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Show there are no integers $a,b,c,d$ such that $$\begin{cases}1=9ac+3ad+3bc-16bd \\ 1=3ad+3bc+2bd \end{cases}$$

Motivation: The ideal $I=(3,1+\sqrt{-17})$ in $R=\mathbb{Z}[\sqrt{-17}]$ has the property that that $\omega=1+\sqrt{-17} \in I^2$ but is not a product of two elements of $I$. Writing out the claim (that $\omega$ is a product of two elements of $I$) as a diophantine equation (and noting the integral basis of $I$) gives the equation.

Mild progress: Subtracting the equations gives $0=9(ac-2bd)$ so that $ac=2bd$ and so we get an equivalent system: $$\begin{cases}0=ac-2bd \\ 1=3ad+3bc+2bd \end{cases}$$

Another method: The norm $N(a+b\sqrt{-17}) = a^2+17b^2$ is multiplicative, and the minimum value of $N$ on $I\setminus \{0\}$ is $9$, so the minimum value of a nonzero product of two elements of $I$ is $81$, while the value on $\omega$ is only $18$. However, I'm not familiar with low-tech methods of finding the minimum nonzero norm of a set. In particular how does one show: $$\min\left(\{ 3a^2+2ab+6b^2 : a,b\in\mathbb{Z}; (a,b) \neq 0\}\right) = 3$$ or at least that it is strictly greater than $1$? [ I've shown it is at least equal to 1. ]

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@DietrichBurde, if a binary quadratic form is positive and reduced in the sense of Gauss, the "minimum" is the first coefficient. –  Will Jagy Dec 30 '13 at 20:06
    
@DietrichBurde: thanks! I actually tried such ideas for several minutes with no luck. Can you post it as an answer too? I'm reading over the other methods too, but so far yours is simplest. –  Jack Schmidt Dec 30 '13 at 20:11
    
@DietrichBurde, nothing wrong at all. Just giving the general fact, when your clever technique may be hard to work out (this would happen if the first and third coefficients were $7 \pmod 8$ for example). Sorry that I gave a negative impression. –  Will Jagy Dec 30 '13 at 20:13
1  
@WillJagy: no problem, I was a bit confused myself. You are quite right. –  Dietrich Burde Dec 30 '13 at 20:17

5 Answers 5

up vote 2 down vote accepted

We have $3a^2+2ab+6b^2=(a+b)^2+2a^2+5b^2$, which is as a sum of non-negative squares at least $5$, if not $b=0$. Hence we may assume that $b=0$. Then obviously $3a^2\ge 3$. The general estimates can be found in the nice answer of Will.

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$$ 4A (A x^2 + B x y + C y^2) = 4 A^2 x^2 + 4 AB x y + B^2 y^2 + (4AC-B^2) y^2= (2Ax+By)^2 + (4AC-B^2) y^2 $$

$$ A x^2 + B x y + C y^2 \geq \left( \frac{4 A C - B^2}{4C}\right) x^2 $$ $$ A x^2 + B x y + C y^2 \geq \left( \frac{4 A C - B^2}{4A}\right) y^2 $$ when $A,C > 0$ and $4AC - B^2 > 0.$

$$ 3 x^2 + 2 x y + 6 y^2 \geq \left( \frac{68}{24}\right) x^2 $$ $$ 3 x^2 + 2 x y + 6 y^2 \geq \left( \frac{68}{12}\right) y^2 $$

$68/24 \approx 2.833333,$ so when $x$ is nonzero and integral,................

If the form $A x^2 + B x y + C y^2$ is positive and reduced, meaning $0 < A \leq C$ and $-A < B \leq A,$ then the minimum of the form for integral $(x,y) \neq (0,0)$ is simply $A$ itself.

Indefinite forms are very different; that would be real quadratic fields..There is an entirely successful approach, due to Lagrange and perhaps Gauss, that gives the "minimum" without needing high decimal accuracy in a continued fraction or any computer memory or pattern matching at all. I have answered many MSE questions with that, I'm not sure anyone understood the advantage over continued fractions; the approaches are mathematically equivalent but not computer implementation equivalent.

The previous indefinite forms were mostly Pell, where the minimum is automatically 1. So, here is an example where i did not know the minimum, I randomly picked coefficients. $-4$ appears at steps 80,81.

