From: jmp@vejlehs.dk (Jan Munch Pedersen) Subject: Announcement: One million Amicable Pairs Date: 30 Jan 01 14:17:47 GMT Newsgroups: sci.math.numberthy Summary: Recent progress on amicable pairs of integers Amicable Pair number one million has been found! Two numbers (M,N) are amicable iff sigma(M)=sigma(N)=M+N where sigma is the sum-of-divisors function. The first pair (220,284) is attributed to Phytagoras. Below thousand pairs were found before the event of computers. Using computers the number of amicable pairs has been raised to more than one million. On January 27, 2001 I received a large batch of pairs from Mariano Garcia which brought the count of pairs in my database up to 1,118,555. All the pairs are visible at the web pages starting at: http://www.vejlehs.dk/staff/jmp/aliquot/knwnap.htm It still hasn't been proved that the number of amicable pairs is infinite. However the explosive growth of pairs the last few years leaves no doubt that this conjecture is true. During time several lists of amicable has been collected and printed, starting with Eulers list [1] of 64 pairs in 1750 (including two false pairs), and ending with Herman te Riele et. al's lists [3]+[4] of 11882 pairs in 1986. When I started my own list in 1995 it soon became clear that printing was out of question. Instead the Internet seems to be a great place for this kind of information. The vast majority of the pairs (more than 90%) are found by a special case of the BDE method [5] formulated as theorem 2.6 in [6]. This method uses two numbers a and u (called a special breeder) with the property sigma(a)/a = (u+sigma(u)-1)/sigma(u). Then for all different factors D1, and D2 of D1.D2 = sigma(u).(u+sigma(u)-1) it is checked if the three numbers q := D1+sigma(u)-1, r := D2+sigma(u)-1, s := u + q + r are all prime. If q, r, and s are primes not dividing a then (a.u.q,a.r.s) is an amicable pair. I wish to thank the following for sending me their old lists and new contributions: Herman te Riele, Stefan Battiato, David Moews, Derek Ball, Pat Costello, David Einstein, Yasutoshi Kohmoto, Alexander Gubanov, Frank Zweers, Andrew Walker, Axel vom Stein, and last but not least Mariano Garcia. It is a great pleasure that it was Mariano Garcia, the grand old man of amicable pairs, who sent me pair number one million. Congratulations to him! You are always welcome to send any pairs you find to jmp@vejlehs.dk. Jan Munch Pedersen Vejle Business College Boulevarden 48 DK-7100 Vejle Denmark References: [1] L. Euler, De numeris amicabilibus, Opuscula varii argumetii, pages 23-107, 1750. Reprinted in [2], pp. 86-162. [2] L. Euler, Leonhardi Euleri Opera Omnia, Sub ausp. soc. scient. nat. Helv., Teubner, Leipzig, Series I, Vol. 1915. [3] H. J. J. te Riele, Computation of All the Amicable Pairs Below 10^10, Math. Comp., 47, 175(1986), pp. 361-368 and Supplement pp. S9-S40. [4] H. J. J. te Riele et. al, Table of Amicable Pairs between 10^10 and 10^52, Note NM-N8603, Department of Numerical Mathematics, Centre for Mathematics and Computer Science, Amsterdam, 1986. [5] E. J. Lee, Amicable Numbers and the Bilinear Diophantine Equation, Math. Comp., 22, 101(1968), pp. 181-187. [6] W. Borho and H. Hoffmann, Breeding Amicable Numbers in Abundance, Math. Comp., 46, 173(1986), pp. 281-293. ============================================================================== From: Herman.te.Riele@cwi.nl Subject: addition to Jan Pedersen's message Date: 30 Jan 01 14:17:46 GMT Newsgroups: sci.math.numberthy To complete Jan Pedersen's list of lists of amicable pairs, I would like to mention the following reference: Elvin J. Lee and Joseph S. Madachy, The History and Discovery of Amicable Numbers, J. Recr. Math. 5 (1972) pp. 77-93, 153-173, 231-249. Errata in vol. 6, pp. 53, 164, and 229. Congratulations to Jan for his great initiative to put the known amicable pairs on internet: I am convinced that his efforts have directly and indirectly made known (or in amicable number terminology: breeded) more amicable pairs than anybody else's. Herman te Riele ============================================================================== From: Fred W. Helenius Subject: Re: Small list of friendly numbers Date: Thu, 28 Jun 2001 05:58:57 -0400 Newsgroups: sci.math "carel" wrote: >It looks like there are infinite many friendly numbers. That's what most people conjecture. By the way, the usual term is "amicable pairs". >Small list of friendly numbers >220 284 >1184 1210 >2620 2924 >5020 5564 >6232 6368 >10744 10856 >12285 14595 >17296 18416 >63020 76084 >66928 66992 >67095 71145 >69615 87633 >71145 67095 You already listed 67095,71145. [rest of list, with many more duplicates, snipped] There's a larger list (2122263 pairs) of amicable numbers at http://www.vejlehs.dk/staff/jmp/aliquot/knwnap.htm . You might also be interested in David Moews's article at http://xraysgi.ims.uconn.edu:8080/amicable.html . -- Fred W. Helenius