Re: One of AP's most beautiful proofs to date-- and a waste.

From	James Waldby <not@valid.invalid>
Newsgroups	sci.math
Subject	Re: One of AP's most beautiful proofs to date-- and a waste.
Date	2018-10-08 04:32 +0000
Organization	A noiseless patient Spider
Message-ID	<ppemkl$6oo$1@dont-email.me> (permalink)
References	(12 earlier) <d698a742-e2c8-48e4-ae9d-e9d7898d26ad@googlegroups.com> <c8f18099-eb5e-448f-b317-873972abb7fa@googlegroups.com> <983df434-4045-4a66-a244-e13ac3e01f50@googlegroups.com> <45801449-7131-4021-9923-4a1393a41f46@googlegroups.com> <79d54de3-74fc-44a2-a1ce-a12cd453f3bb@googlegroups.com>

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On Sat, 06 Oct 2018 21:31:52 -0700, eastside.erik wrote:
> On Wednesday, October 3, 2018 at 11:32:48 PM UTC-7, Archimedes Plutonium
> wrote:
...[snip 252 lines]
>> And then, after doing several Columns for Odd Numbers (mind you not
>> even numbers but Odd Numbers) I realized that all odd numbers can be
>> instantly recognized as composite or prime.
>> 
>> 5   0 4   1 3   2
>> 
>> examine horizontals and do division where we discount the division by 1
>> or 0
>> 
>> 7  0 6  1 5  2 4  3
>> 
>> no divisions possible, because well, 7 is prime
>> 
>> 9   0 8   1 7   2 6   3 5   4
>> 
>> Notice we have 6 divided by 3, hence 9 is composite
>> 
>> 11  0 10  1 9    2 8    3 7    4 6    5
>> 
>> No division possible, because 11 is prime
>> 
>> now skip down to 21 because the other primes show the same -- unable to
>> divide
>> 
>> 21    0 20   1 19    2 18    3 17    4 16   5 15   6 14   7 13   8 12  
>> 9 11   10
>> 
>> Now we have several divisions by factors possible we have 18/3
>> outright, but also factoring division to lowest terms 15/6, and 12/9
>> 
>> Now let us test this theory on the nearby prime 23, to see that there
>> are no outright divisions and no factoring to lowest terms division
>> 
>> 23   0 22   1 21   2 20   3 19   4 18   5 17   6 16   7 15   8 14   9
>> 13   10 12   11
>> 
>> So, I see absolutely no outright divisions nor any factoring division
>> into lowest terms, and that is because 23 is prime.
...[snip 37 lines]
> 
> Mathematicians don't do this because it is probably THE most inefficient
> way to test for primes.

In real life, the method AP demos is inefficient [it is an O(n) method]
but in AP-life, where "instantly" is the cost of writing out 2n numbers
in a row and then seeing if two consecutive numbers in the row have a
common divisor > 1, efficiency is perfect.

What I find more surprising than the inefficiency of the method is that
it actually is a method that will work.  To see this, note that AP's
examples imply the claim that if n is composite, there exists an 
integer j giving us two pairs of numbers (j-1, n-j), (j, n-j-1) such 
that gcd(n-j, j) > 1, while if n is prime, there does not exist such j.

To prove 'if' case: Let d>1 be a divisor of n. (d exists because n is
composite.)  Set j=d.  As d|n, we have d|(n-d), QED.

To prove other case:  If n is prime, we have gcd(n, j) = 1 for all j<n, 
j>1.  Now gcd(n-j, j) = gcd(n, j) = 1 for all j<n, j>1. QED.
[Or: In pairs (h,i),(j,k), if gcd(i,j) = d > 1, write i=f*d, j=g*d;
as n = i+j = (f+g)*d, n is composite, contradicting that n is prime.]

