5
$\begingroup$

I have been looking at binary quadratic forms for a question on MSE, If a binary quadratic form primitively represents $n$ and $n^3$, must it be the identity form?, about forms representing a prime (not dividing the discriminant) and primitively representing its cube. I calculated that the order of such a form must be one or two or four. The group under Gauss composition, mostly using Dirichlet's description.

When such a form has nice coefficients as $f=ax^2 + mc xy + ac y^2$, triple $\langle a,mc,ac \rangle,$ some pleasant things happen. The duplicate form has coefficients $\langle a^2,mc,c \rangle$ and the fourth power is evidently the identity, as $c \mid mc$. This is what Dickson calls "ambiguous" as a representative of the class. We define \begin{align*} X ={} & -ax^3 + 3acx y^2 + mc^2 y^3 \\ Y ={} & mx^3 + 3ax^2 y -ac y^3 \end{align*} after which $ F = a X^2 + mcXY + ac Y^2 $ is identically equal to $f^3$ as polynomials in $x$, $y$.

I have a single strange example so far, $\langle 14, 8, 29 \rangle$. It is of order four, and with \begin{align*} f ={} & 14 x^2 + 8 xy+29y^2 , \\ u={} & 6x^3 + 60x^2y - 3xy^2 -42y^3, \\ v ={} & 8x^3- 18x^2y - 60xy^2 +y^3, \\ h ={} & 14 u^2 + 8 uv+29v^2 \end{align*} cause $h=f^3$ identically as polynomials. $u$, $v$ are coprime when $x$, $y$ are coprime and \begin{align*} y \neq{} & 0 \pmod 2, \\ y \neq{} & x \pmod 3 , \\ x+y \neq{} & 0 \pmod 5 , \\ y-3x \neq{} & 0 \pmod {13}. \end{align*}

However, I was unable to put the form $\langle 14, 8, 29 \rangle$ into the desired shape $\langle a,mc,ac \rangle$. For a favorable shape, the duplicate $\langle 10, 0, 39 \rangle$ would need to have a representative either $\langle 10, 20v, 39 + 10 v^2 \rangle$ or $\langle 39, 78v, 10 + 39 v^2 \rangle$ where the final coefficient is to be a square, in particular the square of something represented by $\langle 14, 8, 29 \rangle$. But that does not happen; the proof involves a half dozen Pell type equations.

Question. Why does $\langle 14, 8, 29 \rangle$ have no equivalent expression as $\langle a,mc,ac \rangle$, and where might we find other examples?

$\endgroup$
4
  • $\begingroup$ the integers $\sqrt{39 + 10 v^2}$ and $\sqrt{10 + 39 v^2}$ come out wrong, either $\pmod 3$ or $\pmod 8,$ for values of $14 x^2 + 8 xy + 29 y^2.$ The form is alone in its genus, factoring does not matter...so weird $\endgroup$
    – Will Jagy
    Feb 19, 2021 at 2:12
  • $\begingroup$ right. The values indicated by the square root signs in my comment last night come out $1, 7, 17, 19, 23 \pmod {24}$ while the values of $14 x^2 + 8xy + 29 y^2$ that are coprime with $24$ are $5, 11 \pmod{24}$ $\endgroup$
    – Will Jagy
    Feb 19, 2021 at 17:36
  • $\begingroup$ getting systematic about it. Found one with a quick proof, $\langle 5,4,8 \rangle$ because the duplicated form is $\langle 4,0,9 \rangle$ and there are only finitely many solutions to $w^2 - 9 v^2=4$ or $w^2 - 4 v^2 = 9$ Two more from that page $\langle 5,2,13 \rangle$ $\langle 6,2,11 \rangle$ $\endgroup$
    – Will Jagy
    Feb 20, 2021 at 1:27
  • $\begingroup$ I added a link to the MSE question, and, in the process, converted your manually spaced inline math (which I found very hard to read) into the relevant AMSmath environments. I hope that was all right. $\endgroup$
    – LSpice
    Feb 20, 2021 at 23:05

1 Answer 1

0
$\begingroup$

Finally made an exhaustive program, finds stubborn order four forms; ran it up to absolute value of discriminant 2500.

