June 10, 1999, Melun (France) Dear all, Finally, here is the second newsletter ! First, I would like to thank Rizos Sakellariou, Nuutti Kuosa, Giovanni Resta and Bob Scher for their ideas and algorithms. Their ideas constantly improve Sum95 ! Also many thanks to Achim Flammenkamp who helped me to design the details page. There have been some surprises these last months. The first one came from Nuutti Kuosa who found the solution of (8,1,10) using his own program. Then Edward Brisse discovered (6,2,5) with Sum95 and a lot of patience. The next one came from Giovanni Resta who found the following (6,2,5) using his own program: 1117^6+770^6=1092^6+861^6+602^6+212^6+84^6 2041^6+691^6=1893^6+1468^6+1407^6+1302^6+1246^6 2441^6+752^6=2184^6+2096^6+1484^6+1266^6+1239^6 2827^6+151^6=2653^6+2296^6+1488^6+1281^6+390^6 2959^6+2470^6=2954^6+2481^6+850^6+798^6+420^6 If you want more informations, you can look at Giovanni Resta's article. Then Nuutti Kuosa, using AVL, found great solutions for some powers above 10. All these new results allowed to improve Sum95. Following Rizos Sakellariou's suggestion to use moduli, Sum95 is now much faster on (k,1,n) with k even. Nuutti Kuosa showed that (8,1,10) could be reached very easily with these moduli. Another big improvement came from Giovanni Resta, which technique improves (k,m,n) with k even, m>=2 and n>m. Using all these refinements, I designed a new Sum99 (version 3 beta 4) which speeds up computations on (6,1,6), (6,2,4), (8,1,9) and (8,2,7). Also, Bob Scher wrote a program to search 5 5's, and the first version is already included into Sum99. This program computes solutions to: a^5 +/- b^5 +/- c^5 +/- d^5 +/- e^5 = 0 Eric W. Weisstein kindly updated his pages with our results: http://www.treasure-troves.com/math/EulersSumofPowersConjecture.html http://www.astro.virginia.edu/~eww6n/math/EulersSumofPowersConjecture.html The most wanted results that were found since the last newsletter: (6,2,5) 1117+770=1092+861+602+212+84 (Edward Brisse, 04/20/1999) (7,4,4) 343+281+46+35=354+112+52+19 (Michael Lau, 05/12/1999) (third solution) (8,1,10) 235=226+184+171+152+142+66+58+34+16+6 (Nuuti Kuosa, 04/06/1999, own program) (11,2,19) 53+24=51+45+43+39+37*2+32+31+23*3+19+18+17+10+9+8+4*2 (Luigi Morelli, 04/13/1999) (11,9,9) 40+39+30+21*2+18+6+4+2=41+37+31+24+20+14+10+3+1 (Nuutti Kuosa, 05/09/1999) (12,10,11) 27*3+17*2+12+6+3*3=29+26+19+10+7+4+1*5 (Nuutti Kuosa, 05/18/1999) (12,9,10) 38+32+31+28+22+15+12+3+2=36*2+33+26+21+20+10+8*2+7 (Nuutti Kuosa, 05/25/1999) (13,11,11) 31+27+23*2+21+19+10*3+6*2=29*2+28+25+24+22+15+5*2+3+1 (Nuutti Kuosa, 06/07/1999) The other results: (12,1,26) 52=50+43*2+42+41+35*2+33*2+32*2+31*2+25+23*4+18+16+14+12+11+10+9+6 (Jean-Charles Meyrignac, 04/15/1999) (12,4,22) 40+36+18+16=38+35*3+33+29+26*3+19*2+17*2+13*2+11*2+9+7+5+4+2 (Mac Wang, 06/06/1999) (12,6,18) 38+34+26+12+10+6=35*3+31+29+25*4+23+21+17+15+14+9+3*2+2 (Nuutti Kuosa, 05/12/1999) (12,7,18) 32+24+17+13*2+3+1=30+29+28+25+22+14*2+12*2+11+10*3+7+6+5+4*2 (Nuutti Kuosa, 06/04/1999) (12,8,17) 30*2+27+26*2+13*2+8=28*6+21+18*2+16*3+14+10+7+4+1 (Nuutti Kuosa, 