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First Proof That Infinitely Many Prime Numbers Come in Pairs
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Sam Wormley
May 16, 2013, 12:03:24 PM5/16/13
to
First Proof That Infinitely Many Prime Numbers Come in Pairs
> http://www.scientificamerican.com/article.cfm?id=first-proof-that-infinite-many-prime-numbers-come-in-pairs
> That goal is the proof to a conjecture concerning prime numbers.
> Those are the whole numbers that are divisible only by one and
> themselves. Primes abound among smaller numbers, but they become less
> and less frequent as one goes towards larger numbers. In fact, the
> gap between each prime and the next becomes larger and larger — on
> average. But exceptions exist: the ‘twin primes’, which are pairs of
> prime numbers that differ in value by 2. Examples of known twin
> primes are 3 and 5, or 17 and 19, or 2,003,663,613 × 2^195,000 − 1 and
> 2,003,663,613 × 2^195,000 + 1.
>
> The twin prime conjecture says that there is an infinite number of
> such twin pairs. Some attribute the conjecture to the Greek
> mathematician Euclid of Alexandria, which would make it one of the
> oldest open problems in mathematics.
> http://www.scientificamerican.com/article.cfm?id=first-proof-that-infinite-many-prime-numbers-come-in-pairs
> That goal is the proof to a conjecture concerning prime numbers.
> Those are the whole numbers that are divisible only by one and
> themselves. Primes abound among smaller numbers, but they become less
> and less frequent as one goes towards larger numbers. In fact, the
> gap between each prime and the next becomes larger and larger — on
> average. But exceptions exist: the ‘twin primes’, which are pairs of
> prime numbers that differ in value by 2. Examples of known twin
> primes are 3 and 5, or 17 and 19, or 2,003,663,613 × 2^195,000 − 1 and
> 2,003,663,613 × 2^195,000 + 1.
>
> The twin prime conjecture says that there is an infinite number of
> such twin pairs. Some attribute the conjecture to the Greek
> mathematician Euclid of Alexandria, which would make it one of the
> oldest open problems in mathematics.
Pubkeybreaker
May 16, 2013, 7:31:20 PM5/16/13
to
On May 16, 12:03 pm, Sam Wormley <sworml...@gmail.com> wrote:
> First Proof That Infinitely Many Prime Numbers Come in Pairs
>
>
>
> >http://www.scientificamerican.com/article.cfm?id=first-proof-that-inf...
> First Proof That Infinitely Many Prime Numbers Come in Pairs
>
>
>
> > That goal is the proof to a conjecture concerning prime numbers.
> > Those are the whole numbers that are divisible only by one and
> > themselves. Primes abound among smaller numbers, but they become less
> > and less frequent as one goes towards larger numbers. In fact, the
> > gap between each prime and the next becomes larger and larger -- on
> > Those are the whole numbers that are divisible only by one and
> > themselves. Primes abound among smaller numbers, but they become less
> > and less frequent as one goes towards larger numbers. In fact, the
> > average. But exceptions exist: the 'twin primes', which are pairs of
> > prime numbers that differ in value by 2. Examples of known twin
> > primes are 3 and 5, or 17 and 19, or 2,003,663,613 × 2^195,000 - 1 and
> > prime numbers that differ in value by 2. Examples of known twin
> > 2,003,663,613 × 2^195,000 + 1.
>
> > The twin prime conjecture says that there is an infinite number of
> > such twin pairs. Some attribute the conjecture to the Greek
> > mathematician Euclid of Alexandria, which would make it one of the
> > oldest open problems in mathematics.- Hide quoted text -
>
> > The twin prime conjecture says that there is an infinite number of
> > such twin pairs. Some attribute the conjecture to the Greek
> > mathematician Euclid of Alexandria, which would make it one of the
>
> - Show quoted text -
This is a gross misstatement of the proof. It did NOT prove that there
were infinitely many prime pairs. What it did prove was that the gap
between primes is FINITELY BOUNDED infinitely often. The bound is 70
x 10^6.
While this will probably be improved it is a long way to proving a
bound of
2.
Wally W.
