Let $n\equiv 1\pmod 8$. Do there exist $x,y,z\in\mathbb{Z}$ with $x\equiv \pm3\pmod 8$ and $x^2+4y^2+4z^2=n$?

Question

Let me preface this by saying I know very little about quadratic forms and most of what I know is about quadratic forms in two variables, whilst this question is about a quadratic form in three variables.

Let $n\ge 1$ (Actually, $n> 1$, to make it interesting, see the edit) be a positive integer, $n\equiv 1\pmod 8$. Then there exist $x,y,z\in\mathbb{Z}$ $$ n = x^2 + 4y^2+4z^2. $$ Question: Can we always find such $(x,y,z)$ with $x\equiv \pm 3\pmod 8$?

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My attempt: Suppose $n\equiv 9\pmod{16}$ and $n$ is represented over the integers by $u^2+16v^2+16w^2$. Then we can take $(x,y,z)=(u,2v,2w)$. Now, according to this, the form $u^2+16v^2+16w^2$ is regular, which I believe means such a representation of $n$ does indeed exist.

For $n\equiv 1\pmod{16}$, the existence of a representation with $x\equiv \pm 3\pmod{8}$ is equivalent to $n$ being represented over the integers by $$ u^2 + (4v+2u)^2 + (4w+2u)^2 = 9u^2+16v^2+16w^2+16uv+16uw. $$ However, I know basically nothing about this quadratic form. Its discriminant is $256$ and does not show up in the aforementioned list of regular forms.

Any pointers in the right direction are appreciated.

Edit (added after ThibautOphelia's answer): There is no such solution when $n=1$, but I'm mainly interested in the case $n>1$.

Answer 1

I have most of it. All that remains are certain squares.

The other form in the genus of $f=x^2 + 16 y^2 + 16 z^2$ has reduced representative $g = 4 x^2 + 9 y^2 + 9 z^2 + 2 y z + 4zx + 4xy.$ In Jones and Pall (1939), this is written as (4,9,9,1,2,2) on the last page, as they halve the last three coefficients.

The form $f$ is regular, and does take care of $9 \pmod{16}.$ ADDED: meanwhile, if a square is $9 \pmod{16}$ then its square root is an integer and is $\pm 3 \pmod{16}$

The form $g$ is what we call spinor regular. It represents the same things as $f$ with the exception of certain squares; it does not represent any $m^2$ when no prime $q \equiv 3 \pmod 4$ divides $m.$ But it does represent all other numbers $n \equiv 1 \pmod{16},$ including any such number that is not a square, or a square that is divisible by some $q \equiv 3 \pmod 4.$

We take such $n$ as $$ n = 4 x^2 + 9 y^2 + 9 z^2 + 2 y z + 4zx + 4xy. $$

We may write this using what Kap called a homothety, $$ n = (2x+y+z)^2 + 4 (y-z)^2 + 4 (y+z)^2 $$ Now: since $n$ is odd, we know $2x+y+z$ is odd, and so are $y \pm z$ But then $(y \pm z)^2 \equiv 1 \pmod 8,$ next $$4(y \pm z)^2 \equiv 4 \pmod {32},$$ then $4(y - z)^2 + 4(y + z)^2 \equiv 8 \pmod {32},$ so $$4(y - z)^2 + 4(y + z)^2 \equiv 8 \pmod {16}.$$

It now follows from $$ n = (2x+y+z)^2 + 4 (y-z)^2 + 4 (y+z)^2 \equiv 1 \pmod{16}$$ that $(2x+y+z)^2 \equiv 9 \pmod{16}$ and $$ 2x+y+z \equiv \pm 3 \pmod 8 $$

To repeat, your desired representation is always possible when $n \equiv 1 \pmod 8$ is not a spinor exception for the genus of $x^2 + 16 y^2 + 16 z^2$

Jones and Pall(1939)

Earnest and Haensch, preprint at arxiv

Answer 2

Take $n=1$, the solutions are $(\pm 1, 0, 0)$, and $\pm 1 \equiv \pm 1 \pmod{8}$.

