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Link to the intro graphic and tutorial from Study.com
My standard Disclaimer: Jamess is NOT a professional Mathematician.
I pursue math recreationally, to exercise my mind, and inspire my sense of wonder.
Many are the Mathematically-trained who will want to correct my loose usage of terms and jargon. If you are in that critics corner, may I suggest: Be kind. Or better yet, try to enlighten and not shame, try to teach and not screech — if you find yourself compelled to correct those under-informed souls, tromping around your neck of the woods. As with most public lands, Math should be the domain of us all.
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Here is the Riddle of the Twin Primes, that started me on my Exploration of Math, many many years ago:
Why is it that all Twin Primes* when added together will be a multiple of 12?
(* for Twin Primes starting with n >= 5).
Well is that “Theorem” even true, when doing some off the cuff calculations?
5 + 7 = 12 = 12 * 1
11 + 13 = 24 = 12 * 2
17 + 19 = 36 = 12 * 3
71 + 73 = 144 = 12 * 12
197 + 199 = 396 = 12 * 33
599 + 601 = 1200 = 12 * 100
Well it certainly seems to be true, when randomly selected Twin Primes, are added together.
Is there any systematic way to explain this common “multiple property” of Twin Primes?
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Well being an Excel-power user, I went to the tool, that I know best … to explore the terrain:
Observations about {N} when n’s are distributed in rows that are 6 numbers long, starting with 2 in the first row[t]. In this Sieve-corralling orientation, some interesting properties can be discovered, that are shared by the n-members in each of the columns. (see table below)
Integers in column 1: always divisible by 2. (technically the first row, has Prime 2 which = 2*1)
Integers in column 2: always divisible by 3. (technically the first row, has Prime 3 which = 3*1)
Integers in column 3: always divisible by 2. (this column has no Primes)
Integers in column 5: always divisible by 6. (this column has no Primes)
The n of {N} which are divisible by 2 are shown with a light green background.
The n of {N} which are divisible by 3 are shown with a darker green background.
The n of {N} which are divisible by 6 are shown with a darker blue background.
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Integers in columns 1, 2, 3, and 5, can be considered Trivial Composites — and will never have Primes, with the exceptions the Primes 2 and 3, found in the first row.
Integers in columns 4 and 6 are the “interesting” numbers:
• All the Primes > 3 are found here in columns 4 and 6.
• All the Twin Primes > 3 are also found here in columns 4 and 6. They are the numbers in bold with the light blue background
• All the Twin Primes > 3 will have a Middle number divisible by 6. These midpoint numbers are found in column 5, with the darker blue background.
• Any Composite found in columns 4 and 6, can never have a factor of 2 or 3. (Can you explain why?)
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So back to the Riddle of the Twin Primes:
Why is it that all Twin Primes* when added together will be a multiple of 12?
(* for Twin Primes starting with n >= 5).
Here is the answer to that Riddle, scroll down. (If you rather answer it yourself, in the comments, jump to those now, and do so.)
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Answer:
Since Twin Primes are always 2 numbers apart, and since they are both Prime, this means that the number between them (the middle number, column 5) CANNOT be Prime. Nowhere in the Numberline does there exist 3 consecutive Prime numbers. (And there is only one case of 2 consecutive Primes [2,3].)
That midpoint number between Twin Primes MUST also be a multiple of 6, since in any consecutive set of 3 numbers (n, n+1, n+2) — at least one of those numbers must be divisible by 2, and at least one of those numbers must be divisible by 3. This implies that in a set of 3 consecutive numbers containing Twin Primes (TP1, t+1, TP2) — it MUST be the Middle number, that is BOTH a multiple of 2 and a multiple of 3, at the same time. As shown in the table above, that Middle number between Twin Primes has the algebraic form of 6r. This is the same as (2*3)*r.
This implies the number preceding it will have the algebraic form 6r -1, and the number immediately following it will have the algebraic form 6r +1.
SO ... when adding those 2 numbers together (Twin Prime 1 plus Twin Prime 2) you get:
6r -1
+ 6r +1
12r + 0
which is the same thing as saying: The Sum of Twin Primes is always a multiple of 12.
(when Twin Primes start with n >= 5).
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This was the just the beginning of my Journey into the exploration of the many wondrous Properties of Twin Primes, a very special subset of set of {N}, the Natural Numbers.
The views to be discovered there include the Symmetry and Reflection properties of the 4 Complement sets of Twin Composites (these are the clear cells, intermixed with the tan cells of the table); the compact “modular” equations, which predict those {TC} members; the recursion planes of Twin Composites {TC}, the recursion plane Quad Composites {TC} [yes, Quad Primes really do exist]. And finally the smooth Asymptotic density trend-line of Quad Prime occurrence Events — when measured with the correctly-sized “yardstick” — in all likelihood vouch for the Infinity of Quad Primes.
If there is enough interest in visiting these “topographical features”, of the not-so-imaginary members of {N}, I could be persuaded to lead these various ‘ephemeral’ tours, as time and willpower allow. … that future has yet to be written. We. shall. see.
(If interested, let me know in comments.)
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Extra Credit:
If we let k represent the columns in the Twin Primes Sieve Table, can you come up with the algebraic forms for Columns 1, 2, and 3, in terms of r, k, a, b, and c? (scroll down for those answers)
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Answers:
(where r = row number in the TP Sieve; and y = the n of {N} that “belongs to” the Column k, in the TP Sieve.)
Column 1: 2*a = 6r -4 get the equivalent form from Excel Trend-line chart
2*a = 2*(3r -2) factor out the 2
a = (3r -2) establish the equality
therefore: y = 2*a restate original equation, and substitute for a
y = 2*(3r -2) = 6r -4
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Column 3: 2*c = 6r -2 get the equivalent form from Excel Trend-line chart
2*c = 2*(3r -1) factor out the 2
c = (3r -1) establish the equality
therefore: y = 2*c restate original equation, and substitute for c
y = 2*(3r -1) = 6r -2
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Column 2: 3*b = 6r -3 get the equivalent form from Excel Trend-line chart
3*b = 3*(2r -1) factor out the 3
b = (2r -1) establish the equality
therefore: y = 3*b restate original equation, and substitute for b
y = 3*(2r -1) = 6r -3
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and for the sake of completeness:
Column 4:
y = 6r -1 the term (6r-1) cannot be systematic factored in terms of r.
Column 5:
y = 6r the term (6r-0) cannot be systematic factored in terms of r.
Column 6:
y = 6r +1 the term (6r+1) cannot be systematic factored in terms of r.
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Qed … Whew, my grey-matter is tired!
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See ya … on the flip side.
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