May 16, 2013, 9:19:22 PM5/16/13
to
On Thu, 16 May 2013 16:31:20 -0700 (PDT), Pubkeybreaker wrote:
>On May 16, 12:03 pm, Sam Wormley <sworml...@gmail.com> wrote:
>> First Proof That Infinitely Many Prime Numbers Come in Pairs
>>
>>
>>
>> >http://www.scientificamerican.com/article.cfm?id=first-proof-that-inf...
>> > That goal is the proof to a conjecture concerning prime numbers.
>> > Those are the whole numbers that are divisible only by one and
>> > themselves. Primes abound among smaller numbers, but they become less
>> > and less frequent as one goes towards larger numbers. In fact, the
>> > gap between each prime and the next becomes larger and larger -- on
>> > average. But exceptions exist: the 'twin primes', which are pairs of
>> > prime numbers that differ in value by 2. Examples of known twin
>> > primes are 3 and 5, or 17 and 19, or 2,003,663,613 � 2^195,000 - 1 and
>> > 2,003,663,613 � 2^195,000 + 1.
>>
>> > The twin prime conjecture says that there is an infinite number of
>> > such twin pairs. Some attribute the conjecture to the Greek
>> > mathematician Euclid of Alexandria, which would make it one of the
>> > oldest open problems in mathematics.- Hide quoted text -
>>
>> - Show quoted text -
>
>This is a gross misstatement of the proof.
A gross misstatement by Sam? Really?!
>On May 16, 12:03 pm, Sam Wormley <sworml...@gmail.com> wrote:
>> First Proof That Infinitely Many Prime Numbers Come in Pairs
>>
>>
>>
>> >http://www.scientificamerican.com/article.cfm?id=first-proof-that-inf...
>> > That goal is the proof to a conjecture concerning prime numbers.
>> > Those are the whole numbers that are divisible only by one and
>> > themselves. Primes abound among smaller numbers, but they become less
>> > and less frequent as one goes towards larger numbers. In fact, the
>> > gap between each prime and the next becomes larger and larger -- on
>> > average. But exceptions exist: the 'twin primes', which are pairs of
>> > prime numbers that differ in value by 2. Examples of known twin
>> > primes are 3 and 5, or 17 and 19, or 2,003,663,613 � 2^195,000 - 1 and
>> > 2,003,663,613 � 2^195,000 + 1.
>>
>> > The twin prime conjecture says that there is an infinite number of
>> > such twin pairs. Some attribute the conjecture to the Greek
>> > mathematician Euclid of Alexandria, which would make it one of the
>> > oldest open problems in mathematics.- Hide quoted text -
>>
>> - Show quoted text -
>
>This is a gross misstatement of the proof.
Oh ... I guess not.
He didn't actually write what he posted.
Maybe he'll want to take credit for vetting it ... or not.
Richard Tobin
May 17, 2013, 6:42:33 AM5/17/13
to
In article <4c7758a5-2b88-4756...@d6g2000yqi.googlegroups.com>,
but it appears to be using "prime pairs" to mean "successive primes",
and "twin primes" to mean "prime pairs where the difference is 2".
That makes the headline misleading because we already knew there were
infinitely many pairs of successive primes.
-- Richard
Pubkeybreaker <pubkey...@aol.com> wrote:
>This is a gross misstatement of the proof. It did NOT prove that there
>were infinitely many prime pairs. What it did prove was that the gap
>between primes is FINITELY BOUNDED infinitely often. The bound is 70
>x 10^6.
I agree that the article (quoted from Scientific American) is unclear,
>This is a gross misstatement of the proof. It did NOT prove that there
>were infinitely many prime pairs. What it did prove was that the gap
>between primes is FINITELY BOUNDED infinitely often. The bound is 70
>x 10^6.
but it appears to be using "prime pairs" to mean "successive primes",
and "twin primes" to mean "prime pairs where the difference is 2".
That makes the headline misleading because we already knew there were
infinitely many pairs of successive primes.
