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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?

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  • $\begingroup$ 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
    Commented 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
    Commented 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
    Commented 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
    Commented Feb 20, 2021 at 23:05

1 Answer 1

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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
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