March 21, 1999, Melun (France) Dear all, This is the first newsletter. Please, excuse me for my bad english. I will try to summarize all the work done on Sum95 and the results collected by all the searchers. First, I would like to thank Edward Brisse, Patrick Fossano and Eric Bainville for being the early beta-testers of the first version of Sum95. They also helped me a lot on improving the whole program. Secondly, I would like to thank everybody who participate to the computation. Some of them had some success, and some not, but everybody's work is important ! Thirdly, I would like to thank George Woltman (author of Prime95) and Ben Pfaff (author of GnuAVL) for their programming contributions to Sum95. Ok, that's all for the greetings. And now for some results. Here are the new equations found by the project since its beginning. The most wanted that were found are: (7,1,7) 568=525+439+430+413+266+258+127 (Mark Dodrill, 03/20/1999) (9,2,10) 137+69=121+116*2+115+89+52+28+26+14+9 (Luigi Morelli, 03/12/1999) (10,4,12) 53+44*2+22=51+49+43+39+29+28+17*2+16+13+7+4 (Eric Bainville) (10,1,15) 108=100+94+91+77*2+76+63+62+52+45+35+33+16+10+1 (Jean-Charles Meyrignac, 03/22/1999) (11,2,20) 44+18=43+38+30+27+26+23*2+16+15+14+12*2+11*3+7+3+2+1*2 (Luigi Morelli) The others are: (12,1,27) 73=72+62+49+43+41+36+35+33+32*2+31+27+25*3+24+23+16*2+14+12+9*2+7*2+5*2 (JC Meyrignac) (12,2,27) 31+1=30+26*2+25+18*4+17+15*2+14+13*10+8+7*3+5 (JC Meyrignac) (12,3,28) 24+13+6=22+21*3+17*2+15*3+14+12+10+8+5+4*4+3*2+2*2+1*6 (JC Meyrignac) (12,6,24) 24+18+8*2+4+2=22+21+20*3+19*2+16*2+14+13*5+11*5+7*2+6+1 (JC Meyrignac) (13,1,42) 917=916+660+471+344+240+162+105+86+70+52+43+35+33+25+24+23*2+22*3+21+20*2+19 +17+16*2+15*3+14+13+11+8*2+7+6*4+5*2 (JC Meyrignac) (13,2,41) 967*2=966*2+731+529+356+209+145+119+81+68+56+44+39+35+32+27+23+19+16+15*3+12 *3+10+9*7+8*4+5*2+4+3 (Patrick Fossano) (14,1,47) 128=127+108+92+80+66+55+46+40+35+31+26+25*3+24+23*3+22+21*2+19*2+17+16+15*4+ 12*3+11*2+9*6+7*3+4+3+2+1 (JC Meyrignac) (14,2,46) 71*2=70*2+66+41+31+28+25*2+24+22*10+20+18*2+17+10*6+7*2+6*2+4*2+3*8+1*3 (JC Meyrignac) (14,4,46) 104*4=103*4+99+70+60+48+36+31+28+26*4+25+22*3+21*3+20*3+18+17+16*2+15+14+12* 5+11+8+7+6*3+4+3+2+1 (JC Meyrignac) (14,5,46) 146*5=145*5+138+99+64+52+45+37+33+30+29*2+26*2+24*4+23+21*4+18+17+13*2+10*3+ 9*4+8*4+6*2+4*2+1 (JC Meyrignac) (15,1,52) 25=24+23+21*2+19+18+16+14*3+12*2+11+10*2+9*5+7+5*3+4*5+3*3+2*6+1*14 (JC Meyrignac) (18,2,118) 3190*2=3189*2+2486+1719+1115+850+645+505+407+313+267+213+182+142+107+90+75+6 3+51+48+44+38+35+33+31+30+26+23*2+20+19+14*2+13*4+12*29+11*5+10*2+8*7+7*6+6* 4+5*5+4+3*16+2*6 (Patrick Fossano) (18,4,119) 3476*4=3475*4+2802+1828+1395+986+721+567+465+364+306+243+206+173+122+97+87+7 2+64+59+51+47+44+37+34+24+22*2+19+17*2+15*2+14+13*6+12*8+11*2+10*3+9+8*10+7* 13+6*3+5*18+4*2+3*10+2*6+1 (Patrick Fossano, 03/15/1999) (18,7,102) 776*7=775*7+701+471+379+308+244+191+151+126+106+80+69+60+55+48+44+40+37+32+2 9+27+26+24+23+20+17+16+14+13+12*8+11*5+10+9*4+7*8+6*17+5*3+4*10+3*5+2*3+1*3 (Patrick Fossano,03/19/1999) (20,1,209) 2955=2954+2301+1753+1373+1072+717+567+456+379+323+267+226+199+177+153+124+10 