August 2002 - Fractint - mailman.xmission.com

Re: [Fractint] Extended Fermat's Last Theorem - Slightly Off-topic
by Andrew Coppin 23 Aug '02

23 Aug '02

IIRC, a pair of Japanise (?) students proposed that all elliptic curves are also modular forms. Someone else pointed out that, assuming Fermat was wrong, his equation x^n + y^n = z^n would be an elliptic equation, and the corresponding modular form is decidedly weird. Yet ANOTHER mathematition later proved that this modular form is in fact too weird to actually exist. In other words, assuming that EVERY elliptic curve REALLY IS a modular form, Fermat's Last Theorum MUST be true. Andrew Wiles proved that elliptic curves ARE modular, and as a side effect proved Fermat right. Andrew. PS. I've read the book. And I'm kinda bored today... >From: "Morgan L. Owens" <packrat(a)nznet.gen.nz> >Reply-To: fractint(a)mailman.xmission.com >To: fractint(a)mailman.xmission.com, fractint(a)mailman.xmission.com >Subject: Re: [Fractint] Extended Fermat's Last Theorem - Slightly >Off-topic >Date: Mon, 19 Aug 2002 13:43:43 +1200 > >At 01:09 19/08/2002, dirving(a)box.net.au wrote: >>"Ricardo M. Forno" wrote: >> > >> > Does someone in the list know if the following conjecture, that >>includes as >> > a particular case Fermat's Last Theorem, has ever been posed, and >>proved >> > false or true? >> > x[1]^p + x[2]^p + ... x[n]^p = z^p, for x[i] > 0 and 1 < n < p, has no >> > integer solutions. >> > TIA. >> > >> > _______________________________________________ >> > Fractint mailing list >> > Fractint(a)mailman.xmission.com >> > http://mailman.xmission.com/cgi-bin/mailman/listinfo/fractint >> >>I would conjecture ... no. > >Reasonable: a proof of this extended conjecture would in turn provide an >alternatve proof of FLT as a special case. > >So if there has been a proof, it has been found more recently than (and >probably develops the ideas of) Wiles' tour de force. > > >Morgan L. Owens >"Yes, Andrew; a corollary :-)" > > >_______________________________________________ >Fractint mailing list >Fractint(a)mailman.xmission.com >http://mailman.xmission.com/cgi-bin/mailman/listinfo/fractint _________________________________________________________________ Chat with friends online, try MSN Messenger: http://messenger.msn.com

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Re: [Fractint] Extended Fermat's Last Theorem - Slightly Off-topic
by Franktal Gallery 22 Aug '02

22 Aug '02

Morgan, thanks for the explanation and the link ! Regards, Miguel >From: "Morgan L. Owens" <packrat(a)nznet.gen.nz> >Reply-To: fractint(a)mailman.xmission.com >To: fractint(a)mailman.xmission.com, fractint(a)mailman.xmission.com >Subject: Re: [Fractint] Extended Fermat's Last Theorem - Slightly >Off-topic >Date: Fri, 23 Aug 2002 00:48:14 +1200 > >At 23:46 22/08/2002, Franktal Gallery wrote: >>Could you please explain the "semistable" concept ? > >Oof. Where to start (and where to finish?) > >See http://mathworld.wolfram.com/Semistable - following is a paraphrase. >[snip] >[snip] >[snip] _________________________________________________________________ Chat with friends online, try MSN Messenger: http://messenger.msn.com

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Re: [Fractint] Extended Fermat's Last Theorem - Slightly Off-topic
by Franktal Gallery 22 Aug '02

22 Aug '02

Could you please explain the "semistable" concept ? Thanks, Miguel >From: "Morgan L. Owens" <packrat(a)nznet.gen.nz> > >[snip] > >The two theorems run: >If FLT is false, then y^2=x(x-z^n)(x+y^n) is semistable, but not modular. >(Conjectured in 1982 by Gerhard Frey; proven in 1986 by Ken Ribet via >Jean-Pierre Serre). > >and > >All semistable elliptic curves are modular. (Conjectured by Yutaka Taniyama >in 1955, later generalised by Goro Shimura and Andre Weil by dropping the >"semistable" bit; proven in 1994 by Andrew Wiles). > _________________________________________________________________ Send and receive Hotmail on your mobile device: http://mobile.msn.com

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FOTD 20-08-02 (Fractal Kernels [7])
by Jim Muth 20 Aug '02