-- Richard
Pubkeybreaker
May 17, 2013, 9:34:46 AM5/17/13
to
On May 17, 6:42 am, rich...@cogsci.ed.ac.uk (Richard Tobin) wrote:
> In article <4c7758a5-2b88-4756-91f1-d59d43bcd...@d6g2000yqi.googlegroups.com>,
> In article <4c7758a5-2b88-4756-91f1-d59d43bcd...@d6g2000yqi.googlegroups.com>,
>
> Pubkeybreaker <pubkeybrea...@aol.com> wrote:
> >This is a gross misstatement of the proof. It did NOT prove that there
> >were infinitely many prime pairs. What it did prove was that the gap
> >between primes is FINITELY BOUNDED infinitely often. The bound is 70
> >x 10^6.
>
> I agree that the article (quoted from Scientific American) is unclear,
> but it appears to be using "prime pairs" to mean "successive primes",
> and "twin primes" to mean "prime pairs where the difference is 2".
>
> That makes the headline misleading because we already knew there were
> infinitely many pairs of successive primes.
Not with a finitely bounded gap between them we didn't.
> Pubkeybreaker <pubkeybrea...@aol.com> wrote:
> >This is a gross misstatement of the proof. It did NOT prove that there
> >were infinitely many prime pairs. What it did prove was that the gap
> >between primes is FINITELY BOUNDED infinitely often. The bound is 70
> >x 10^6.
>
> I agree that the article (quoted from Scientific American) is unclear,
> but it appears to be using "prime pairs" to mean "successive primes",
> and "twin primes" to mean "prime pairs where the difference is 2".
>
> That makes the headline misleading because we already knew there were
> infinitely many pairs of successive primes.
Richard Tobin
May 17, 2013, 10:42:21 AM5/17/13
to
In article <0545f14d-ed7c-48a1...@k6g2000yqh.googlegroups.com>,
-- Richard
Pubkeybreaker <pubkey...@aol.com> wrote:
>> That makes the headline misleading because we already knew there were
>> infinitely many pairs of successive primes.
>Not with a finitely bounded gap between them we didn't.
Yes, that's why the headline is wrong - it doesn't say that.
>> That makes the headline misleading because we already knew there were
>> infinitely many pairs of successive primes.
>Not with a finitely bounded gap between them we didn't.
-- Richard
dull...@sprynet.com
May 17, 2013, 11:54:23 AM5/17/13
to
On Thu, 16 May 2013 16:31:20 -0700 (PDT), Pubkeybreaker
the difficulty of factoring large primes.
Makes you wonder how accurate they are on topics where you
can't immediately see the errors...
Aatu Koskensilta
May 17, 2013, 12:45:37 PM5/17/13
to
dull...@sprynet.com writes:
> Some years ago there was something in Scientific American about
> the difficulty of factoring large primes.
I once read it somewhere that factoring large primes is what is known
> Some years ago there was something in Scientific American about
> the difficulty of factoring large primes.
as an NP ("Non-Polynomial") complete problem.
--
Aatu Koskensilta (aatu.kos...@uta.fi)
"Wovon man nicht sprechen kann, dar�ber muss man schweigen"
- Ludwig Wittgenstein, Tractatus Logico-Philosophicus
1treePetrifiedForestLane
May 17, 2013, 8:17:08 PM5/17/13
to
bounded?... does that mean that there are no primes,
further than taht, apart?... seems, I don't know, but
I prefer the twin primes, viz
Brun's constant is not transcendental.
further than taht, apart?... seems, I don't know, but
I prefer the twin primes, viz
Brun's constant is not transcendental.
Bill Taylor
May 18, 2013, 1:41:15 AM5/18/13
to
On May 18, 4:45 am, Aatu Koskensilta <aatu.koskensi...@uta.fi> wrote:
-- Bantering Bill
> dullr...@sprynet.com writes:
> > Some years ago there was something in Scientific American about
> > the difficulty of factoring large primes.
>
> I once read it somewhere that factoring large primes is what is known
> as an NP ("Non-Polynomial") complete problem.
It's also at the very top of the oracle hierarchy!
> > Some years ago there was something in Scientific American about
> > the difficulty of factoring large primes.
>
> I once read it somewhere that factoring large primes is what is known
> as an NP ("Non-Polynomial") complete problem.
-- Bantering Bill
Richard Tobin
May 18, 2013, 9:06:55 AM5/18/13
to
In article <bc48b311-68c0-44cd...@fq2g2000pbb.googlegroups.com>,
are always more the 70 million apart.