    =========================================

        jagy@phobeusjunior:~/old drive/home/jagy/Cplusplus$ ./indefCycle
        Input three coefficients a b c for indef f(x,y)= a x^2 + b x y + c y^2 
        173 567 -1020

          0  form            173         567       -1020  delta      0
          1  form          -1020        -567         173  delta      1
          2  form            173         913        -280


                  -1          -1
                   0          -1

        To Return  
                  -1           1
                   0          -1

        0  form   173 913 -280   delta  -3
        1  form   -280 767 392   delta  2
        2  form   392 801 -246   delta  -3
        3  form   -246 675 581   delta  1
        4  form   581 487 -340   delta  -2
        5  form   -340 873 195   delta  4
        6  form   195 687 -712   delta  -1
        7  form   -712 737 170   delta  5
        8  form   170 963 -147   delta  -6
        9  form   -147 801 656   delta  1
        10  form   656 511 -292   delta  -2
        11  form   -292 657 510   delta  1
        12  form   510 363 -439   delta  -1
        13  form   -439 515 434   delta  1
        14  form   434 353 -520   delta  -1
        15  form   -520 687 267   delta  3
        16  form   267 915 -178   delta  -5
        17  form   -178 865 392   delta  2
        18  form   392 703 -340   delta  -2
        19  form   -340 657 438   delta  1
        20  form   438 219 -559   delta  -1
        21  form   -559 899 98   delta  9
        22  form   98 865 -712   delta  -1
        23  form   -712 559 251   delta  3
        24  form   251 947 -130   delta  -7
        25  form   -130 873 510   delta  1
        26  form   510 147 -493   delta  -1
        27  form   -493 839 164   delta  5
        28  form   164 801 -588   delta  -1
        29  form   -588 375 377   delta  1
        30  form   377 379 -586   delta  -1
        31  form   -586 793 170   delta  5
        32  form   170 907 -301   delta  -3
        33  form   -301 899 182   delta  5
        34  form   182 921 -246   delta  -3
        35  form   -246 555 731   delta  1
        36  form   731 907 -70   delta  -13
        37  form   -70 913 692   delta  1
        38  form   692 471 -291   delta  -2
        39  form   -291 693 470   delta  1
        40  form   470 247 -514   delta  -1
        41  form   -514 781 203   delta  4
        42  form   203 843 -390   delta  -2
        43  form   -390 717 329   delta  2
        44  form   329 599 -508   delta  -1
        45  form   -508 417 420   delta  1
        46  form   420 423 -505   delta  -1
        47  form   -505 587 338   delta  2
        48  form   338 765 -327   delta  -2
        49  form   -327 543 560   delta  1
        50  form   560 577 -310   delta  -2
        51  form   -310 663 474   delta  1
        52  form   474 285 -499   delta  -1
        53  form   -499 713 260   delta  3
        54  form   260 847 -298   delta  -3
        55  form   -298 941 119   delta  8
        56  form   119 963 -210   delta  -4
        57  form   -210 717 611   delta  1
        58  form   611 505 -316   delta  -2
        59  form   -316 759 357   delta  2
        60  form   357 669 -406   delta  -2
        61  form   -406 955 71   delta  13
        62  form   71 891 -822   delta  -1
        63  form   -822 753 140   delta  6
        64  form   140 927 -300   delta  -3
        65  form   -300 873 221   delta  4
        66  form   221 895 -256   delta  -3
        67  form   -256 641 602   delta  1
        68  form   602 563 -295   delta  -2
        69  form   -295 617 548   delta  1
        70  form   548 479 -364   delta  -2
        71  form   -364 977 50   delta  19
        72  form   50 923 -877   delta  -1
        73  form   -877 831 96   delta  9
        74  form   96 897 -580   delta  -1
        75  form   -580 263 413   delta  1
        76  form   413 563 -430   delta  -1
        77  form   -430 297 546   delta  1
        78  form   546 795 -181   delta  -4
        79  form   -181 653 830   delta  1
        80  form   830 1007 -4   delta  -252
        81  form   -4 1009 578   delta  1
        82  form   578 147 -435   delta  -1
        83  form   -435 723 290   delta  2
        84  form   290 437 -721   delta  -1
        85  form   -721 1005 6   delta  168
        86  form   6 1011 -217   delta  -4
        87  form   -217 725 578   delta  1
        88  form   578 431 -364   delta  -1
        89  form   -364 297 645   delta  1
        90  form   645 993 -16   delta  -62
        91  form   -16 991 707   delta  1
        92  form   707 423 -300   delta  -2
        93  form   -300 777 353   delta  2
        94  form   353 635 -442   delta  -1
        95  form   -442 249 546   delta  1
        96  form   546 843 -145   delta  -6
        97  form   -145 897 384   delta  2
        98  form   384 639 -403   delta  -2
        99  form   -403 973 50   delta  19
        100  form   50 927 -840   delta  -1
        101  form   -840 753 137   delta  6
        102  form   137 891 -426   delta  -2
        103  form   -426 813 215   delta  4
        104  form   215 907 -238   delta  -4
        105  form   -238 997 35   delta  28
        106  form   35 963 -714   delta  -1
        107  form   -714 465 284   delta  2
        108  form   284 671 -508   delta  -1
        109  form   -508 345 447   delta  1
        110  form   447 549 -406   delta  -1
        111  form   -406 263 590   delta  1
        112  form   590 917 -79   delta  -12
        113  form   -79 979 218   delta  4
        114  form   218 765 -507   delta  -1
        115  form   -507 249 476   delta  1
        116  form   476 703 -280   delta  -3
        117  form   -280 977 65   delta  15
        118  form   65 973 -310   delta  -3
        119  form   -310 887 194   delta  4
        120  form   194 665 -754   delta  -1
        121  form   -754 843 105   delta  8
        122  form   105 837 -778   delta  -1
        123  form   -778 719 164   delta  5
        124  form   164 921 -273   delta  -3
        125  form   -273 717 470   delta  1
        126  form   470 223 -520   delta  -1
        127  form   -520 817 173   delta  5
        128  form   173 913 -280


