Are there folks on this list interested in prime numbers?
While there are interesting applications of prime numbers to, say, music
theory and other investigations, a formula for generating all primes
successfully (and ONLY primes) still eludes mathematicians, and remains one
of the great remaining mysteries of simple arithmetic, now that Fermat's
last theorem was solved a few years ago, after centuries. (I still have
not seen a particularly elegant proof, or the simple one that Fermat
suggested existed but was "too big for the margins" of the page on which he
was writing.) In fact, I'll wager that at some point the undiscovered
formula for primes and Fermat's last theorem are integrally related, but
that's another discussion for another time.
I want to propose a different approach -- i.e., I want to reframe the way
we look at primes. Instead of defining a prime number as a positive integer
having no whole factors besides itself and the number 1, I'd propose a
different definition: Look at it via ADDITION: A prime is the SUM of two
and only two consecutive positive integers. All other positive integers can
be expressed as the sum of at least three or more consecutive positive
Unlike multiplication (which is repetitive addition), addition enables us
to perform a function -- in this case adding consecutive numbers -- that
are of the same type. In multiplication we multiply a cardinal number by an
ordinal number -- by the number of times we want to ADD, say, a set of
three items. It is this self-referencing dimension (quality) of numbers
that multiplication -- unlike simple addition -- depends on.
So in multiplying 3 x 4, for instance, we are starting with a set of 3
items and adding 3 more items, and then 3 more, and then 3 more. The "4",
therefore, is serving a different kind of function in the equation than the
"3". They are not performing the same tasks.
In simple addition, however, we are adding directly items in the same
category without the self-referential quality. We add directly the number
of items to another number of items. I believe that reliance on
multiplication without recognizing the different categories involved has
prevented us from achieving a clear understanding of primes. The use of
these two categories of numbers inherent in multiplying "tricks" us. By
using simple addition we gain new insights into the nature of primes that
are concealed by addition.
I hypothesize (but cannot yet prove) that a prime number has an additional
quality to it that tells other numbers how to behave in certain
circumstances or sequences, just as under particular conditions certain DNA
sequences tell the rest of the chain (of which it is a part) how to behave,
to turn on or off, etc. This is one avenue in which the approach I suggest
may open new vistas.
In addition and unlike multiplication, the reason why a number that sums to
three or more consecutive integers cannot be prime is simple: Any (n-1) + n
+ (n+1) will sum to 3n and thus 3 and n will be factors of the sum. This
holds for EVERY odd number of consecutive integers-- obvious in addiition,
but opaque when looked at via multiplication.
For every even number of consecutive integers, pair off the integers [(n-3)
+ (n-2)] + [(n-1) + (n)] + [(n+1) + (n+2)] . For the case of odd number of
pairs, the sum will be a multiple of the center pair multiplied by the
number of pairs, and thus not prime. Or, in summing the example above
containing 3 pairs, the sum comes to 6n - 3, which can be divided by 3, and
thus cannot be prime. The same can also be done for every number of
consecutive integers -- pair them off and they become divisible by the
number of pairs. Again, this simple aspect of addition is concealed by
multiplication, and it shows WHY those numbers that are the sum of only two
consecutive numbers are primes.
(We can then examine how many sets of different consecutive sequences a
particular sum may contain.)
Primes have fascinated me since I was a kid, and like many folks I've
played around with them, tried them out in different bases, etc. (My
brother the musician says that base 5 generates interesting results.)
And then, of course, is the question -- why haven't mathematicians gone
back to examining what has kept them from generating a formula for primes,
and used that "unasking" of the question to developing a simple reframing
-- a new approach -- such as the one above.
What do you think?