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Re: Factorization theory wrong? Or algorithmic error?
Posted:
Apr 6, 2012 12:49 PM
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On 05/04/2012 23:19, barker wrote:
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> We (mathematicians) have grown to accept the primality checkers as
> gospel. So did I, until recently.
>
> 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.
>
> 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:
>
> 3634908448770161716619462884730373820150226880205007030541419827683585
> 7931761274740311086713549497603607279611408949613526779622187756741117
> 9048935484829402996681944342388178421558785023331981868685440034884277
> 9396792124395994336764804183754455993340622344242614470170379064513230
> 0552661368276733695867117608484513671228954258971153834928109857741
>
> 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.
>
> 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:
>
> 3246726736489147307461784686107468324672673648914730746178468610746834
> 6821883878114173728372983219193183717113173468218838781141737283729832
> 1919318371711317
>
> ** 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).
>
> 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.
>
> 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.
>
> {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.}
>
> 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).
>
> Now check C's primality. C should be prime, per the lemma above. Right?
>
> 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
>
> Well, I can tell you that I have factorized C... and hand-checked it, as
> at first I could not believe the fluke finding.
>
> 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.
>
> 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.
>
> I do hope I have not overlooked anything. Your assistance is appreciated.
>
> Thank you,
>
> "barker" (associate of the late falsified non-dullrich Dr Pertti Lounesto)
>
> 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.
>
>
> -----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-----
>
For what it's worth a quick cut and paste produced..
http://www.mathwarehouse.com/arithmetic/numbers/prime-number/prime-test.php?number=111956094361694734740061540900257528436988746514301060213050630976617907530060726713223042028923487695623178805395619821799868743856430058731438452818437316840959014392166803390411010978334873
Results of Prime Factorization test
111956094361694734740061540900257528436988746514301060213050630976617907530060726713223042028923487695623178805395619821799868743856430058731438452818437316840959014392166803390411010978334873
is not a prime number
Obviously if you ask it to look for factors it will time out.
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