20 Aug '02

FOTD -- August 20, 2002 (Rating 7) Fractal visionaries and enthusiasts: Today's fractal image is corny. This does not mean that it is trite and old-fashioned, it means that it is filled with corn kernels. The corn in the image appears in neat rows, almost the way corn kernels grow on the corncob. This corniness inspired the name "Fractal Kernels". The effect of corn kernels is produced by iterating the formula Z^(-2)-0.01(Z^2)+(1/C), and doing the calculation with the bailout radius set to 99.999, (which is critical), and the outside set to < fmod >. The small range of colors in the image was not intentional. It exists because the image uses only a small part of the 256-color palette. Unwilling to have yet another consecutive 6-rated FOTD, I rated today's image at a rather liberal 7. Perhaps a 6-1/2 would have been more appropriate, but my rating scale does not have half- step increments. The 4-minute render time is acceptable, but downloading the completed GIF image from: <http://home.att.net/~Paul.N.Lee/FotD/FotD.html> or from: <http://sdboyd.dyndns.org/~sdboyd/fotd/index.html> is more acceptable. The fractal weather Monday here at Fractal Central was hot and humid enough to keep the fractal cat duo housebound. The high temperature reached 97F 36C at 4pm, and as is usually the case these days, the promised cooling showers once again failed to appear. One thing that had better appear however is the work that's due, and the only way to finish the work is to start it. So without further fanfare, I'd best get started. Until next time, on the 22nd, take care, and watch your bailout. Jim Muth jamth(a)mindspring.com jimmuth(a)aol.com START 20.0 PAR-FORMULA FILE================================ Fractal_Kernels { ; time=0:04:00.35--SF5 on a P200 reset=2002 type=formula formulafile=allinone.frm formulaname=MandelbrotMix4 function=recip passes=1 center-mag=-4.43910986029936400/-5.965792592977377\ 00/1.175254e+008/1/-42.4999968823281691/-4.8222949\ 2321933906e-006 params=1/-2/-0.01/2/0/-0.001 float=y maxiter=1000 inside=0 proximity=0.5 outside=fmod logmap=yes periodicity=10 colors=000vtzvtztszqrzqqzopznoznozlnzilzgkzeizegzd\ gzafz_dz_czYazXazV_zTZzTXzQVxOTvMTqMQoKOnIOlHMgFKe\ FKdDIaBHY9FX9FV7DT5BO3BM29K27I07F05D03B03905H27S39\ Z5Bc7Df9FiBHkDIzFKzHMzIOzKQzKQzKQzKQzKTzKTzKTzKTzK\ VzKVxMVvMVqMXoMXnMXiMXgMYeMYaMY_MYYKX_KX_IXaIXaHXd\ HXdFXeFXeFXgDXgDXiBXiBXl9Xl9Xn9Xn7Xo7Xo5Xq5Xq3Xt3X\ t3Xt5Yq7Yo7Yn9YlB_iB_gD_gF_eF_dHaaIa_IaYKaXKaXMdVO\ dTOdQQdOTeMTeKVeKXeIXeHYgFYgD_gBagBag9di7ei5ei3gi2\ il0il0ll0ll0nl0on0on0qn0tn0tn0vo0xo0xo0zo0zo0zv0zz\ 0zz0zz0zz3zz7zzBzzDzzDzzFzvFzoHtiHndHgXIaQIXKKQDKK\ 7MF2M90MH3KM9KTFKYMIdTIiYIoeHvlHzqHzzFzzFzzFzz7zz0\ zz0zz0nz0vz_zzazxdzxdzzatzaaz_Kz_5z_IqQVgIeX9tM2zB\ 0z20z00zT3zO5zM7zK9zI9zHBxFDqDFlBFTIDBQB0X93_IDaTM\ d_VeidgqnizvlzzIzz0zz0zv0zzIzzVzzezzqzxzzqzqlziezd\ dviaIoqD_z7Kz37z7FzBMzFVvI_oMgiQodVx_YzOizDtz2zz0z\ z0zz0zz0zo3ze5zV9zKBzBxtz } frm:MandelbrotMix4 {; Jim Muth a=real(p1), b=imag(p1), d=real(p2), f=imag(p2), g=1/f, h=1/d, j=1/(f-b), z=(-a*b*g*h)^j, k=real(p3)+1, l=imag(p3)+100, c=fn1(pixel): z=k*((a*(z^b))+(d*(z^f)))+c, |z| < l } END 20.0 PAR-FORMULA FILE==================================