  144:  < 5, 4, 8>     STUBBORN      144:  < 4, 0, 9>     144:  < 1, 0, 36>  144 = 2^4 * 3^2
 256:  < 5, 2, 13>     STUBBORN      256:  < 4, 4, 17>     256:  < 1, 0, 64>  256 = 2^8
 260:  < 6, 2, 11>     STUBBORN      260:  < 9, 8, 9>     260:  < 1, 0, 65>  260 = 2^2 * 5 * 13
 360:  < 9, 6, 11>     STUBBORN      360:  < 9, 0, 10>     360:  < 1, 0, 90>  360 = 2^3 * 3^2 * 5
 384:  < 7, 6, 15>     STUBBORN      384:  < 4, 4, 25>     384:  < 1, 0, 96>  384 = 2^7 * 3
 468:  < 7, 6, 18>     STUBBORN      468:  < 9, 0, 13>     468:  < 1, 0, 117>  468 = 2^2 * 3^2 * 13
 528:  < 7, 2, 19>     STUBBORN      528:  < 4, 0, 33>     528:  < 1, 0, 132>  528 = 2^4 * 3 * 11
 576:  < 5, 2, 29>     STUBBORN      576:  < 9, 0, 16>     576:  < 1, 0, 144>  576 = 2^6 * 3^2
 576:  < 9, 6, 17>     STUBBORN      576:  < 9, 0, 16>     576:  < 1, 0, 144>  576 = 2^6 * 3^2
 640:  < 7, 2, 23>     STUBBORN      640:  < 4, 4, 41>     640:  < 1, 0, 160>  640 = 2^7 * 5
 735:  < 12, 9, 17>     STUBBORN      735:  < 15, 15, 16>     735:  < 1, 1, 184>  735 = 3 * 5 * 7^2
 768:  < 13, 8, 16>     STUBBORN      768:  < 4, 4, 49>     768:  < 1, 0, 192>  768 = 2^8 * 3
 819:  < 5, 1, 41>     STUBBORN      819:  < 9, 9, 25>     819:  < 1, 1, 205>  819 = 3^2 * 7 * 13
 832:  < 11, 2, 19>     STUBBORN      832:  < 16, 16, 17>     832:  < 1, 0, 208>  832 = 2^6 * 13
 896:  < 9, 2, 25>     STUBBORN      896:  < 4, 4, 57>     896:  < 1, 0, 224>  896 = 2^7 * 7
 896:  < 15, 8, 16>     STUBBORN      896:  < 4, 4, 57>     896:  < 1, 0, 224>  896 = 2^7 * 7
 900:  < 9, 6, 26>     STUBBORN      900:  < 9, 0, 25>     900:  < 1, 0, 225>  900 = 2^2 * 3^2 * 5^2
 900:  < 13, 6, 18>     STUBBORN      900:  < 9, 0, 25>     900:  < 1, 0, 225>  900 = 2^2 * 3^2 * 5^2
 912:  < 11, 10, 23>     STUBBORN      912:  < 4, 0, 57>     912:  < 1, 0, 228>  912 = 2^4 * 3 * 19
1035:  < 7, 1, 37>     STUBBORN     1035:  < 9, 9, 31>    1035:  < 1, 1, 259>  1035 = 3^2 * 5 * 23
1040:  < 9, 2, 29>     STUBBORN     1040:  < 4, 0, 65>    1040:  < 1, 0, 260>  1040 = 2^4 * 5 * 13
1088:  < 9, 8, 32>     STUBBORN     1088:  < 16, 0, 17>    1088:  < 1, 0, 272>  1088 = 2^6 * 17
1152:  < 11, 6, 27>     STUBBORN     1152:  < 4, 4, 73>    1152:  < 1, 0, 288>  1152 = 2^7 * 3^2
1224:  < 9, 6, 35>     STUBBORN     1224:  < 9, 0, 34>    1224:  < 1, 0, 306>  1224 = 2^3 * 3^2 * 17
1224:  < 18, 12, 19>     STUBBORN     1224:  < 9, 0, 34>    1224:  < 1, 0, 306>  1224 = 2^3 * 3^2 * 17
1240:  < 17, 16, 22>     STUBBORN     1240:  < 10, 0, 31>    1240:  < 1, 0, 310>  1240 = 2^3 * 5 * 31
