BABYL OPTIONS: -*- rmail -*- Version: 5 Labels: Note: This is the header of an rmail file. Note: If you are seeing it in rmail, Note: it means the file has no messages in it.  Supplemental Material for "Amicable Pairs..." This webpage contains supplemental material for the article:

Amicable Pairs and Aliquot Cycles for Elliptic Curves
Joseph H. Silverman and Katherine E. Stange
Experimental Mathematics 20 (2011), 329-357.


Section 1: Amicable pairs for y2 + y = x3 + x up to 1012

We used PARI-GP to compute all normalized amicable pairs (p,q) on the curve y2 + y = x3 + x up to p < 1011, and Andrew Sutherland extended our list up to p < 1012. Sutherland's computations took about 23 hours on a quad-core 3 GHz AMD. The results are given in the following two tables.

(853,883) (77761,77999)
(1147339,1148359) (1447429,1447561)
(82459561,82471789) (109165543,109180121)
(253185307,253194619) (320064601,320079131)
(794563993,794571803) (797046407,797057473)
(2185447367,2185504261) (2382994403,2383029443)
(4101180511,4101190039) (4686466159,4686510971)
(5293671709,5293749623) (6677602471,6677694539)
(7074693823,7074704971) (7806306133,7806380963)
(9395537549,9395559011) (9771430993,9771433303)
(9849225103,9849306373) (10574564857,10574619851)
(12657210407,12657303353) (13003880317,13003900901)
(13789895011,13790023199) (14436076927,14436180091)
(14976551207,14976590371) (15597047659,15597075937)
(15679549877,15679688491) (16322301811,16322366867)
(17725049203,17725142719) (17841395323,17841406601)
(20780607817,20780797927) (23338053773,23338135543)
(28358243743,28358411071) (29859516131,29859782089)
(31615097957,31615194739) (33266376239,33266419807)
(33963999907,33964128017) (34525477799,34525684639)
(39287748091,39287808559) (40136806357,40137038941)
(46438194193,46438453213) (52011956957,52012184953)
(51838270219,51838493561) (51881025571,51881167549)
(55823622193,55823919169) (57920520199,57920640709)
(62765305697,62765625749) (62995853671,62996152237)
(66252308051,66252349753) (67177409329,67177631771)
(69449506103,69449741239) (75002612911,75002660263)
(77264683829,77264993327) (77635421531,77635670141)
(79067605783,79067881429) (81263083703,81263204563)
(94248260597,94248586591)
Amicable pairs (p,q) on the curve y2 + y = x3 + x up to p < 1011


(104544108049,104544364087) (111287830573,111288274567)
(118206158729,118206360829) (120791219099,120791323493)
(132962516737,132962703661) (142574237383,142574369533)
(144750903551,144751137469) (155467666099,155467836031)
(161226480901,161227124081) (173164057399,173164630033)
(178633373617,178633516081) (213013688359,213013931239)
(218475851959,218475922267) (222335132807,222335345521)
(225529688431,225529987157) (232349609983,232349658979)
(234896302009,234896350369) (240677586449,240678201091)
(241352193611,241352273849) (265340194039,265340401483)
(277515892207,277516507711) (287800715711,287801137609)
(299486604371,299487430807) (302166243187,302166581251)
(323643851647,323644499221) (356299878281,356300493907)
(378008294449,378008508961) (383399841217,383399894341)
(392864677427,392865349441) (415381769743,415381922953)
(421953112561,421953604103) (425072615243,425073437039)
(438722917471,438723215947) (475655912713,475656729419)
(477171588461,477171935243) (509779650181,509780267947)
(519205252403,519205488493) (580562183213,580562489173)
(605229610571,605229758977) (614484897889,614485486079)
(637355743513,637356846673) (649999477469,649999993999)
(655455388397,655456255439) (658459698947,658460090441)
(662097699853,662098655233) (705006602177,705006769807)
(723299067853,723299355619) (775857545861,775859048443)
(793725967891,793727339077) (794925473327,794926023761)
(811569419461,811569591827) (838059794239,838061257667)
(851273574199,851274251683) (885227547847,885227943451)
(916134576373,916134747943) (948135054247,948136458277)
(954115635797,954115645823) (977575750447,977576865637)
Amicable pairs (p,q) on the curve y2 + y = x3 + x with 1011 < p < 1012

Section 2: Amicable Pair Counts on Selected Curves

In this section we give Andrew Sutherland's counts on the number of amicable pairs on three curves not having CM. The curves are described as [a1,a2,a3,a4,a6] to represent the elliptic curve
y2 + a1 x y + a3 y = x3 + a2 x2 + a4 x + a6.