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Re: [Fractint] Extended Fermat's Last Theorem - Slightly Off-topic
by Rupert Millard 19 Aug '02

19 Aug '02

Ricardo, >Does someone in the list know if the following conjecture, that includes as >a particular case Fermat's Last Theorem, has ever been posed, and proved >false or true? >x[1]^p + x[2]^p + ... x[n]^p = z^p, for x[i] > 0 and 1 < n < p, has no >integer solutions. >TIA. Yes, this conjecture was made by Euler in 1769. He conjectured "it is impossible to exhibit three fourth powers whose sum is a fourth power, four fifth powers whose sum is a fifth power, and similarly for higher powers." This was first disproven in 1966 by L.J.Lander & T.R.Parkin with: 27^5+84^5+110^5+133^5 =144^5 The first found with fourth powers was done by Noam Elkies in 1988 and is: 20615673^4 = 2682440^4 + 15365639^4 + 18796760^4 Then Roger Frye found: 422481^4 = 95800^4 + 217519^4 + 414560^4 which is the smallest solution for fourth powers. From, Rupert _________________________________________________________________ Chat with friends online, try MSN Messenger: http://messenger.msn.com

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Extended Fermat's Last Theorem - Slightly Off-topic
by Ricardo M. Forno 19 Aug '02

19 Aug '02

Does someone in the list know if the following conjecture, that includes as a particular case Fermat's Last Theorem, has ever been posed, and proved false or true? x[1]^p + x[2]^p + ... x[n]^p = z^p, for x[i] > 0 and 1 < n < p, has no integer solutions. TIA.

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FOTD 18-08-02 (Iterative Teardrops [6])
by Jim Muth 18 Aug '02