1275:  < 13, 5, 25>     STUBBORN     1275:  < 19, 13, 19>    1275:  < 1, 1, 319>  1275 = 3 * 5^2 * 17
1332:  < 9, 6, 38>     STUBBORN     1332:  < 9, 0, 37>    1332:  < 1, 0, 333>  1332 = 2^2 * 3^2 * 37
1344:  < 17, 4, 20>     STUBBORN     1344:  < 16, 16, 25>    1344:  < 1, 0, 336>  1344 = 2^6 * 3 * 7
1344:  < 11, 8, 32>     STUBBORN     1344:  < 16, 16, 25>    1344:  < 1, 0, 336>  1344 = 2^6 * 3 * 7
1360:  < 11, 2, 31>     STUBBORN     1360:  < 4, 0, 85>    1360:  < 1, 0, 340>  1360 = 2^4 * 5 * 17
1420:  < 17, 12, 23>     STUBBORN     1420:  < 5, 0, 71>    1420:  < 1, 0, 355>  1420 = 2^2 * 5 * 71
1440:  < 9, 6, 41>     STUBBORN     1440:  < 9, 0, 40>    1440:  < 1, 0, 360>  1440 = 2^5 * 3^2 * 5
1440:  < 11, 10, 35>     STUBBORN     1440:  < 9, 0, 40>    1440:  < 1, 0, 360>  1440 = 2^5 * 3^2 * 5
1536:  < 11, 2, 35>     STUBBORN     1536:  < 4, 4, 97>    1536:  < 1, 0, 384>  1536 = 2^9 * 3
1560:  < 14, 8, 29>     STUBBORN     1560:  < 10, 0, 39>    1560:  < 1, 0, 390>  1560 = 2^3 * 3 * 5 * 13
1575:  < 8, 5, 50>     STUBBORN     1575:  < 9, 9, 46>    1575:  < 1, 1, 394>  1575 = 3^2 * 5^2 * 7
1580:  < 19, 4, 21>     STUBBORN     1580:  < 5, 0, 79>    1580:  < 1, 0, 395>  1580 = 2^2 * 5 * 79
1600:  < 13, 8, 32>     STUBBORN     1600:  < 16, 0, 25>    1600:  < 1, 0, 400>  1600 = 2^6 * 5^2
1600:  < 17, 10, 25>     STUBBORN     1600:  < 16, 0, 25>    1600:  < 1, 0, 400>  1600 = 2^6 * 5^2
1640:  < 18, 4, 23>     STUBBORN     1640:  < 10, 0, 41>    1640:  < 1, 0, 410>  1640 = 2^3 * 5 * 41
1664:  < 15, 14, 31>     STUBBORN     1664:  < 4, 4, 105>    1664:  < 1, 0, 416>  1664 = 2^7 * 13
1680:  < 11, 6, 39>     STUBBORN     1680:  < 4, 0, 105>    1680:  < 1, 0, 420>  1680 = 2^4 * 3 * 5 * 7
1680:  < 13, 6, 33>     STUBBORN     1680:  < 4, 0, 105>    1680:  < 1, 0, 420>  1680 = 2^4 * 3 * 5 * 7
1680:  < 19, 12, 24>     STUBBORN     1680:  < 4, 0, 105>    1680:  < 1, 0, 420>  1680 = 2^4 * 3 * 5 * 7
1683:  < 7, 5, 61>     STUBBORN     1683:  < 9, 9, 49>    1683:  < 1, 1, 421>  1683 = 3^2 * 11 * 17
1700:  < 22, 18, 23>     STUBBORN     1700:  < 21, 8, 21>    1700:  < 1, 0, 425>  1700 = 2^2 * 5^2 * 17
1764:  < 9, 6, 50>     STUBBORN     1764:  < 9, 0, 49>    1764:  < 1, 0, 441>  1764 = 2^2 * 3^2 * 7^2
1768:  < 22, 16, 23>     STUBBORN     1768:  < 17, 0, 26>    1768:  < 1, 0, 442>  1768 = 2^3 * 13 * 17
1780:  < 19, 14, 26>     STUBBORN     1780:  < 5, 0, 89>    1780:  < 1, 0, 445>  1780 = 2^2 * 5 * 89
1792:  < 11, 10, 43>     STUBBORN     1792:  < 4, 4, 113>    1792:  < 1, 0, 448>  1792 = 2^8 * 7
