# Positive binary quadratic form plus univariate monic cubic (giving Hilbert class field)

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

• Just out of curiosity: who is Noam here? I didn't see any indication from the web page you linked to. – Todd Trimble Oct 27 '14 at 2:47
• @Todd, Noam Elkies. I can edit in where those comments occurred. Give me a few minutes. – Will Jagy Oct 27 '14 at 2:50
• @Will Jagy, This is a really nice question; but could you please put in a better title for it? Thanks. – Amritanshu Prasad Oct 27 '14 at 5:00