# Looking for proof of no solution to 4-variable quadratic diophantine equation

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
@WillJagy: no problem, I was a bit confused myself. You are quite right. – Dietrich Burde Dec 30 '13 at 20:17

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.

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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