It's easy to see that there are arbitrarily large gaps between some
consecutive primes. Consider N factorial: the N-1 integers N!-N ... N!-2
are all composite.
-- Richard
1treePetrifiedForestLane <Spac...@hotmail.com> wrote:
>bounded?... does that mean that there are no primes,
>further than taht, apart?...
No, it means that there isn't a point after which consecutive primes
>bounded?... does that mean that there are no primes,
>further than taht, apart?...
are always more the 70 million apart.
It's easy to see that there are arbitrarily large gaps between some
consecutive primes. Consider N factorial: the N-1 integers N!-N ... N!-2
are all composite.
-- Richard
Graham Cooper
May 18, 2013, 4:48:19 PM5/18/13
to
On May 18, 4:00 am, Gus Gassmann <no...@nospam.com> wrote:
> On 17/05/2013 1:45 PM, Aatu Koskensilta wrote:
> P = NP. Very impressive.
>
No Oracles allowed!
Herc
--
IS DOG+1 a NUMBER ?
http://blockprolog.com/nat-s-dog.png
> On 17/05/2013 1:45 PM, Aatu Koskensilta wrote:
>
> > dullr...@sprynet.com writes:
>
> >> Some years ago there was something in Scientific American about
> >> the difficulty of factoring large primes.
>
> > I once read it somewhere that factoring large primes is what is known
> > as an NP ("Non-Polynomial") complete problem.
>
> ... which also settles that other notorious problem, proving that
> > dullr...@sprynet.com writes:
>
> >> Some years ago there was something in Scientific American about
> >> the difficulty of factoring large primes.
>
> > I once read it somewhere that factoring large primes is what is known
> > as an NP ("Non-Polynomial") complete problem.
>
> P = NP. Very impressive.
>
No Oracles allowed!
Herc
--
IS DOG+1 a NUMBER ?
http://blockprolog.com/nat-s-dog.png
1treePetrifiedForestLane
May 18, 2013, 5:36:47 PM5/18/13
to
ah, so, there could still be infinity of twin-primes.
Graham Cooper
May 18, 2013, 6:18:01 PM5/18/13
to
On May 19, 7:36 am, 1treePetrifiedForestLane <Space...@hotmail.com>
wrote:
N!-N
---- = (N-1)! - 1
N
^_^
wrote:
---- = (N-1)! - 1
N
^_^
Richard Tobin
May 19, 2013, 6:16:29 AM5/19/13
to
In article <1a2603a5-a24b-4b98...@k8g2000pbf.googlegroups.com>,
Very good, but what is your point?
-- Richard
Very good, but what is your point?
-- Richard
Charlie-Boo
May 19, 2013, 12:40:22 PM5/19/13
to
Charlie-Boo
May 19, 2013, 12:46:11 PM5/19/13
to
On May 18, 4:48 pm, Graham Cooper <grahamcoop...@gmail.com> wrote:
Let Q be (all A)(exists B) LT(A,B) ^ P(B) ^ P(f(B))
Could Q be true and unprovable? Could Q be false and unrefutable?
For which P and f [specific or general] is Q one or the other of these
conditions?
C-B
Charlie-Boo
May 19, 2013, 12:58:00 PM5/19/13
to
heck with silly ZF, and we aren't questioning sets anyway, say we are
using PA.
Then if Q = = PA is consistent the 1st condition holds (for us.)
Now what can be proven in PA? This question is really sad, because
apparently people don't know - does anyone? I say this because when
talking about when certain theorems known to be true in PA apply, they
always say "If it has Peano's axioms" or some dinky "You can express
arithmetic in it." PA means a+b=c = = |- a+b=c , a*b=c = = |- a*b=c
and |-(allx)TRUE(x) where TRUE(a) = = a is a natural number. ( TRUE
means everything as in {x|TRUE} not like a "true sentence".)
In general, if Q is a Godel sentence of PA. In other words . . . etc.
