Numbers & such

[H:O:R:A:C:E]

the process we've begun is to discover a percentage of the odd pairs set that are prime pairs.
if we can assume that the notion is valid, i suppose we've already demonstrated that polignac's conjecture is true.
[since a finite division of an infinite group will yield an infinite group (still)]
what i'd like to do is introduce a particular irrational number into the world; the sika.
a mathematical expression for it is what i'm needing.

[H:O:R:A:C:E]

in a similar manner that the "3" multiples were removed from (soop), we can extract the pairs that
contain "5" multiples. i believe that we should apply the process to the 'interim' set, rather than (soop) proper.
(since the "3" multiples have all already been accounted for... we don't need to be extracting "15" a second time).
then again, we extract 2/7 of what remains, then 2/11, etc.
so, something like:
[a value] = (soop) - 2/3 (soop) - 2/5 (soop) - 2/7 (soop) - 2/11 (soop) ...etc

i think i need to find my notes, lol

[H:O:R:A:C:E]

okay, it seems i was trying to out think myself there; the associative
property of algebra is axiomatic and it declares that the order of operation won't matter.

(soop):
(3,5), (5,7), (7,9), (9,11), (11,13), (13,15), (15,17), (17,19), (19,21), (21,23), (23,25), (25,27), (27,29), (29,31), (31,33), (33,35), (35,37), (37,39), (39,41), (41,43), (43,45), (45,47), (47,49), (49,51), (51,53), (53,55), (55,57), (57,59), (59,61), (61,63), (63,65), (65,67), (67,69), (69,71), (71,73), (73,75), (75,77), (77,79), (79,81), (81,83), (83,85), (85,87), (87,89), (89,91), (91,93), (93,95), (95,97), (97,99),...etc

(soop) with 2/3 indicated as removable:
(3,5), (5,7), (7,9), (9,11), (11,13), (13,15), (15,17), (17,19), (19,21), (21,23), (23,25), (25,27), (27,29), (29,31), (31,33), (33,35), (35,37), (37,39), (39,41), (41,43), (43,45), (45,47), (47,49), (49,51), (51,53), (53,55), (55,57), (57,59), (59,61), (61,63), (63,65), (65,67), (67,69), (69,71), (71,73), (73,75), (75,77), (77,79), (79,81), (81,83), (83,85), (85,87), (87,89), (89,91), (91,93), (93,95), (95,97), (97,99),...etc

(soop) with 2/5 indicated as removable:
(3,5), (5,7), (7,9), (9,11), (11,13), (13,15), (15,17), (17,19), (19,21), (21,23), (23,25), (25,27), (27,29), (29,31), (31,33), (33,35), (35,37), (37,39), (39,41), (41,43), (43,45), (45,47), (47,49), (49,51), (51,53), (53,55), (55,57), (57,59), (59,61), (61,63), (63,65), (65,67), (67,69), (69,71), (71,73), (73,75), (75,77), (77,79), (79,81), (81,83), (83,85), (85,87), (87,89), (89,91), (91,93), (93,95), (95,97), (97,99),...etc

(soop) with 2/5 indicated as removable after 2/3 have been preindicated as removable:
(3,5), (5,7), (7,9), (9,11), (11,13), (13,15), (15,17), (17,19), (19,21), (21,23), (23,25), (25,27), (27,29), (29,31), (31,33), (33,35), (35,37), (37,39), (39,41), (41,43), (43,45), (45,47), (47,49), (49,51), (51,53), (53,55), (55,57), (57,59), (59,61), (61,63), (63,65), (65,67), (67,69), (69,71), (71,73), (73,75), (75,77), (77,79), (79,81), (81,83), (83,85), (85,87), (87,89), (89,91), (91,93), (93,95), (95,97), (97,99),...etc

with all pairs containing multiples of "3" removed, (soop) becomes 1/3 of what it began. removing all multiples of "5" (instead) would reduce (soop) to 60% (3/5) of what it had begun at. when applying the removal of "5" multiples after the "3" multiples had been extracted, the extraction process appears to be closer to 50% than to 40%.

[H:O:R:A:C:E]

a recap (already):
given: the twin prime conjecture:
There are infinitely many primes p such that p + 2 is also prime.
given: the set of prime pairs is a subset of the set of "odd pairs" (soop)
i submit that if the set of prime pairs were not an infinite set,
then the ratio of (set of prime pairs)/(soop) would tend toward zero
(since any finite set divided infinitely would approach zero)
the challenge then becomes one of demonstrating that the ratio is
not tending toward zero, but rather some other value.
discovering that number is what interests me, and the means to express it.

