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Re: Factorization theory wrong? Or algorithmic error?
Posted:
Apr 7, 2012 11:50 AM
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On Apr 7, 10:46 am, hagman <goo...@von-eitzen.de> wrote:
> Am Freitag, 6. April 2012 22:46:50 UTC+2 schrieb William Hughes:
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> > On Apr 5, 7:19 pm, barker
> > <name.temporarily.withh...@antispamming.harvard.edu> wrote:
> > > -----BEGIN PGP SIGNED MESSAGE-----
> > > Hash: SHA512
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> > > We (mathematicians) have grown to accept the primality checkers as
> > > gospel. So did I, until recently.
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> > > This could be big, or it could be I've overlooked something, though I
> > > have hunted for 3 days for a flaw. I'd appreciate if you could check
> > > this over for me. This post is digitally signed in case I need to prove
> > > ownership, should my discovery (if it is a discovery) be stolen.
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> > > As part of my research into improving factorization algorithms, I
> > > encountered this composite number, 347 decimal digits long (>1150 bits),
> > > which I'll call A:
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> > > 3634908448770161716619462884730373820150226880205007030541419827683585
> > > 7931761274740311086713549497603607279611408949613526779622187756741117
> > > 9048935484829402996681944342388178421558785023331981868685440034884277
> > > 9396792124395994336764804183754455993340622344242614470170379064513230
> > > 0552661368276733695867117608484513671228954258971153834928109857741
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> > > I won't tell you how I generated A, because if there's no flaw in what
> > > I've done (I intend to make real money out of this, if it is possible),
> > > the way I came up with A is a giveaway to the whole process.
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> > > I won't ask you to factorize A, because you may not be able to. Here is
> > > its "smaller prime factor"** ("B"), which is 156 decimal digits long:
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> > > 3246726736489147307461784686107468324672673648914730746178468610746834
> > > 6821883878114173728372983219193183717113173468218838781141737283729832
> > > 1919318371711317
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> > > ** that is, smaller as identified by all the factorization algorithms
> > > that I have encountered. If you are not professional mathematicians
> > > and do not have access to factorization tools, I recommend you use:
> > > http://www.alpertron.com.ar/ECM.HTM
> > > which will work on any modern web browser, to confirm what I have
> > > just stated (i.e., that A is composite, B is prime and that A/B is an
> > > integer; whether A/B is prime is moot).
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> > > ECM's author Dario Alpern has diligently implemented factorization
> > > algorithms. His implementations are not in question (I assume they are
> > > accurate, as do my colleagues) - it is the theory itself that is now
> > > in question.
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> > > Divide A by B to get the 192 decimal digit number C. Since 192/2 < 156,
> > > it follows that if B was the smaller prime factor of A, then C must be
> > > prime.
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> > > {Lemma: Assume C was non-prime. Then it must have at least one prime
> > > factor that is less than 97 (= 192/2 + 1) decimal digits long. This
> > > would falsify the algorithmic result that B, at 156 decimal digits, is
> > > the smallest prime factor of A. Therefore C must be prime.}
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> > > I didn't want to give you C (= A/B) as I want you to (trivially) compute
> > > it yourself (but for the lazy, it appears at the end of this post).
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> > > Now check C's primality. C should be prime, per the lemma above. Right?
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> > > Indeed, all the primality checkers I have tested show that C is prime.
> > > Including the java one at:
> > > http://www.alpertron.com.ar/ECM.HTM
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> > > Well, I can tell you that I have factorized C... and hand-checked it, as
> > > at first I could not believe the fluke finding.
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> > > C's smaller factor is almost 2^300, so C's decomposition is non-trivial.
> > > In the time window before you can brute-force this, I will disclose its
> > > factors, and the methods that:
> > > 1) got me to A (Hint: diagonalization, Cantor), and
> > > 2) factorized C.
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> > > But at this point, I do not want to disclose C's factors, until I have
> > > heard the more competent fellow mathematicians here confirm C's alleged
> > > primality, according to the algorithms we all becomed conditioned to
> > > believing are true.
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> > > I do hope I have not overlooked anything. Your assistance is appreciated.
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> > > Thank you,
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> > > "barker" (associate of the late falsified non-dullrich Dr Pertti Lounesto)
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> > > Footnote: For the lazy, here is the 192 decimal digit number C:
> > > 1119560943616947347400615409002575284369887465143010602130506309766179
> > > 0753006072671322304202892348769562317880539561982179986874385643005873
> > > 1438452818437316840959014392166803390411010978334873
> > > which tested algorithms suggest is prime, but which I have factorized.
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> > > -----BEGIN PGP SIGNATURE-----
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> > > VQEcBAEBCgAGBQJOe7YRAAoJEAjjY4weks8oA7QIAK3ELb/+NKP1vLPT8f7HQTaf
> > > Ym-qnG0TdO44RMJdbqpxsp6DoMx5JkMgluha8y6LIV3rBHHDKGQx3YwKzVTT5r81
> > > DOOQ-r3LQdLgmoemhdot2Dse16XQ7OoWzvJw-qvvYYBZ0S/J2SsrAFUAoQAe35/4
> > > 9NkVg3-JSzV+AFPQyv5hpS780v0cObSPl7yz32MypgvZkYZupC3xP/3Pdl8Fg205
> > > NkiDEaDl-JcIKM8ARJJtndd7cfNBKZ3Bh1OEQ1NwPFEMZ6uAR3S/DLdF0dY1MMxr
> > > RRcluph+ML-mTRZngA8NG9qRCBQT2IgTZNatjnZv2pcwgC0MddUnyS07bNypHg8=
> > > =87KU
> > > -----END PGP SIGNATURE-----
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> > I predict that if and when you provide your putative factors, one of
> > them will have a prime factor less that 1,000,000
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> > - William Hughes
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> Since it is readily checked that C has no prime factors less than 1,000,000
> (or even less than 1,000,000,000), your prediction amounts to barker
> not revealing factors at all. :)
>
> hagman
Nope. The above is only valid if the putative factors are in fact
factors of C.
Since C does not have any proper factors they will not be.
- William Hughes
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