C-B
Graham Cooper
May 19, 2013, 4:31:08 PM5/19/13
to
My Defn:
if THM is an axiom then THM is provable
if A is provable and B is provable
and A^B->THM
then THM is provable
Charlie-Boo
May 19, 2013, 11:15:21 PM5/19/13
to
On May 19, 4:31 pm, Graham Cooper <grahamcoop...@gmail.com> wrote:
> On May 20, 2:46 am, Charlie-Boo <shymath...@gmail.com> wrote:
>
>
>
>
>
>
>
>
>
> > On May 18, 4:48 pm, Graham Cooper <grahamcoop...@gmail.com> wrote:
>
> > > On May 18, 4:00 am, Gus Gassmann <no...@nospam.com> wrote:
>
> > > > On 17/05/2013 1:45 PM, Aatu Koskensilta wrote:
>
> > > > > dullr...@sprynet.com writes:
>
> > > > >> Some years ago there was something in Scientific American about
> > > > >> the difficulty of factoring large primes.
>
> > > > > I once read it somewhere that factoring large primes is what is known
> > > > > as an NP ("Non-Polynomial") complete problem.
>
> > > > ... which also settles that other notorious problem, proving that
> > > > P = NP. Very impressive.
>
> > > No Oracles allowed!
>
> > > Herc
>
> > > --
>
> > > IS DOG+1 a NUMBER ?
>
> > >http://blockprolog.com/nat-s-dog.png
>
> > Let P be a recursive set and f(I) be a recursive function.
>
> > Let Q be (all A)(exists B) LT(A,B) ^ P(B) ^ P(f(B))
>
> > Could Q be true and unprovable? Could Q be false and unrefutable?
>
> > For which P and f [specific or general] is Q one or the other of these
> > conditions?
>
> > C-B
>
> Define provable!
Pay 'tension! I said "say we are using PA. . . . Now what can be
> On May 20, 2:46 am, Charlie-Boo <shymath...@gmail.com> wrote:
>
>
>
>
>
>
>
>
>
> > On May 18, 4:48 pm, Graham Cooper <grahamcoop...@gmail.com> wrote:
>
> > > On May 18, 4:00 am, Gus Gassmann <no...@nospam.com> wrote:
>
> > > > On 17/05/2013 1:45 PM, Aatu Koskensilta wrote:
>
> > > > > dullr...@sprynet.com writes:
>
> > > > >> Some years ago there was something in Scientific American about
> > > > >> the difficulty of factoring large primes.
>
> > > > > I once read it somewhere that factoring large primes is what is known
> > > > > as an NP ("Non-Polynomial") complete problem.
>
> > > > ... which also settles that other notorious problem, proving that
> > > > P = NP. Very impressive.
>
> > > No Oracles allowed!
>
> > > Herc
>
> > > --
>
> > > IS DOG+1 a NUMBER ?
>
> > >http://blockprolog.com/nat-s-dog.png
>
> > Let P be a recursive set and f(I) be a recursive function.
>
> > Let Q be (all A)(exists B) LT(A,B) ^ P(B) ^ P(f(B))
>
> > Could Q be true and unprovable? Could Q be false and unrefutable?
>
> > For which P and f [specific or general] is Q one or the other of these
> > conditions?
>
> > C-B
>
> Define provable!
proven in PA? This question is really sad, . .."
This may make a constructive proof actually useful (Martin Davis
demanded that I make my axiomatized proofs of incompleteness in logic
constructive despite the loss from ignoring nonconstructive proofs,
much to my chagrin.)
If we can construct Godel sentences (as opposed to only proving they
exist), then as a bonus we can see if they are equivalent to our Q
Now what's the formula for constructing a Godel sentence? And don't
forget that while Godel thought of only 1, there are 3 in the seminal
"18 Word Proof of the Incompleteness Theorems of Godel, Rosser and
Smullyan".
C-B
Charlie-Boo
May 19, 2013, 11:30:30 PM5/19/13
to
On May 18, 4:48 pm, Graham Cooper <grahamcoop...@gmail.com> wrote:
http://forums.cannabisculture.com/forums/ubbthreads.php?ubb=showflat&Number=576091
> http://blockprolog.com/nat-s-dog.png
Graham Cooper
May 20, 2013, 1:28:48 AM5/20/13
to
A0 based on the defn I gave you.