1815:  < 19, 3, 24>     STUBBORN     1815:  < 15, 15, 34>    1815:  < 1, 1, 454>  1815 = 3 * 5 * 11^2
1860:  < 14, 10, 35>     STUBBORN     1860:  < 10, 10, 49>    1860:  < 1, 0, 465>  1860 = 2^2 * 3 * 5 * 31
1860:  < 21, 18, 26>     STUBBORN     1860:  < 10, 10, 49>    1860:  < 1, 0, 465>  1860 = 2^2 * 3 * 5 * 31
1872:  < 11, 8, 44>     STUBBORN     1872:  < 4, 0, 117>    1872:  < 1, 0, 468>  1872 = 2^4 * 3^2 * 13
1920:  < 13, 2, 37>     STUBBORN     1920:  < 4, 4, 121>    1920:  < 1, 0, 480>  1920 = 2^7 * 3 * 5
1920:  < 17, 16, 32>     STUBBORN     1920:  < 4, 4, 121>    1920:  < 1, 0, 480>  1920 = 2^7 * 3 * 5
1924:  < 10, 6, 49>     STUBBORN     1924:  < 25, 24, 25>    1924:  < 1, 0, 481>  1924 = 2^2 * 13 * 37
1975:  < 22, 15, 25>     STUBBORN     1975:  < 25, 25, 26>    1975:  < 1, 1, 494>  1975 = 5^2 * 79
1980:  < 16, 2, 31>     STUBBORN     1980:  < 9, 0, 55>    1980:  < 1, 0, 495>  1980 = 2^2 * 3^2 * 5 * 11
1980:  < 9, 6, 56>     STUBBORN     1980:  < 9, 0, 55>    1980:  < 1, 0, 495>  1980 = 2^2 * 3^2 * 5 * 11
2020:  < 22, 2, 23>     STUBBORN     2020:  < 5, 0, 101>    2020:  < 1, 0, 505>  2020 = 2^2 * 5 * 101
2040:  < 21, 12, 26>     STUBBORN     2040:  < 15, 0, 34>    2040:  < 1, 0, 510>  2040 = 2^3 * 3 * 5 * 17
2064:  < 23, 12, 24>     STUBBORN     2064:  < 4, 0, 129>    2064:  < 1, 0, 516>  2064 = 2^4 * 3 * 43
2088:  < 9, 6, 59>     STUBBORN     2088:  < 9, 0, 58>    2088:  < 1, 0, 522>  2088 = 2^3 * 3^2 * 29
2100:  < 11, 10, 50>     STUBBORN     2100:  < 21, 0, 25>    2100:  < 1, 0, 525>  2100 = 2^2 * 3 * 5^2 * 7
2100:  < 17, 12, 33>     STUBBORN     2100:  < 21, 0, 25>    2100:  < 1, 0, 525>  2100 = 2^2 * 3 * 5^2 * 7
2112:  < 19, 4, 28>     STUBBORN     2112:  < 16, 0, 33>    2112:  < 1, 0, 528>  2112 = 2^6 * 3 * 11
2112:  < 17, 8, 32>     STUBBORN     2112:  < 16, 0, 33>    2112:  < 1, 0, 528>  2112 = 2^6 * 3 * 11
2115:  < 13, 11, 43>     STUBBORN     2115:  < 9, 9, 61>    2115:  < 1, 1, 529>  2115 = 3^2 * 5 * 47
2139:  < 5, 1, 107>     STUBBORN     2139:  < 25, 19, 25>    2139:  < 1, 1, 535>  2139 = 3 * 23 * 31
2180:  < 21, 16, 29>     STUBBORN     2180:  < 5, 0, 109>    2180:  < 1, 0, 545>  2180 = 2^2 * 5 * 109
2223:  < 8, 7, 71>     STUBBORN     2223:  < 9, 9, 64>    2223:  < 1, 1, 556>  2223 = 3^2 * 13 * 19
2244:  < 15, 6, 38>     STUBBORN     2244:  < 25, 16, 25>    2244:  < 1, 0, 561>  2244 = 2^2 * 3 * 11 * 17
2244:  < 19, 6, 30>     STUBBORN     2244:  < 25, 16, 25>    2244:  < 1, 0, 561>  2244 = 2^2 * 3 * 11 * 17