18 Aug '02

FOTD -- (Rating 6) Fractal visionaries and enthusiasts: It's Sunday mid-morning here at Fractal Central, and all is well. I have just finished emptying my e-mail box of two large undeliverable e-mails that were returned to me. This is well enough, except for the fact that I never sent the letters in the first place. It's probably another virus, and a clever one at that -- one that works like sending a letter via normal mail with no postage and writing the name of the address you want it delivered to in the 'from' space. Then when the post office 'returns' the letter for lack of postage, they are actually delivering it. I also feel a bit cruel about failing to reply to Baerbel's request for information about unsubscribing from the Fractint list. But it's a bit amusing to see such a request (from someone who already knew how to subscribe) appear three lines below the unsubscribe information in the letter's header. As for today's fractal, (which I have rated at a harmless 6), it's a scene in the fractal that results when 0.01 part of Z^2 is added to Z^(-2), and (1/C) is added to the combination, then the whole thing is iterated with a bailout radius of 100. That 0.01 part of Z^2 is not much, but it makes a world of difference in the final fractal. The parent fractal, which is enclosed in a circle, looks surprisingly like a fractal planet. Changing the logmap of today's image to 0 and doing about 10 out-zooms will reveal this planet in its full glory. As is the case with most fractals that contain a negative exponent of Z, the bailout radius has a great effect on the appearance of the fractal. With the parent fractal on the screen, redraw the image with imag(p3) set to 2400. A totally different image appears. I named today's image "Iterative Teardrops" because of the teardrop shape of the elements surrounding the central midget. These disconnected elements are typical of fractals whose formula contains negative powers of Z. The rating of 6 equals a little above normal on my rating scale -- about the mildest praise I could give to such a mundane image. The redeeming feature is the render time of 6 minutes, and the fact that the completed GIF image file may be downloaded from either of the following web addresses: <http://home.att.net/~Paul.N.Lee/FotD/FotD.html> <http://sdboyd.dyndns.org/~sdboyd/fotd/index.html> The fractal weather Saturday here at Fractal Central was uneventful. The partly cloudy day with a temperature of 95F 35C and moderately high humidity was notable only because it continues the warmer and drier than normal pattern that has prevailed most all summer. Also it was the fourth day in a row on which the weather experts had predicted showers, and the fourth day in a row on which the predicted showers failed to arrive. Those experts are far better at explaining why something did not happen than forecasting what will happen. The fractal cats, who are unconcerned with weather experts, passed the day doing what cats do best -- lazing around the fractal grounds. As for me, it's Sunday morning, and if I really work hard, I might find a way to get through the day without working. And don't laugh. After all, working to avoid work is no more anti- productive than fighting for peace. Until next time, on the 20th, take care, and don't be too lazy. Jim Muth jamth(a)mindspring.com START 20.0 PAR-FORMULA FILE================================ IterativeTeardrops { ; time=0:06:12.45--SF5 on a P200 reset=2002 type=formula formulafile=allinone.frm formulaname=MandelbrotMix4 function=recip passes=1 center-mag=+3.31533979732890300/+3.380854530622418\ 00/2.718068e+010/1/17.4927412805414235/0.007870415\ 83158384211 params=1/-2/0.01/2/0/0 float=y maxiter=1000 inside=0 logmap=130 periodicity=10 colors=000GLKFNKNUTV`WbgZjnhtml90M_pwcipgbgkWZnPQe\ TNXXKOXHRbJTecVhLfkm_nNmqOctPzvQYqRQlSIgTBcUFdXJe_\ NfbRgeVhhTqpSYwRZwQOwbzocmidhcdcZe0TeUO_TPVTQPTRKT\ SFRTnGwjJtgMndPhaSbZVXWYRT`LQcGWaOa`WgZclYjghsbrwn\ auzLpjKmVJkFIi0Ig6RoBZvHmwMnwJcwHTwFIwD8wSCgfGTuJE\ tLFsMFsOGrPGrRHqSHqTH9WbDYeG_gJajMblPdnSfqVhsYiurk\ wzhtseiscZtf_pdbmbej`gg_jdYlaWoZVqWTtTRwQPwNOwKMwH\ KwEJwDIwCHwBGwAFw9Eu8Ds7Cp6Bm5Ak49h38e38cBDfVEkWCp\ XAuX9wO5pG2gK9lNGpQMtUTwXZw_ewbkwb_wbOwbCwb1wd2we3\ wf4wg5wh6wj6wk7wl8wm9wnAwoAwjRtegjaxadscfnehigjdil\ _knVmpQorLqtGsvBux6wz2wrDujOqbZnVijNsgQUwNWwKYwI_w\ FawDcvAet7gs5ir2kq0lp4fs7`uAVwDUwGVwJWwMXwPYwSZwU_\ wW`wZaw`bwbcwedwgewifwlgwnhwpiwsjwukwwlwymwvnwtowr\ pwpqsnrnlsiitdgu`evWcwRaxM_yIYzMWzQUzTSzXQz_OzcMzg\ LzjJznHzqFzuDzwBzw9zw8zwJzwUzwdzwozwnzwnzwnzwnzwnz\ wnzwnzwozwpzwpzspzopzkpzg } frm:MandelbrotMix4 {; Jim Muth a=real(p1), b=imag(p1), d=real(p2), f=imag(p2), g=1/f, h=1/d, j=1/(f-b), z=(-a*b*g*h)^j, k=real(p3)+1, l=imag(p3)+100, c=fn1(pixel): z=k*((a*(z^b))+(d*(z^f)))+c, |z| < l } END 20.0 PAR-FORMULA FILE==================================

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(no subject)
by Baerbel Strauch 18 Aug '02

18 Aug '02

how can i stop getting e-mails about fracint? _________________________________________________________________ Chat with friends online, try MSN Messenger: http://messenger.msn.com

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Fractal Tone-Poems (music?)
by Bill Jemison 17 Aug '02

17 Aug '02

Thanks Miguel! I remember that you had posted some music pars a while ago, in fact I have them saved in a special directory. so please do post some more. I like Nocturn as well. the orbits are rather hypnotic along with the music. Lotus is probably my favorite, due to the way that the unfolding of the "petals" follows the tonal patterns. It's not particularly "melodic" but I feel that it has merit anyway. That's why I am inclined to call them tone poems rather than music, although one might be able to argue that there isn't much difference. Bill

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Re: [Fractint] Fractal Tone-Poems (music?)
by Franktal Gallery 16 Aug '02

16 Aug '02

Bill, I believe Nocturn is one of the best music PARs I ever heard ! I used to post music PARs a couple of years ago, and I'll do it again when inspiration strikes me... ;-) For those of you who may never heard them, here's a link to my music PAR collection : http://www.geocities.com/franktal2002/sounds.html Regards, Miguel >From: Bill Jemison <fishburnIII(a)compuserve.com> >Reply-To: fractint(a)mailman.xmission.com >To: fractint(a)mailman.xmission.com >Subject: [Fractint] Fractal Tone-Poems (music?) _________________________________________________________________ Send and receive Hotmail on your mobile device: http://mobile.msn.com

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