!a0(a1)
8203215
Note a 2nd line is required A1=8203215 (in binary)
A FULLY CODED GODEL STATEMENT
https://groups.google.com/group/alt.bible/browse_thread/thread/4f4dc0f290a90d12/aef683a9f8504edd
*******************************
a0(a11)=a11
a0(a11)=a0(a111)^a0(a110)^(!(a111^a110)va11)
a1=11111010010101111001111
!a0(a1)
*******************************
This is just my definition above encoded.
if THM is an axiom then THM is provable
if A is provable and B is provable
and A^B->THM
then THM is provable
proof( a11 ) <- proof( a111 ) & proof( a111 ) & not( a111 or a11 )
then
a11 = 8203215
not(proof(8203215)
Herc
Graham Cooper
May 20, 2013, 1:31:17 AM5/20/13
to
...
> a1 = 8203215
> not(proof(8203215)
>
The above formula refers to itself !
Herc
> a1 = 8203215
> not(proof(8203215)
>
The above formula refers to itself !
Herc
Charlie-Boo
May 20, 2013, 6:48:45 AM5/20/13
to
We have neither digits, numerals nor constant considerations. We have
only sets and functions. We have Q, defined to be (all A)(exists B)
LT(A,B) ^ P(B) ^ P(f(B)) where P be a recursive set and f(I) be a
recursive function. You have the recursive relation x proves y, the
r.e. set x is provable, the recursive function f(x)=y iff x proves y.
Let us pay homage to Godel, for He created the undecidable sentence
and decided it is true. How did he construct that temple from the
wisdom in his head? When will the Godel Sentence G be the same as the
C-B Sentence Q?
You must express "wff number x with x substituted for its free
variable is not provable", or, expanding your possibilities, realize
that,
"When any one of these sets [the true, provable and unrefutable
sentences] P, is expressible or representable, the sentence
that expresses or represents, respectively, 'This is in P.' is
undecidable." This includes:
1. Since unprovability is expressible: The sentence that expresses
"This is not provable." is undecidable.
2. Since refutability is expressible: The sentence that expresses
"This is refutable." is undecidable.
3. Since refutability is representable: The sentence that represents
"This is refutable." is undecidable.
http://www.cs.nyu.edu/pipermail/fom/2010-July/014933.html
When is each of these Q?
C-B
Graham Cooper
May 20, 2013, 5:21:22 PM5/20/13
to
On May 20, 8:48 pm, Charlie-Boo <shymath...@gmail.com> wrote:
> On May 20, 1:31 am, Graham Cooper <grahamcoop...@gmail.com> wrote:
>
> > ...
>
> > > a1 = 8203215
> > > not(proof(8203215)
>
> > The above formula refers to itself !
>
> > Herc
>
> Yes, and a fine job indeed. But let us abstract UP and generalize.
> We have neither digits, numerals nor constant considerations. We have
> only sets and functions.
OK but you need SOME functionality in your parameters.
> On May 20, 1:31 am, Graham Cooper <grahamcoop...@gmail.com> wrote:
>
> > ...
>
> > > a1 = 8203215
> > > not(proof(8203215)
>
> > The above formula refers to itself !
>
> > Herc
>
> Yes, and a fine job indeed. But let us abstract UP and generalize.
> We have neither digits, numerals nor constant considerations. We have
> only sets and functions.
Be it
call by name sqrt(X)
call by value sqrt(2)
call by line number, godel number..
20 not proof(20)
or
G<->not(proof(G))
are fine too, it depends what TYPE the domain of your functions are.
> We have Q, defined to be (all A)(exists B)
> LT(A,B) ^ P(B) ^ P(f(B)) where P be a recursive set and f(I) be a
> recursive function. You have the recursive relation x proves y, the
> r.e. set x is provable, the recursive function f(x)=y iff x proves y.
> Let us pay homage to Godel, for He created the undecidable sentence
> and decided it is true. How did he construct that temple from the
> wisdom in his head? When will the Godel Sentence G be the same as the
> C-B Sentence Q?
>
> You must express "wff number x with x substituted for its free
> variable is not provable", or, expanding your possibilities, realize
> that,
>
> "When any one of these sets [the true, provable and unrefutable
> sentences] P, is expressible or representable, the sentence
> that expresses or represents, respectively, 'This is in P.' is
> undecidable." This includes:
>
> 1. Since unprovability is expressible: The sentence that expresses
> "This is not provable." is undecidable.
