
#1
Jul2707, 12:27 AM

P: 120

I posted the following on my blog (http://fooledbyprimes.blogspot.com/2...lyprimes.html)
Not until recently has the whole prime number "culture" become a distraction to me. While a child the primes never really caught my attention. Even in college there was not much drawing me to the subject beyond the occasional newspaper headline proclaiming the exuberance of the mathematics community as some rather skinny, unkempt math geek held a new largest prime in high esteem. One of the things that bothered me about primes is how messy they are. From the perspective of where they are on the number line one can't help but get the feeling that any equation related to their distribution is going to be ugly. Maybe I am a sucker for simplicity just call it an eye for elegance! Taking a look at the math culture's definition of a prime we find something like: "..a natural number that has exactly two (distinct) natural number divisors, which are 1 and the prime number itself." Oh how boring! Of course the mathematicians tell us that primes build all the other numbers. Digging around one will find this formal statement called the fundamental theorem of arithmetic. It says, "every natural number greater than 1 can be written as a unique product of prime numbers." It appears to be very, very important to mathematics afterall, it is the fundamental theorem of arithmetic! I must admit I didn't investigate the prime number sequence at all other than taking a quick peek at the first 100 primes. Instead, I became intensely focused on the two related definitions given above. Take a look at the words in the definition and convince yourself which words convey the most "action" the meat of the definitions so to speak. I came up with "natural number divisors" and "unique product." Now, I must say right away that I failed calculus II so I do not profess to be a brilliant mathematician (don't worry, I took the class again with a different professor and got an passing grade). There is one thing that I do know about math and it is this: multiplication is just repeated addition. So, I wondered what would happen if the math culture rewrote the fundamental theorem of arithmetic without using the word "product." Wouldn't that be cool a simplified version of the definition! Maybe... just maybe... we might find some new way to think about prime numbers and make some progress on the stubborn topic. Personally, I believe that a number which is "prime" is just highlighing a side effect of shortcut addition. We have to have shortcuts otherwise we humans would count to each other when we simply wanted to say "I'll pay you 25 copper coins to feed my camels." Think about the axioms of arithmetic. List them on paper and then erase the ones related to multiplication and division. Now, tell me what a prime number is! I feel that we have been duped by the math community at large because they told us for so long that primes are super important even godly. I challenge everyone to go back to the basics for the sake of progress! (I know you're just as tired of the centuriesold unsolved prime number mysteries) What I am saying is that the prime numbers are not mystical. What is mystical is the relationship between the algorithmic process of counting and the notion of shortcuts (multiplication). Are the two different? Yes. Shortcuts require some sort of memory. The memory is in the form of additional "wiring"... like defining new kinds of number systems. Think about it: the Egyptians, Babylonians, Greeks, Hebrews, Hindus, they all count the same. But their short cut methods are what are different. Counting is simple, just repeat after me: "da, da, da, da, da, da, da....." Shortcutting and communicating about where the counting stops is a completely different ballgame and it is what produces the "mysterious" properties that we perceive in the primes. I would be interested in literature about the primes from the perspective above. Thanks, Philip R. Dutton Columbia, SC, USA http://fooledbyprimes.blogspot.com/ http://forum.wolframscience.com/memb...ster&forumid=4 



#2
Jul2807, 05:17 PM

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I'm sorry: "the fundamental theorem of arithmetic without using the word 'product.'"? I can't imagine how it could be stated more easily! In general, multiplication is NOT a "shortcut" for addition. Thinking it is misses the whole point.




#3
Jul2807, 05:28 PM

P: 1,705

how old are you? your writing is worse than high school quality and i don't see your point at all. seems like someone just trying to use a lot of big words.




#4
Jul2807, 05:44 PM

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whole prime number
For what it's worth I see multiplication as the core relation, with addition as a more complex and loess natural system that makes numbers more complex. For me, the primes are quite literally the atom of the natural numbers.
Perhaps there is a version of the fundamental theorem using just gcds and its like? 



#5
Jul2807, 05:47 PM

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In response to your third blog post's challenge: "Try to define a prime number without using the word 'product' nor the word 'multiplication.'"
A prime number is a number with a nonzero residue modulo all numbers 1 < k < n. 



#6
Jul2807, 06:05 PM

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#7
Jul2807, 06:08 PM

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#8
Jul2807, 06:09 PM

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#9
Jul2807, 06:27 PM

P: 120

The funny thing about the Peano axioms is that most of them start out with "If b is a natural number..." Well, basically Peano states in his assumptions that you are given all the natural numbers. So, all the natural numbers that happen to be in the position of primes are there too. But if you stop writing down axioms before you define the successor function, then you can not have the notional of primality. I am having trouble explaining all this. Basically I can create a system for counting with no fluffy extra axioms related to operations. Heck, let me just count to the 100th prime number: "da,da,da,da,da,da,da,da,da,....,da,da,da" There, you see that last "da"? That is in the same position on the number line as the 100th prime number as defined already. However, in my "da, da,da" counting system, I can not tell you what it means to be "prime." 



#10
Jul2807, 06:30 PM

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#11
Jul2807, 06:40 PM

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If you want to read up on abstract algebra, here are some basic notes from the Web: http://www.math.niu.edu/~beachy/aaol/contents.html 



#12
Jul2807, 06:49 PM

P: 120

Primes, squares, addition, fractions, etc. all have to do with permitted "operations." But I still believe the natural numbers are still implicitly defined and do sit in place on the number line whether the operational axioms are defined yet or not. 



#13
Jul2807, 06:51 PM

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#14
Jul2807, 07:03 PM

P: 120

"Counting is all too easy. Figuring out how to talk about where you stopped is the hard part."  Philip Ronald Dutton




#15
Jul2807, 07:11 PM

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#16
Jul2807, 07:14 PM

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The successor function is the only way to create new numbers in this system. The last property makes each number 1, S(1), S(S(1)), ... different. There's really no chickenegg problem  unless you remove the successor operation. If you do that you'll need to add in a lot of tools to do most anything. 



#17
Jul2807, 07:19 PM

P: 120

PS: thanks for chatting thus far! 



#18
Jul2807, 07:24 PM

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