minimum was   4
rep 
177003899612243834005256403731983 
608947905015187166087607559710201 
disc   1027329 dSqrt 1013.572395  M_Ratio  34.32554
Automorph, written on right of Gram matrix:  
3635174031481176012669428632236695796012068290929256206913  20241115434222530087909412012011999770317408825378480215040
12506117750430348947172600993135985572374684738537418132864  69635668286642497335031189942832966475654190639395514908097
=========================================


        =========================================
        jagy@phobeusjunior:~/old drive/home/jagy/Cplusplus$
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Can you explain the (4AC-B^2)/C inequality? (briefly is probably fine) Can you link one of your answers to the indefinite forms? (no use to me on this problem, but surely indefinite forms come up) –  Jack Schmidt Dec 30 '13 at 20:17
    
Thanks! I gave DB the check (since my goal was to explain why in easiest terms), but I suspect your answer will get the most votes since it answers many more questions than just mine. –  Jack Schmidt Dec 30 '13 at 20:58

You only have to show it doesn't equal $1$. Solving the equation for $a$ gives $$ b=\frac{-a\pm \sqrt{6-17a^2}}6 $$ If $a\neq 0$, the root is imaginary, so $a=1$. But then, $b$ is not an integer, so this doesn't have any solutions.

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Suppose we want to make $3a^2+2ab+6b^2$ small, where neither $a$ nor $b$ is $0$. It is clearly best to choose $a$ and $b$ of opposite sign. So we want to make $3x^2-2xy+6y^2$ small, with $x$ and $y$ positive. Note that $$3x^2-2xy+6y^2=3(x-\sqrt{2}y)^2+(6\sqrt{2}-2)xy.$$ For positive $x$ and $y$, the right-hand side is greater than $6$.

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Simpler than my version. –  André Nicolas Dec 30 '13 at 20:08

Not sure you need all this high power machinery. Here is my analysis

From $ac = 2 bd$ we have $$ a/b = 2d /c ~\text{ (say) } = s/t$$ Then $$ a = u s \\ b=u t \\ d =v s/2 \\ c= v t$$ Substitute in the second and solve for $u$ $$ \frac{2}{\left( 6\,{t}^{2}+2\,s\,t+3\,{s}^{2}\right) \,v}$$ So $$ v = \pm 1 ~\text{ or } \pm 2$$ and $$ 6\,{t}^{2}+2\,s\,t+3\,{s}^{2} = \pm1 ~\text{ or } \pm 2$$

Argue that $6\,{t}^{2}+2\,s\,t+3\,{s}^{2} $ can never be $\pm 1$ or $\pm 2$ by setting it to $\pm 1$ and $\pm 2$ and solving for $t$.

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Thanks! I like how the norm just reappears in your answer. –  Jack Schmidt Dec 30 '13 at 20:12
    
Yeah. I forgot to mention that when I wrote the answer. You should see $\sqrt{-17}$ pop up –  user44197 Dec 30 '13 at 20:19

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