-- 
jiw

Thread

4.3- nested Primitive Pythagorean Triples// AP wants to do one better on Ulam's Prime Spiral Archimedes Plutonium <plutonium.archimedes@gmail.com> - 2018-09-30 23:31 -0700
  Re: 4.3- nested Primitive Pythagorean Triples// AP wants to do one better on Ulam's Prime Spiral Archimedes Plutonium <plutonium.archimedes@gmail.com> - 2018-10-01 00:02 -0700
    research taken a new turn-- Heavily Dense Composites predicting future primes//nested Primitive Pythagorean Triples// AP wants to do one better on Ulam's Prime Spiral Archimedes Plutonium <plutonium.archimedes@gmail.com> - 2018-10-01 15:31 -0700
    stuck on 67// nested Primitive Pythagorean Triples// AP wants to do one better on Ulam's Prime Spiral Archimedes Plutonium <plutonium.archimedes@gmail.com> - 2018-10-01 23:45 -0700
      forced primes Re: stuck on 67// nested Primitive Pythagorean Triples// AP wants to do one better on Ulam's Prime Spiral Archimedes Plutonium <plutonium.archimedes@gmail.com> - 2018-10-02 23:57 -0700
        Perhaps found a pattern//forced primes Re: stuck on 67// AP wants to do one better on Ulam's Prime Spiral Archimedes Plutonium <plutonium.archimedes@gmail.com> - 2018-10-03 00:36 -0700
          Perhaps found a pattern//forced primes Re: stuck on 67// AP wants to do one better on Ulam's Prime Spiral Archimedes Plutonium <plutonium.archimedes@gmail.com> - 2018-10-03 00:44 -0700
            Perhaps found a pattern//forced primes Re: stuck on 67// AP wants to do one better on Ulam's Prime Spiral Archimedes Plutonium <plutonium.archimedes@gmail.com> - 2018-10-03 01:02 -0700
              Perhaps found a pattern//forced primes Re: stuck on 67// AP wants to do one better on Ulam's Prime Spiral Archimedes Plutonium <plutonium.archimedes@gmail.com> - 2018-10-03 01:27 -0700
                Perhaps found a pattern//forced primes Re: stuck on 67// AP wants to do one better on Ulam's Prime Spiral Archimedes Plutonium <plutonium.archimedes@gmail.com> - 2018-10-03 01:34 -0700
                Perhaps found a pattern//forced primes Re: stuck on 67// AP wants to do one better on Ulam's Prime Spiral Archimedes Plutonium <plutonium.archimedes@gmail.com> - 2018-10-03 01:41 -0700
                Perhaps found a pattern//forced primes Re: stuck on 67// AP wants to do one better on Ulam's Prime Spiral Archimedes Plutonium <plutonium.archimedes@gmail.com> - 2018-10-03 01:51 -0700
                Re: Perhaps found a pattern//forced primes Re: stuck on 67// AP wants to do one better on Ulam's Prime Spiral Archimedes Plutonium <plutonium.archimedes@gmail.com> - 2018-10-03 02:04 -0700
                Re: Perhaps found a pattern//forced primes Re: stuck on 67// AP wants to do one better on Ulam's Prime Spiral Archimedes Plutonium <plutonium.archimedes@gmail.com> - 2018-10-03 02:11 -0700
                Re: Perhaps found a pattern//forced primes Re: stuck on 67// AP wants to do one better on Ulam's Prime Spiral Archimedes Plutonium <plutonium.archimedes@gmail.com> - 2018-10-03 02:24 -0700
                Re: Perhaps found a pattern//forced primes Re: stuck on 67// AP wants to do one better on Ulam's Prime Spiral Archimedes Plutonium <plutonium.archimedes@gmail.com> - 2018-10-03 02:31 -0700
                Re: Perhaps found a pattern//forced primes Re: stuck on 67// AP wants to do one better on Ulam's Prime Spiral Archimedes Plutonium <plutonium.archimedes@gmail.com> - 2018-10-03 05:06 -0700
                Re: Perhaps found a pattern//forced primes Re: stuck on 67// AP wants to do one better on Ulam's Prime Spiral Archimedes Plutonium <plutonium.archimedes@gmail.com> - 2018-10-03 10:17 -0700
                Formats to eliminate odd composites 9+6N // AP wants to do one better on Ulam's Prime Spiral Archimedes Plutonium <plutonium.archimedes@gmail.com> - 2018-10-03 10:41 -0700
                still think it all hinges on 2*a*b Re: Formats to eliminate odd composites 9+6N // AP wants to do one better on Ulam's Prime Spiral Archimedes Plutonium <plutonium.archimedes@gmail.com> - 2018-10-03 14:11 -0700
                summary at this point-- good-- but all depends on differentiation // AP wants to do one better on Ulam's Prime Spiral Archimedes Plutonium <plutonium.archimedes@gmail.com> - 2018-10-03 15:24 -0700
                the fingerprint of being prime // AP wants to do one better on Ulam's Prime Spiral Archimedes Plutonium <plutonium.archimedes@gmail.com> - 2018-10-03 19:16 -0700
                One of AP's most beautiful proofs to date-- the entire Pattern of Primes Re: the fingerprint of being prime // AP wants to do one better on Ulam's Prime Spiral Archimedes Plutonium <plutonium.archimedes@gmail.com> - 2018-10-03 23:32 -0700
                Re: Archimedes "Village Idiot" Plutonium flunked the Math Test of a lifetime-generation test moroney@world.std.spaamtrap.com (Michael Moroney) - 2018-10-07 03:31 +0000
                Re: One of AP's most beautiful proofs to date-- and a waste. eastside.erik@gmail.com - 2018-10-06 21:31 -0700
                Re: One of AP's most beautiful proofs to date-- and a waste. James Waldby <not@valid.invalid> - 2018-10-08 04:32 +0000
                Re: One of AP's most beautiful proofs to date-- and a waste. Archimedes Plutonium <plutonium.archimedes@gmail.com> - 2018-10-07 21:59 -0700
                concept of prime should include concept of coprime Re: One of AP's most beautiful proofs to date-- Archimedes Plutonium <plutonium.archimedes@gmail.com> - 2018-10-07 22:35 -0700
                Re: concept of prime should include concept of coprime Re: One of AP's most beautiful proofs to date-- Archimedes Plutonium <plutonium.archimedes@gmail.com> - 2018-10-07 22:41 -0700
                Re: One of AP's most beautiful proofs to date-- the entire Pattern of Primes Re: the fingerprint of being prime // AP wants to do one better on Ulam's Prime Spiral qbwrfmlx@gmail.com - 2018-10-06 22:40 -0700
                bean counter gives a test Re: One of AP's most beautiful proofs to date-- the entire Pattern of Primes Archimedes Plutonium <plutonium.archimedes@gmail.com> - 2018-10-07 22:28 -0700
                Re: bean counter gives a test Re: One of AP's most beautiful proofs to date-- the entire Pattern of Primes qbwrfmlx@gmail.com - 2018-10-07 22:43 -0700
                Re: Archimedes "Village Idiot" Plutonium flunked the Math Test of a lifetime-generation test moroney@world.std.spaamtrap.com (Michael Moroney) - 2018-10-08 13:58 +0000
                AP's tool to determine if prime or composite-- Odd Rectangles Re: the fingerprint of being prime // AP wants to do one better on Ulam's Prime Spiral Archimedes Plutonium <plutonium.archimedes@gmail.com> - 2018-10-08 14:12 -0700
                Re: Archimedes "Village Idiot" Plutonium flunked the Math Test of a lifetime-generation test moroney@world.std.spaamtrap.com (Michael Moroney) - 2018-10-09 06:58 +0000

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