[H:O:R:A:C:E]

to reduce (soop) to the set of prime pairs, we need to remove all terms which contain non prime members.
lets go up the number line.
pairs containing numbers divisible by "1" are permitted, this is the "identity factor".
pairs containing numbers divisible by "2" are presorted from the set beforehand (inapplicable).
pairs containing numbers divisible by "3" must be discarded. this process begins at 3x3, since 3x1 and 3x2 are accounted for already.
so, beginning with the first occurrence of "9", all subsequent (remaining) "3" factors require elimination. all prime multiples of "3",
beginning with 3x3 and proceeding upwards along the "prime list" need extraction (since all lower valued factors have been addressed already).
pairs containing numbers divisible by "4" are nonexistant. this and all subsequent nonprime values are to be skipped over as irrelevant.
pairs containing numbers divisible by "5" must be discarded. this begins with the first occurrence of "25" (5x5), since lower values are accounted for.
the terms containing 5x5, 5x7, 5x11, 5x13, 5x17, etc are to be removed.
pairs containing numbers divisible by "6" are irrelevant (as before, with "4").
pairs containing numbers divisible by "7" must be discarded; beginning with 7x7.
8,9,10 ~ skipped
pairs containing numbers divisible by "11" must be discarded. (starting with the first occurrence of "121")
12 ~ skipped
pairs containing numbers divisible by "13" must be discarded. (starting with "169"s appearance)

we may want to introduce a notation that indicates a function somewhat similar to the factorial.
[factorial = the product of an integer and all the integers below it; e.g., factorial four ( 4! ) is equal to 24.]
this new notation would indicate multiplying prime numbers equal to and greater than a listed prime value.
what would be a good symbol or set of symbols for this?

[H:O:R:A:C:E]

i have found my notes, and there is the notion that as we progress along the (soop), removing ever more terms,
there is a certain percentage being removed... but even at the time of my writing them i had realized that the
actual percentages weren't accurately depicted through my model.
in the expression to follow (Pn) refers to a prime value, and (Pn+1) refers to the next higher prime (Pn-1) is the prior prime.
the idea had been to express the percentage of pairs to be removed from (soop) by examining a specific portion
(namely that between where (Pn)^2 and (Pn+1)^2 occur).

(2/3) + (2/5) (1/3) + (2/7) (3/5) (1/3) + ... + [(2/(Pn) x (((Pn-1)-2))/((Pn-1)) x (((Pn-2)-2)/(Pn-2)) x ... x (5/7) x (3/5) x (1/3)]

as the (Pn) value increases, the additional percentage of terms removed becomes less and less pronounced
(approaching zero additional percentage) approaching a fixed percentage [which is greater than zero and less than 100%]

[H:O:R:A:C:E]

(soop) > (set of prime pairs)
(sopp)/(soop) = ?, where ? < 1
to demonstrate that (sopp)/(soop) > 0 would effectively prove the twin prime conjecture

although it does not yield precise percentages, i feel that the division process is a valid method for reducing (soop) into (sopp).
beginning with a term of P^2 (and proceeding toward larger paired values), removing an additional 2/P of our 'process' (soop) [from (soop)] appears sound.
at P = 3, we remove 2/3 of all remaining (soop) pairs... this leaves 1/3.
at P = 5, we remove 2/5 of what remains of 'process' (soop)... this leaves (subsequently/henceforth) [1/3 x 3/5] = 1/5 [of (soop), at this step in the process].
at P = 7, we remove 2/7 more... [1/3 x 3/5 x 5/7] = 1/7
at P = 11, we remove 2/11 more... [1/3 x 3/5 x 5/7 x 9/11] = 9/77
at P = 13, ... ... 9/91
at P = 17, ... ... 135/1547

[H:O:R:A:C:E]

i need calculus at this point.
to express an infinite summation.
an infinite summation of terms which are approaching a value of zero; thereby approaching a "fixed" value.

maybe i need to discuss sets and subsets... ?
although the set of all whole numbers and the set of all even numbers are each infinite, the one is
clearly a "subset" of the other, and not the other way around. evens are "lesser" in that manner.

[H:O:R:A:C:E]

[H:O:R:A:C:E]

okay, a discussion of set theory appears necessary.
there are dimensions to sets.
some are finite, and some are infinite.
there are distinctions of infinities; the set of rational numbers is infinite, the set of irrational numbers is infinitely larger.
if a set can be put into a one-to-one correspondence with another, their degrees of infinity are equal.
[the set of whole numbers can be placed in a one-to-one correspondence with the set of evens... they are of equal cardinality]


although the set of whole numbers are equally infinite with the set of evens, it is plain to see that the set of whole numbers contains
precisely double the number of terms as the set of evens. mind-boggled? perhaps i need some language that has not been invented yet.
it is obvious to viewers that odds and evens match up one-to-one, and that the set of wholes contains equally as many odds as evens.
also, the set of evens is a clear subset of wholes, but wholes are NOT a subset of evens.
i propose that in any examination relating members of one set to another, if we can establish a fixed value 'factorability', we can be certain
to have sets of the same cardinality.

the set of prime pairs is clearly a subset of the set of odd-pairs, and since the set of odd-pairs is infinite, establishing that the fractional
portion of odd-pairs which constitute prime pairs is greater than zero effectively proves that the set of prime pairs is equally infinite.
i simply desire to find the calculus expression which will describe the sika.
it is more 'involved' than (pi) = (circumference)/(diameter).