14
$\begingroup$

We have the Lucas numbers, $$ L_1 = 1, \; L_2 = 3, \; L_3 =4, \; L_4 = 7, L_5 = 11, \; L_{n+2} = L_{n+1}+ L_n \; . $$

Question: is it the case that $$ f(x,y,z) = 4 x^2 + 3 x y + 9 y^2 + z^3 + 3 z $$ integrally represents every integer except all $$ L_{6n+1}, \; - L_{6n+1}, \; L_{6n+5}, \; - L_{6n+5}? $$

These values of $L$ are odd and (primitively) satisfy $L^2 - 5 F^2 = -4.$ Put another way, they satisfy $$ \operatorname{discriminant}_z \left(z^3 + 3 z + L \right) = -135 F^2. $$

Quirks: $4 x^2 + 3 x y + 9 y^2$ is a positive quadratic form. The evident $\pm$ symmetry (in the numbers represented) is not well explained except for $ \operatorname{discriminant}_z \left(z^3 + 3 z + L \right) = -135 F^2. $ Next, the imprimitive solutions to $L^2 - 5 F^2 = -4$ are taken care of by the cubic part, that is $$ L_{2n+1}^3 + 3 L_{2n+1} = L_{6n+3}. $$ What else, the monic cubic $z^3 + 3 z -1$ describes a field, discriminant $-135,$ same as the quadratic form, while $4 x^2 + 3 xy + 9 y^2$ is not a cube in its class group. That is the whole game, right there. If the field is not the Hilbert class field of something, maybe someone will tell me a correct name for it. My original examples, years ago, used class number restricted to 3, so my name was probably strictly correct then. This time, class number is six. Oh, and we always strip off the constant term, $z^3 + 3 z - 1$ is stripped to $z^3 + 3 z.$ Works much better this way.

This all works for target numbers between $-4,000,000$ and $4,000,000.$

Oh, the class group of positive binary quadratic forms of discriminant $-135$ is $$ \langle 1,1,34 \rangle, \; \langle 4,3,9 \rangle, \; \langle 4,-3,9 \rangle $$ in the principal genus and $$ \langle 5,5,8 \rangle, \; \langle 2,1,17 \rangle, \; \langle 2,-1,17 \rangle $$ in the other genus.

  4 x^2 + 3 x y + 9 y^2 + z^3 + 3 z      z    4 x^2 + 3 x y + 9 y^2
   -674947                             -368       49162189     to go  34
  -2425241                             -375       50310259     to go  33
  -2138969                             -375       50596531     to go  32
  -2074751                             -375       50660749     to go  31
  -3046381                             -380       51826759     to go  30
  -3201569                             -381       52105915     to go  29
  -2324091                             -379       52116985     to go  28
  -3613827                             -388       54798409     to go  27
  -3356493                             -391       56421151     to go  26
  -3985891                             -395       57645169     to go  25
  -2674819                             -395       58956241     to go  24
  -3796009                             -410       65126221     to go  23
  -2745141                             -430       76763149     to go  22

   Targets between  -4,000,000  and  4,000,000
   that appear to have no integer expression as  
   4 x^2 + 3 x y + 9 y^2 + z^3 + 3 z   : 

  -3010349 =  -1 * 3010349 
  -1149851 =  -1 * 59 * 19489 
  -167761 =  -1 * 11 * 101 * 151 
  -64079 =  -1 * 139 * 461 
  -9349 =  -1 * 9349 
  -3571 =  -1 * 3571 
  -521 =  -1 * 521 
  -199 =  -1 * 199 
  -29 =  -1 * 29 
  -11 =  -1 * 11 
  -1 =  -1 *  1  
  1 =  1  
  11 = 11 
  29 = 29 
  199 = 199 
  521 = 521 
  3571 = 3571 
  9349 = 9349 
  64079 = 139 * 461 
  167761 = 11 * 101 * 151 
  1149851 = 59 * 19489 
  3010349 = 3010349 

  Sun Oct 26 18:04:33 PDT 2014


  max binary 85,123,123
  poscount  11
  negcount  11
   number binary values saved  7,423,220

  -2745141           -430       76763149          

  jagy@phobeusjunior

Note: a key ingredient in this was playing with the field website, mentioned by Noam Elkies in this answer, in this case degree 3 and absolute value of discriminant 135. The lucky parts were how the polynomial given had 1 as the constant term and a common factor of 27 divided out of everything in the part about the discriminant with different constant term being required a square multiple of the discriminant with constant term 1.

Earlier questions on the same theme were What numbers are integrally represented by $4 x^2 + 2 x y + 7 y^2 - z^3$ and Integers not represented by $ 2 x^2 + x y + 3 y^2 + z^3 - z $

Here is the beginning of the file of Lucas numbers from OEIS, indexing agrees with what I am using above:

  0 2
  1 1
  2 3
  3 4
  4 7
  5 11
  6 18
  7 29
  8 47
  9 76
  10 123
  11 199
  12 322
  13 521
  14 843
  15 1364
  16 2207
  17 3571
  18 5778
  19 9349
  20 15127
  21 24476
  22 39603
  23 64079
  24 103682
  25 167761
  26 271443
  27 439204
  28 710647
  29 1149851
  30 1860498
  31 3010349
  32 4870847
  33 7881196
  34 12752043
  35 20633239
  36 33385282
  37 54018521
  38 87403803
  39 141422324
  40 228826127
  41 370248451
  42 599074578
  43 969323029
  44 1568397607
  45 2537720636
  46 4106118243
  47 6643838879
  48 10749957122
  49 17393796001
  50 28143753123
  51 45537549124
  52 73681302247
  53 119218851371
  54 192900153618
  55 312119004989
  56 505019158607
  57 817138163596
  58 1322157322203
  59 2139295485799
  60 3461452808002
  61 5600748293801
  62 9062201101803
  63 14662949395604
  64 23725150497407
  65 38388099893011
  66 62113250390418
$\endgroup$
  • $\begingroup$ Just out of curiosity: who is Noam here? I didn't see any indication from the web page you linked to. $\endgroup$ – Todd Trimble Oct 27 '14 at 2:47
  • $\begingroup$ @Todd, Noam Elkies. I can edit in where those comments occurred. Give me a few minutes. $\endgroup$ – Will Jagy Oct 27 '14 at 2:50
  • 20
    $\begingroup$ @Will Jagy, This is a really nice question; but could you please put in a better title for it? Thanks. $\endgroup$ – Amritanshu Prasad Oct 27 '14 at 5:00

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.