> 2. Since refutability is expressible: The sentence that expresses
> "This is refutable." is undecidable.
> 3. Since refutability is representable: The sentence that represents
> "This is refutable." is undecidable.
>
> http://www.cs.nyu.edu/pipermail/fom/2010-July/014933.html
>
> When is each of these Q?
>
> C-B
GODEL = no PROVE(X) predicate
TARSKI = and no TRUE(X) either
isn't helping the poor old programmer!
As far as PROGRAMMING IN LOGIC is concerned.
2+1=3 ?
and
PROVE( 2+1=3 )
is just selecting the [TRACE=ON ] button!
----------------
That is why I simplified my provability predicate in Provable Set
Theory.
OLD
XeS<->p(X) <-> provable( XeS<->p(X) )
not(provable(X)) <-> provable(not(X))
provable(THM) <-> provable(A) ^ provable(B) ^ provable(A^B->THM)
-----------------
Godel was right in a way - YOU JUST CANT PROGRAM PROVABLE()!
it reduces down to the same as:
TRUE-WFF(X)
THEOREM(X)
DERIVE(X)
TRUE(X)
SOLVE(X)
THM(X)
T(X)
So THM(X) <-> NOT(PROVABLE(NOT(X)))
can just be done with double negation.
F<->F
~(F<->~F) %NO CONTRADICTIONS!
..
XeS<->p(X) <-> XeS<->p(X)
RUSSELLSET <-> (CONTRADICTION)
RUSSELLSET <-> FALSE
~EXIST(rs) rs<->x~ex
---------------------
I can't help any further because I simply don't have a predicate for
PROVE(THM)
t(R,z(Z)) :- if(L,R) , t(L,Z).
Where does it go?
Herc
Phil Carmody
May 23, 2013, 5:54:46 AM5/23/13
to
Pubkeybreaker <pubkey...@aol.com> writes:
> On May 17, 6:42ï¿œam, rich...@cogsci.ed.ac.uk (Richard Tobin) wrote:
> > In article <4c7758a5-2b88-4756-91f1-d59d43bcd...@d6g2000yqi.googlegroups.com>,
> >
> > Pubkeybreaker ￜ<pubkeybrea...@aol.com> wrote:
> > >This is a gross misstatement of the proof. It did NOT prove that there
> > >were infinitely many prime pairs. ï¿œWhat it did prove was that the gap
> > >between primes is FINITELY BOUNDED infinitely often. ï¿œThe bound is 70
as meaning that there is a finite gap such that there are infinitely many
consecutive primes with that gap. At no point did I interpret "prime pairs"
to mean "twin primes" rather than "consecutive primes a fixed, but unspecified,
distance apart".
Phil
--
"In a world of magnets and miracles"
-- Insane Clown Posse, Miracles, 2009. Much derided.
"Magnets, how do they work"
-- Pink Floyd, High Hopes, 1994. Lauded as lyrical geniuses.
> On May 17, 6:42ï¿œam, rich...@cogsci.ed.ac.uk (Richard Tobin) wrote:
> > In article <4c7758a5-2b88-4756-91f1-d59d43bcd...@d6g2000yqi.googlegroups.com>,
> >
> > Pubkeybreaker ￜ<pubkeybrea...@aol.com> wrote:
> > >This is a gross misstatement of the proof. It did NOT prove that there
> > >between primes is FINITELY BOUNDED infinitely often. ï¿œThe bound is 70
> > >x 10^6.
> >
> > I agree that the article (quoted from Scientific American) is unclear,
> > but it appears to be using "prime pairs" to mean "successive primes",
> > and "twin primes" to mean "prime pairs where the difference is 2".
> >
> > That makes the headline misleading because we already knew there were
> > infinitely many pairs of successive primes.
>
> Not with a finitely bounded gap between them we didn't.
I interpreted the headline (having already heard the story via other routes)
> >
> > I agree that the article (quoted from Scientific American) is unclear,
> > but it appears to be using "prime pairs" to mean "successive primes",
> > and "twin primes" to mean "prime pairs where the difference is 2".