Answer 3

We cannot find such $(x, y, z)$ with $x\equiv \pm 3 \pmod{8}$ for $n=289$, or $625$ for example.

Considering $y^2+z^2 = (n-x^2) / 4$ with the same assumptions as you made, the sum of two squares theorem tells you that this equation has a solution if and only if its prime decomposition contains no factor $p^k$ where $p\equiv 3 \pmod{4}$ and $k$ is odd. Knowing that, I could not relate the distribution of the sums of two squares with congruence modulo 8 for $x$. So I tried to find a counterexample numerically.

Answer 4

Better list. Up to $10^7,$ the failures are all $1 \pmod {16},$ all squares, and never divisible by any prime $q \equiv 3 \pmod 4$

Thu 08 Feb 2024 10:59:14 AM PST
1 =   1    sixteen   1
289 =  17^2   sixteen   1
625 =  5^4   sixteen   1
1681 =  41^2   sixteen   1
4225 =  5^2 13^2   sixteen   1
5329 =  73^2   sixteen   1
7921 =  89^2   sixteen   1
9409 =  97^2   sixteen   1
12769 =  113^2   sixteen   1
18769 =  137^2   sixteen   1
21025 =  5^2 29^2   sixteen   1
28561 =  13^4   sixteen   1
34225 =  5^2 37^2   sixteen   1
37249 =  193^2   sixteen   1
54289 =  233^2   sixteen   1
58081 =  241^2   sixteen   1
66049 =  257^2   sixteen   1
70225 =  5^2 53^2   sixteen   1
78961 =  281^2   sixteen   1
83521 =  17^4   sixteen   1
93025 =  5^2 61^2   sixteen   1
97969 =  313^2   sixteen   1
113569 =  337^2   sixteen   1
124609 =  353^2   sixteen   1
142129 =  13^2 29^2   sixteen   1
160801 =  401^2   sixteen   1
167281 =  409^2   sixteen   1
180625 =  5^4 17^2   sixteen   1
187489 =  433^2   sixteen   1
201601 =  449^2   sixteen   1
208849 =  457^2   sixteen   1
231361 =  13^2 37^2   sixteen   1
255025 =  5^2 101^2   sixteen   1
271441 =  521^2   sixteen   1
297025 =  5^2 109^2   sixteen   1
323761 =  569^2   sixteen   1
332929 =  577^2   sixteen   1
351649 =  593^2   sixteen   1
361201 =  601^2   sixteen   1
380689 =  617^2   sixteen   1
390625 =  5^8   sixteen   1
410881 =  641^2   sixteen   1
452929 =  673^2   sixteen   1
474721 =  13^2 53^2   sixteen   1
485809 =  17^2 41^2   sixteen   1
555025 =  5^2 149^2   sixteen   1
579121 =  761^2   sixteen   1
591361 =  769^2   sixteen   1
616225 =  5^2 157^2   sixteen   1
628849 =  13^2 61^2   sixteen   1
654481 =  809^2   sixteen   1
707281 =  29^4   sixteen   1
734449 =  857^2   sixteen   1
748225 =  5^2 173^2   sixteen   1
776161 =  881^2   sixteen   1
819025 =  5^2 181^2   sixteen   1
863041 =  929^2   sixteen   1
877969 =  937^2   sixteen   1
908209 =  953^2   sixteen   1
954529 =  977^2   sixteen   1
970225 =  5^2 197^2   sixteen   1
1018081 =  1009^2   sixteen   1
1050625 =  5^4 41^2   sixteen   1
1067089 =  1033^2   sixteen   1
1100401 =  1049^2   sixteen   1
1151329 =  29^2 37^2   sixteen   1
1203409 =  1097^2   sixteen   1
1221025 =  5^2 13^2 17^2   sixteen   1
1274641 =  1129^2   sixteen   1
1311025 =  5^2 229^2   sixteen   1
1329409 =  1153^2   sixteen   1
1423249 =  1193^2   sixteen   1
1442401 =  1201^2   sixteen   1
1481089 =  1217^2   sixteen   1
1540081 =  17^2 73^2   sixteen   1
1560001 =  1249^2   sixteen   1
1661521 =  1289^2   sixteen   1
1682209 =  1297^2   sixteen   1
1723969 =  13^2 101^2   sixteen   1
1745041 =  1321^2   sixteen   1
1809025 =  5^2 269^2   sixteen   1
1852321 =  1361^2   sixteen   1
1874161 =  37^4   sixteen   1