05/13/1999) (12,12,12) 18+17*2+12*2+10*4+5*2+3=19+15+9*3+2*7 (Nuutti Kuosa, 05/18/1999) (13,12,12) 24+20+19+15+14+12+8+7*2+6*2+3=22*3+21+13+11*3+2*4 (Nuutti Kuosa, 05/24/1999) (14,1,45) 34=31*2+28*5+27+25*3+24*2+23*4+22*2+21*3+19*2+18*2+17*2+14*2+13*4+11+10*2+8+ 7*2+4+2*4 (Nuutti Kuosa, 04/29/1999) (15,1,46) 50=49+45+40+37+35+31*3+30*4+28*2+26*2+24*3+18+17+16+14*2+13+11*3+10*2+8*2+6* 2+5*3+3*4+2*2+1*3 (Nuutti Kuosa, 04/26/1999) (15,2,42) 33+15=32+28*2+27*3+26+24+23*2+22+21*3+20+18+17*2+14*2+11*4+10+9+8+7+6*4+5*4+ 4*5+2 (David Alten, 06/10/1999) (15,3,47) 25+24+9=23*3+22*4+19*5+17*4+15*3+12*5+10*8+8+7+6*2+4+3*2+1*8 (Jean-Charles Meyrignac, 04/22/1999) (16,1,77) 32=31+30+25+23*2+22+21*8+19*2+17*8+16*2+15*3+14+13*8+12*5+11+10+9*21+6*2+5*5 +3*3+1 (Nuutti Kuosa, 04/24/1999) (16,2,76) 24*2=23+22*2+21*7+19*6+17*6+16*2+14+13*2+12*3+11*5+10+9*12+8*2+7*7+5*8+4+3*3 +1*7 (Norman Ho, 05/21/1999) (17,1,74) 51=50+47+41+39+34+32+29+27+23+21+20*5+19*5+18*16+16+15*2+14+13*2+12+11+10*6+ 8+7*5+3*2+2*11+1*5 (Larry Hays, 05/05/1999) (18,1,96) 26=24*2+23+22*6+21*4+20+18*18+17+16+15*10+14*9+13*6+12*2+11*6+10+9+8*3+7*5+6 *3+5*5+4+3+2+1*8 (Nuutti Kuosa, 04/25/1999) (19,1,144) 386=385+329+230+196+171+152+130+114+98+84+75+68+60+53+50+44+42+37+35+31+29+2 7+25+24+22*2+17*4+16+15*17+14*3+13+12*3+11*4+10*14+9*5+8*23+7*5+6*6+5*9+4*19 +3*3+1 (Jean-Charles Meyrignac, 05/25/1999) (20,1,170) 568=567+480+362+301+258+190+161+139+119+108+91+71+66+62+49+47+41+36+34+32+31 +29+28+27+26+25+21+18+17*7+16*8+15+14*18+13*4+12*17+11*3+10*12+9*2+8*8+6*2+5 *3+4*15+3*18+2*13+1*11 (Jean-Charles Meyrignac, 06/04/1999) (21,1,270) 28=27*2+24+23+22*2+18+17*2+16+14*7+13*6+12*21+11*34+10*34+9*13+8*78+7*6+6+5* 19+4*4+3*18+2*10+1*9 (Jean-Charles Meyrignac, 05/02/1999) (22,1,300) 56=55+53+47+45+37+34+33+31+30+26+25+24*2+22*2+19+18*2+17*2+16+15+13*17+12*43 +11*23+10*9+9*46+8*60+7*8+6*20+5*2+4*6+3*11+1*33 (Jean-Charles Meyrignac, 05/31/1999) (23,1,447) 58=57+55+50+44+40+37+36+32+31*2+25*2+23*2+20*2+19*2+18*3+17+16+14+13*4+12*3+ 11*24+10*111+9*15+8*39+7*15+6*13+5*74+4*44+3*27+2*47+1*7 (Jean-Charles Meyrignac, 06/04/1999) (24,1,512) 685=684+595+495+432+292+262+231+208+181+158+144+127+116+106+92+84+75+69+57+5 2+47+44+40+38+36+35+32+30+29+28+27+26+24*2+23+22+21+16*2+14*2+13+12*41+11*3+ 10*96+9*39+8*48+7*6+6*69+5*5+4*23+3*32+2*15+1*93 (Jean-Charles Meyrignac, 06/07/1999) (24,2,589) 225*2=224*2+210+187+148+129+119+105+95+88+83+74+70+64+61+56+52+47+42+41+37+3 6+34+32+30+28*2+22+20*2+19*2+18*3+16*4+14*2+13+12*2+11*26+10+9*27+8*6+7*22+6 *45+5*71+4*16+3*173+2*8+1*150 (Jean-Charles Meyrignac, 05/19/1999) Now, the most wanted equations below 10th power are: (6,1,6) (6,2,4) (7,1,6) (7,2,5) (7,3,4) (8,1,9) (8,2,7) (8,3,6) (8,4,4) (9,1,11) (9,2,9) (9,3,8) (9,4,5) Future plans for Sum95 include a tutorial, an Internet client/server (with the help of Nuutti Kuosa), and the hunt for 7 7's (which will be written by Bob Scher). The hunt for 7 7's will try to solve the equation: a^7 +/- b^7 +/- c^7 +/- d^7 +/- e^7 +/- f^7 +/- g^7 = 0 Jean-Charles Meyrignac http://www.chez.com/powersum/index.htm Mirror page: http://www.multimania.com/powersum/index.htm