Bound on p [0,1,1,0,0] [0,0,1,-1,0] [0,0,0,1,1]
108 5 1 2
109 10 3 8
1010 21 16 21
1011 59 33 53
1012 117 115 129
Number of amicable pairs on y 2 + a 1 x y + a 3 y = x 3 + a 2 x 2 + a 4 x + a 6 with p up to the given bound

Section 3:  Evaluating the Formula for Mk[1] when k = 1 (mod 3)

In this section we give the PARI-GP script that we used to compute Mk[1](S,1) via the formulas (42), (43), (44), and (45). In these formulas we treat k, pi, and epsilon as indeterminates and formally set pi-bar = k/pi and epsilon-bar = 1/epsilon.  The value of Mk[1](S,1) turns out to be independent of k mod 4 and is always a quadratic polynomial in k.  The output from this PARI script is displayed in Table 2 of the article.

/* The function TestFormulaForSplitPrimes computes M_k^{[1]}
   for k = 1 (mod 3), using the formula as a sum of products of #C - e terms.
   The following global variables must be left as indeterminates:
   k, pi, e2
   Here e2 represents the cubic residue of 2 modulo pi.
   The conjugates of these are given by
   pibar = k/pi  and   e2bar = 1/e2.
   Further, w is assigned the value quadgen(-3), and wbar = conj(w) = 1/w
*/

{
TestFormulaForSplitPrimes() =
  local(m,ISets,IndexSet);
  print("\\begin{align*}");
  forstep(kmod4 = 1, 3, 2,
    print("  k\\equiv ",kmod4," \\pmod{4} \\\\");
    for (i = 0, 5,
      m = MMSum([i],kmod4);
      print1("  \\#M_k^{[1]}(\\o^",i,",1) &= ");
      print1("(",content(content(m)),")(",m/content(content(m)),")");
      if (i < 5 || kmod4 == 1, print(" \\\\"), print);
    );
    if (kmod4 == 1, print("  \\\\"));
  );
  print("\\end{align*}");
  print("\\begin{align*}");
  ISets = [[1,5], [1,3,5], [1,2,4,5], [0,1,2,3,4,5]];
  for (j = 1, #ISets,
    IndexSet = ISets[j];
    m = MMSum(IndexSet,1);
    print1("  \\#M_k^{[1]}\\bigl(\\{");
    for (i = 1, #IndexSet,
      print1("\\o^",IndexSet[i]);
      if (i < #IndexSet, print1(","));
    );
    print1("\\},1\\bigr)\n     &= (",content(content(m)),")");
    print1("(",m/content(content(m)),")");
    if (j < #ISets, print(" \\\\"), print);
  );
  print("\\end{align*}");
}

w = quadgen(-3);
wbar = conj(w);
pibar = k/pi;
e2bar = 1/e2;

{
CC(u,v,kmod4) =
  if (kmod4 == 0, error("k mod 4 must be 1 or 3"));
  k + 1
     + w^(2*v)*pibar + wbar^(2*v)*pi
     + e2^2*w^(3*u+4*v)*pibar + e2bar^2*wbar^(3*u+4*v)*pi
     + e2*w^(5*u+2*v)*pibar + e2bar*wbar^(5*u+2*v)*pi
     + (-1)^((kmod4-1)/2)*e2*w^(u+2*v)*pibar
     + (-1)^((kmod4-1)/2)*e2bar*wbar^(u+2*v)*pi;
}

{
ee(u,v) =
  if (u % 6 == 0, 6, 0)
    + if((u-v) % 3 == 0, 3, 0)  + if((2*u-v) % 3 == 0, 3, 0);
}

MM(u,v,kmod4) = (1/18) * (CC(u,v,kmod4) - ee(u,v));

{
MMSum(IndexSet,kmod4) =
  local(i,s);
  s = 0;
  for (j = 1, #IndexSet,
    i = IndexSet[j];
    for (u = 0, 5,
    for (v = 0, 2,
      s = s + MM(u,v,kmod4)*MM(u-i,v,kmod4);
    );
    );
  );
  return(s);
}

Section 4: Errata

The proof of Theorem 14 has a gap. In the proof it is asserted that for a given l, if we choose n sufficiently large, then

           qn+lqn – 1 ≤ 2 √qn

This estimate is beyond our current knowledge, even for l = 1. However, it is possible to show that this assertion is true for many sets of l consecutive primes, which suffices for the proof of Theorem 14.

As in the paper, let qj denote the j-th prime, and consider the primes in the interval (x,2x]. Let qs be the smallest prime in this interval, and let qt be the largest prime in this interval. We want to estimate the number of exceptions:

           A(x) = #{ j : qj+l and qj are in (x,2x] and qj+lqj > 2√qj + 1 }.

Since the distance between two primes in the interval (x,2x] cannot exceed x, we find that

           lx > ∑0≤j<l (qtjqs+j) = ∑kjtl (qj+lqj ) > 2√xA(x).

The prime number theorem says that there are on the order of x/log(x) primes in the interval (x,2x]. Since the number of exceptions satisfies A(x) < lx, taking x large enough, we are guaranteed to find many good strings qj,…,qj+l in the interval (x,2x].

(The authors thank Andrew Granville and Ethan Smith for bringing this to their attention and for explaining the above argument that fills the gap.)