2275:  < 19, 9, 31>     STUBBORN     2275:  < 25, 25, 29>    2275:  < 1, 1, 569>  2275 = 5^2 * 7 * 13
2304:  < 5, 4, 116>     STUBBORN     2304:  < 25, 14, 25>    2304:  < 1, 0, 576>  2304 = 2^8 * 3^2
2304:  < 20, 4, 29>     STUBBORN     2304:  < 25, 14, 25>    2304:  < 1, 0, 576>  2304 = 2^8 * 3^2
2304:  < 9, 6, 65>     STUBBORN     2304:  < 9, 0, 64>    2304:  < 1, 0, 576>  2304 = 2^8 * 3^2
2304:  < 13, 6, 45>     STUBBORN     2304:  < 4, 4, 145>    2304:  < 1, 0, 576>  2304 = 2^8 * 3^2
2304:  < 17, 12, 36>     STUBBORN     2304:  < 9, 0, 64>    2304:  < 1, 0, 576>  2304 = 2^8 * 3^2
2320:  < 17, 14, 37>     STUBBORN     2320:  < 4, 0, 145>    2320:  < 1, 0, 580>  2320 = 2^4 * 5 * 29
2331:  < 11, 1, 53>     STUBBORN     2331:  < 9, 9, 67>    2331:  < 1, 1, 583>  2331 = 3^2 * 7 * 37
2331:  < 17, 7, 35>     STUBBORN     2331:  < 25, 13, 25>    2331:  < 1, 1, 583>  2331 = 3^2 * 7 * 37
2340:  < 23, 12, 27>     STUBBORN     2340:  < 10, 10, 61>    2340:  < 1, 0, 585>  2340 = 2^2 * 3^2 * 5 * 13
2368:  < 19, 8, 32>     STUBBORN     2368:  < 16, 16, 41>    2368:  < 1, 0, 592>  2368 = 2^6 * 37
2368:  < 23, 22, 31>     STUBBORN     2368:  < 16, 16, 41>    2368:  < 1, 0, 592>  2368 = 2^6 * 37
2400:  < 21, 6, 29>     STUBBORN     2400:  < 24, 0, 25>    2400:  < 1, 0, 600>  2400 = 2^5 * 3 * 5^2
2400:  < 25, 20, 28>     STUBBORN     2400:  < 24, 0, 25>    2400:  < 1, 0, 600>  2400 = 2^5 * 3 * 5^2
2420:  < 23, 8, 27>     STUBBORN     2420:  < 5, 0, 121>    2420:  < 1, 0, 605>  2420 = 2^2 * 5 * 11^2
2436:  < 10, 2, 61>     STUBBORN     2436:  < 25, 8, 25>    2436:  < 1, 0, 609>  2436 = 2^2 * 3 * 7 * 29
2436:  < 15, 12, 43>     STUBBORN     2436:  < 25, 8, 25>    2436:  < 1, 0, 609>  2436 = 2^2 * 3 * 7 * 29
2448:  < 13, 10, 49>     STUBBORN     2448:  < 4, 0, 153>    2448:  < 1, 0, 612>  2448 = 2^4 * 3^2 * 17
2475:  < 23, 3, 27>     STUBBORN     2475:  < 25, 25, 31>    2475:  < 1, 1, 619>  2475 = 3^2 * 5^2 * 11
2475:  < 25, 15, 27>     STUBBORN     2475:  < 25, 25, 31>    2475:  < 1, 1, 619>  2475 = 3^2 * 5^2 * 11
2484:  < 5, 4, 125>     STUBBORN     2484:  < 25, 4, 25>    2484:  < 1, 0, 621>  2484 = 2^2 * 3^3 * 23
2499:  < 15, 9, 43>     STUBBORN     2499:  < 25, 1, 25>    2499:  < 1, 1, 625>  2499 = 3 * 7^2 * 17
2511:  < 20, 17, 35>     STUBBORN     2511:  < 28, 25, 28>    2511:  < 1, 1, 628>  2511 = 3^4 * 31
$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge that you have read and understand our privacy policy and code of conduct.

Not the answer you're looking for? Browse other questions tagged or ask your own question.