> >
> > That makes the headline misleading because we already knew there were
> > infinitely many pairs of successive primes.
>
> Not with a finitely bounded gap between them we didn't.
as meaning that there is a finite gap such that there are infinitely many
consecutive primes with that gap. At no point did I interpret "prime pairs"
to mean "twin primes" rather than "consecutive primes a fixed, but unspecified,
distance apart".
Phil
--
"In a world of magnets and miracles"
-- Insane Clown Posse, Miracles, 2009. Much derided.
"Magnets, how do they work"
-- Pink Floyd, High Hopes, 1994. Lauded as lyrical geniuses.
Phil Carmody
May 23, 2013, 5:59:30 AM5/23/13
to
Aatu Koskensilta <aatu.kos...@uta.fi> writes:
> dull...@sprynet.com writes:
> > Some years ago there was something in Scientific American about
> > the difficulty of factoring large primes.
>
> I once read it somewhere that factoring large primes is what is known
> as an NP ("Non-Polynomial") complete problem.
My recollection is that it was Bill Gate in /The Road Ahead/ who
> dull...@sprynet.com writes:
> > Some years ago there was something in Scientific American about
> > the difficulty of factoring large primes.
>
> I once read it somewhere that factoring large primes is what is known
> as an NP ("Non-Polynomial") complete problem.
first mentioned how important an advance it would be if we could
factor large primes.
rossum
May 23, 2013, 8:01:49 AM5/23/13
to
On 23 May 2013 12:59:30 +0300, Phil Carmody
factor any large prime you give me. Unfortunately the margins of this
post are too small...
rossum
<thefatphi...@yahoo.co.uk> wrote:
>Aatu Koskensilta <aatu.kos...@uta.fi> writes:
>> dull...@sprynet.com writes:
>> > Some years ago there was something in Scientific American about
>> > the difficulty of factoring large primes.
>>
>> I once read it somewhere that factoring large primes is what is known
>> as an NP ("Non-Polynomial") complete problem.
>
>My recollection is that it was Bill Gate in /The Road Ahead/ who
>first mentioned how important an advance it would be if we could
>factor large primes.
>
>Phil
Well, that's going to be a Fields Medal for me then. I can completely
>Aatu Koskensilta <aatu.kos...@uta.fi> writes:
>> dull...@sprynet.com writes:
>> > Some years ago there was something in Scientific American about
>> > the difficulty of factoring large primes.
>>
>> I once read it somewhere that factoring large primes is what is known
>> as an NP ("Non-Polynomial") complete problem.
>
>My recollection is that it was Bill Gate in /The Road Ahead/ who
>first mentioned how important an advance it would be if we could
>factor large primes.
>
>Phil
factor any large prime you give me. Unfortunately the margins of this
post are too small...
rossum
David Bernier
May 23, 2013, 4:47:25 PM5/23/13
to
On 05/23/2013 05:59 AM, Phil Carmody wrote:
> Aatu Koskensilta <aatu.kos...@uta.fi> writes:
>> dull...@sprynet.com writes:
>>> Some years ago there was something in Scientific American about
>>> the difficulty of factoring large primes.
>>
>> I once read it somewhere that factoring large primes is what is known
>> as an NP ("Non-Polynomial") complete problem.
>
> My recollection is that it was Bill Gate in /The Road Ahead/ who
> first mentioned how important an advance it would be if we could
> factor large primes.
>
> Phil
>
With respect to P =? NP, Scott Aaronson recently wrote:
> Aatu Koskensilta <aatu.kos...@uta.fi> writes:
>> dull...@sprynet.com writes:
>>> Some years ago there was something in Scientific American about
>>> the difficulty of factoring large primes.
>>
>> I once read it somewhere that factoring large primes is what is known
>> as an NP ("Non-Polynomial") complete problem.
>
> My recollection is that it was Bill Gate in /The Road Ahead/ who
> first mentioned how important an advance it would be if we could
> factor large primes.
>
> Phil
>
`` It�s just too hopelessly far beyond our current abilities[...]"
Ref.:
< http://www.scottaaronson.com/blog/?p=1385#comment-73266 > .
dave
--
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