1918225 =  5^2 277^2   sixteen   1
1985281 =  1409^2   sixteen   1
2007889 =  13^2 109^2   sixteen   1
2053489 =  1433^2   sixteen   1
2146225 =  5^2 293^2   sixteen   1
2193361 =  1481^2   sixteen   1
2217121 =  1489^2   sixteen   1
2289169 =  17^2 89^2   sixteen   1
2362369 =  29^2 53^2   sixteen   1
2411809 =  1553^2   sixteen   1
2512225 =  5^2 317^2   sixteen   1
2563201 =  1601^2   sixteen   1
2588881 =  1609^2   sixteen   1
2640625 =  5^6 13^2   sixteen   1
2719201 =  17^2 97^2   sixteen   1
2745649 =  1657^2   sixteen   1
2825761 =  41^4   sixteen   1
2879809 =  1697^2   sixteen   1
2961841 =  1721^2   sixteen   1
3045025 =  5^2 349^2   sixteen   1
3073009 =  1753^2   sixteen   1
3129361 =  29^2 61^2   sixteen   1
3157729 =  1777^2   sixteen   1
3243601 =  1801^2   sixteen   1
3330625 =  5^4 73^2   sixteen   1
3478225 =  5^2 373^2   sixteen   1
3508129 =  1873^2   sixteen   1
3568321 =  1889^2   sixteen   1
3659569 =  1913^2   sixteen   1
3690241 =  17^2 113^2   sixteen   1
3751969 =  13^2 149^2   sixteen   1
3783025 =  5^2 389^2   sixteen   1
3845521 =  37^2 53^2   sixteen   1
3940225 =  5^2 397^2   sixteen   1
3972049 =  1993^2   sixteen   1
4068289 =  2017^2   sixteen   1
4165681 =  13^2 157^2   sixteen   1
4330561 =  2081^2   sixteen   1
4363921 =  2089^2   sixteen   1
4431025 =  5^2 421^2   sixteen   1
4464769 =  2113^2   sixteen   1
4532641 =  2129^2   sixteen   1
4566769 =  2137^2   sixteen   1
4635409 =  2153^2   sixteen   1
4669921 =  2161^2   sixteen   1
4950625 =  5^4 89^2   sixteen   1
5058001 =  13^2 173^2   sixteen   1
5094049 =  37^2 61^2   sixteen   1
5166529 =  2273^2   sixteen   1
5202961 =  2281^2   sixteen   1
5276209 =  2297^2   sixteen   1
5313025 =  5^2 461^2   sixteen   1
5424241 =  17^2 137^2   sixteen   1
5536609 =  13^2 181^2   sixteen   1
5650129 =  2377^2   sixteen   1
5726449 =  2393^2   sixteen   1
5841889 =  2417^2   sixteen   1
5880625 =  5^4 97^2   sixteen   1
5958481 =  2441^2   sixteen   1
6076225 =  5^2 17^2 29^2   sixteen   1
6115729 =  2473^2   sixteen   1
6355441 =  2521^2   sixteen   1
6477025 =  5^2 509^2   sixteen   1
6558721 =  13^2 197^2   sixteen   1
6723649 =  2593^2   sixteen   1
6806881 =  2609^2   sixteen   1
6848689 =  2617^2   sixteen   1
6932689 =  2633^2   sixteen   1
7059649 =  2657^2   sixteen   1
7102225 =  5^2 13^2 41^2   sixteen   1
7230721 =  2689^2   sixteen   1
7317025 =  5^2 541^2   sixteen   1
7360369 =  2713^2   sixteen   1
7447441 =  2729^2   sixteen   1
7579009 =  2753^2   sixteen   1
7711729 =  2777^2   sixteen   1
7756225 =  5^2 557^2   sixteen   1
7845601 =  2801^2   sixteen   1
7890481 =  53^4   sixteen   1
7980625 =  5^4 113^2   sixteen   1
8025889 =  2833^2   sixteen   1
8162449 =  2857^2   sixteen   1
8254129 =  13^4 17^2   sixteen   1
8392609 =  2897^2   sixteen   1
8579041 =  29^2 101^2   sixteen   1
8720209 =  2953^2   sixteen   1
8814961 =  2969^2   sixteen   1
8862529 =  13^2 229^2   sixteen   1
8958049 =  41^2 73^2   sixteen   1
9006001 =  3001^2   sixteen   1
9247681 =  3041^2   sixteen   1
9296401 =  3049^2   sixteen   1
9394225 =  5^2 613^2   sixteen   1
9541921 =  3089^2   sixteen   1
9740641 =  3121^2   sixteen   1
9840769 =  3137^2   sixteen   1
9891025 =  5^2 17^2 37^2   sixteen   1
9991921 =  29^2 109^2   sixteen   1
Thu 08 Feb 2024 10:59:14 AM PST