3+84+72+67+60+54+50+47+45+37+33+30+23+22+21+20*2+17+16*3+14+13*2+12*20+11*8+ 10*8+9*13+8*7+7*13+6*8+5*20+4*30+2*5+1*37 (JC Meyrignac) (21,1,402) 34=33+32+31+29+26+25+24*2+22+21+20+19*2+18*2+17+15*2+12*4+11*8+10*12+9*13+8+ 7*31+6*13+5*118+4*155+3*5+2*11+1*13 (Micha Fleuren) (21,2,467) 58*2=57*2+56+52+49+44+39+38+32+30+28+27+26+25*2+22+21+20+19+16+14*3+13*2+12* 3+11*13+10*29+9*44+8*22+7*80+6*15+5*11+4*70+3*25+2*104+1*26 (Micha Fleuren, 03/12/1999) (21,4,458) 14*4=13*18+12*4+11*7+10+9*9+8*47+7*35+6*31+5*110+4*58+3*72+2*45+1*21 (Micha Fleuren, 03/16/1999) (21,7,418) 53*7=52*10+50+40+30+29+28+27+25+24*2+20*2+18+17+15+13*4+12*2+11*7+10*49+9*10 +8*56+7*2+6*77+5*146+4*2+3*7+2*6+1*26 (Micha Fleuren, 03/16/1999) (22,2,669) 16*2=15*8+14+13+12+11*3+10+9*3+8*50+7*17+6*80+5*395+3*17+2*88+1*4 (Micha Fleuren,03/16/1999) (22,1,553) 124=123+114+98+85+75+67+63+59+56+53+49+47+41+39+36+34+33+31+30+27+25+24+21+1 9*2+18*2+17*3+16+15+14+13+12*5+11*4+10*18+9*120+8+7*5+6*41+5*71+3*48+2*141+1 *65 (Micha Fleuren) (23,1,805) 29=28*2+26+24*2+21+20+18+17+14+13*4+12+11*4+10*4+9*36+8*232+7*13+6*49+5*107+ 4*10+3*25+1*310 (Micha Fleuren) If you forgot the meaning of the notation, go to www.chez.com/powersum/records.htm. I would like to announce a new version of Sum95. This version is able to compute much faster than the previous versions when the number of terms is low. How does it work ? Well, in fact, the new version stores all the sums of n terms. A solution is found when 2 sums are equal. Technically, the algorithm uses AVL trees with a sliding window. All the computations are done in 128 bits. This technique is especially useful when the number of terms is low. For example, it's very efficient for the exploration of k<11. Here is a practical example. I worked on power=4, start=0, end=500, number of terms=2 and 1 megabyte of memory (the new algorithm needs some memory, which is user definable). (4,2,2) at 635318657 (4,2,2) at 3262811042 (4,2,2) at 8657437697 (4,2,2) at 10165098512 (4,2,2) at 52204976672 (4,2,2) at 51460811217 (4,2) 0-500 done S2:2B96F736 JC (4,2,2) at 635318657 means that two sums of two powers of 4 are equal to 635318657. You can easily check that these values are: (4,2,2) 158+59=134+133 (Euler,1772) The solutions listed are all the double sums between 0^4 and 500^4. The new version is still in beta-test, but I'm starting to reserve ranges. Here are the searches that will be started with the new version. The first number is the power, the second is the number of terms, and the third element is the expected solutions. 5 3 (5,2,3) 5 4 (5,1,4) or (5,2,3) 6 4 (6,2,4) or (6,3,4) 6 5 (6,1,5) or (6,2,5) 6 6 (6,1,6) 7 4 (7,3,4) 7 5 (7,2,5) 7 6 (7,1,6) 8 4 (8,4,4) 8 5 (8,4,5) 8 6 (8,2,6) or (8,3,6) 8 7 (8,1,7) or (8,2,7) 8 8 (8,1,8) 8 9 (8,1,9) 9 5 (9,4,5) or (9,5,5) 9 6 (9,4,6) or (9,4,7) or (9,5,6) 10 5 (10,5,5) 10 6 (10,5,6) or (10,4,6) 10 7 (10,3,7) or (10,4,7) Of course, computation on (6,6) will help the computation of (6,5)... If you are interested, mail me the power and number of terms that you want to explore. Jean-Charles Meyrignac