns may be easier to see if we quit assuming 1 can't be prime.
What other assumptions do we have? Primes get further and further apart... Not true... No matter how high you go, there will be twin primes. This tells us that twin primes are not a special case or exception -- but are in fact part of the pattern. And don't forget what we said earlier -- it is a fractal -- so it repeats forever.
In fact, the whole concept that there are 'types of primes' is simply a misnomer. Each type of prime is simply a subset of the fractal.
First, it is commonly believed that there is no pattern to prime numbers. Research studied under this belief might very well be simply falling prey to self-fullfilling prophecy. We know that prime numbers do indeed have a pattern. How? Because color-coding a Pascal Triangle to look like a Sierpinski Gasket shows that prime numbers form perfect fractals, while composite numbers show overlapping fractcals. I discussed this a little while ago. Using this method, with just a couple sentences of explanation, ANYONE can look at a picture (mod some number) and tell you if it is a prime number or not (primes look clean, composites look cluttered). The pattern is visually obvious.
Second, let's tackle the concept that '1' is not a prime number... Some people will tell you that '1' isn't prime because it doesn't fit the rules. Others because it isn't convenient. Still others say it was due to the Greeks thinking number games weren't as challenging with it. Hell, you are probably wondering why we even care... Simple: If '1' IS a prime number, then all the equations that don't quite work change... the math changes... We get rid of "except 1" or ">=2" type of exceptions... but most importantly, because it would force us to look at things differently -- which is what we desperately need if we are going to 'solve' primes.
So, what about the argument that it doesn't fit the rules? What is the definition of a prime number? Generally, people assume it is "any number greater than 1 that has only 1 and itself as factors".. So 1*3 is prime because no other number is a factor. 1*4 is not prime because 2 is a factor. 1*1 has no other factors, therefore it should be considered prime. But, you also see why people have a problem with it. Because it is also a square ;) They always say "1 and itself"... Well, under that definition '1' should be prime. The only reason they say it isn't is because they add the clause "greater than 1" or "except 1" to the rule. Occam's razor fans should be well aware of the problem here. You don't add a single exception (for a number that would otherwise work) just because you don't like or understand it.
You are probably still wondering why we care... Let's say you are writing an equation... f(n)=nth prime. Well, if '1' is prime, then all the current attempts are actually doing f(n)=(n+1)st prime... thus patterns may be easier to see if we quit assuming 1 can't be prime.
What other assumptions do we have? Primes get further and further apart... Not true... No matter how high you go, there will be twin primes. This tells us that twin primes are not a special case or exception -- but are in fact part of the pattern. And don't forget what we said earlier -- it is a fractal -- so it repeats forever.
In fact, the whole concept that there are 'types of primes' is simply a misnomer. Each type of prime is simply a subset of the fractal.
17 January 2007
Slashdot | Largest Twin Prime Yet Discovered
The Twin Internet Prime Search and PrimeGrid have recently discovered the largest known twin prime. A twin prime is a pair of prime numbers separated by the integer two. The pair discovered on January 15th was 2003663613 * 2195,000 ± 1. The two primes are 58,711 digits long. The discoverer was Eric Vautier, from France.
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