Answer 5

Finally, showing all the failures:

Everything you need comes from Jones and Pall(1939), in this case Theorem 4 on page 177. When positive $m >0 \; , \; \; m \equiv 1 \pmod 4$ has all prime factors $p \equiv 1 \pmod 4,$ then every representation $m^2 = x^2 + y^2 + z^2$ with odd $z$ has actually $x \equiv y \equiv 0 \pmod 4.$

Take such a representation. Find $g > 0$ with $g = \gcd(m,x,y,z) .$ Name $m_1 = m/g, \; x_1 = x/g, \; y_1 = y/g, z_1 = z/g.$ Note $g \equiv 1 \pmod 4 $ and $m_1 \equiv 1 \pmod 4. $ Now we have

$$m_1^2 = x_1^2 + y_1^2 + z_1^2$$ in coprime integers; indeed $\gcd(x_1, y_1, z_1) = 1.$ Thus we have, version in Spira (1962), integers $t,u,v,w$ such that $$ x_1= 2(uw-tv) \; , \; \; y_1=2(tu +vw) \; , \; \; z_1 = u^2 + v^2 - w^2 - t^2 \; , \; \; m_1 = u^2 + v^2 + w^2 + t^2 $$

As $m_1$ is odd, we know that we can't have an even number of odd terms in $t,u,v,w.$ As $m_1 \neq 3 \pmod 4$ we cannot have three odd terms in $t,u,v,w.$ So, there is exactly one odd term, and three even variables. It follows from $ x_1= 2(uw-tv) \; , \; \; y_1=2(tu +vw) \; ,$ that both $x_1, y_1 \equiv 0 \pmod 4.$

Finally we return to $m^2 = x^2 + y^2 + z^2$ using $x = g x_1$ and $y = g y_1.$ It follows that both $x,y$ are divisible by $4.$ We have shown that every representation is actually of the form $$ m^2 = 16 \left( \frac{x}{4} \right)^2 + 16 \left( \frac{y}{4} \right)^2 + z^2 $$

Finally, we restrict to $m \equiv 1 \pmod 8.$ We know that $m^2 \equiv 1 \pmod {16}.$

Here is the punchline: when $m >0, \; \;m \equiv 1 \pmod 8,$ it follows that $z^2 \equiv 1 \pmod {16},$ so that

$$ z \equiv \pm 1 \pmod {8}.$$