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This may be a very naive question and I apologize in advance.
Suppose that $n$ is a positive integer which cannot be written as a sum of three squares $a^2+b^2+c^2$ for integers $a,b,c\in\mathbb{Z}$.
Does it follow that $n$ cannot be written as a sum of three squares $a'^2+b'^2+c'^2$ for rationals $a',b',c'\in\mathbb{Q}$? Why?
There is a fairly easy answer; a positive integer is the sum of three squares if and only if it is not of the form $4^k (8n +7).$ Or, more useful for us, if and only if it is not $4^k m$ with $m \equiv 7 \pmod 8.$
If you have the sum of three rational squares equal to such a number, call it $t,$ then we can multiply through by the square of the least common denominator, which forces everything to be integers, so we have all integers in $x^2 + y^2 + z^2 = t L^2.$ Now, $L$ factors as some power of $2$ times an odd number, so let $L = 2^r s $ with odd $s.$ The important bit is that $s^2 \equiv 1 \pmod 8$ (CHECK THIS). So, $L^2 = 4^r s^2.$
Finally, we knew $t = 4^km$ with $m \equiv 7 \pmod 8,$ because $t$ was not integrally the sum of three squares. So $t L^2 = 4^{k+r} m s^2, $ where $m s^2 \equiv 7 \pmod 8.$ Therefore $t L^2$ is also not the sum of three integer squares and we are done.
It is true that positive integers are the sum of three integer squares if and only if they are the sum of three rational squares. This result, for a number of quadratic forms, is generally referred to as the Davenport-Cassels Theorem, but was first proved by Aubry about 1912 I think. It appears in Serre's little book, A Course in Arithmetic, with three squares on pages 45-46, and Davenport-Cassels on pages 46-47.
There are infinitely many quadratic forms with the same property, they represent a number over the integers if and only if they represent it over the rationals. The stronger property used in the Davenpost-Cassels hypothesis, for positive quadratic forms, occurs for just 70 forms of the type called "even lattices." The restriction is that the "covering radius" be strictly below $\sqrt 2.$ A full list is given by G. Nebe see http://www.math.rwth-aachen.de/~Gabriele.Nebe/pl.html and the pdf listed as Even lattices with covering radius stricktly smaller than sqrt{2}. Beiträge zur Algebra und Geometrie, Vol. 44, No. 1, 2003, 229-234 Note that I forgot the case R=A2A1A1 in Theorem 7. This root system yields two further lattices, one with covering radius =sqrt{2} and one with c.r. strictly smaller than sqrt{2}
There are 103 positive forms in three variables that have your property, that they represent an integer over the integers if and only if they represent it over the rationals. Each is summarized by six coefficients preceded by a "discriminant" I call Delta. Any primitive, positive ternary with the property is "equivalent" to one of these 103. ADC stands for Aubry-Davenport-Cassels. Let's see, only thirteen of the forms below belong on Nebe's list. Her hypotheses are far more restrictive than the adc property. Her list has 70 entries because the number of variables gets as large as ten.
$$Q(x,y,z) = a x^2 + b y^2 + c z^2 + r y z + s z x + t x y,$$ and $$\Delta = 4 a b c + r s t - a r^2 - b s^2 - c t^2$$
Delta a b c r s t
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2: 1 1 1 1 1 1 ADC 2 = 2
3: 1 1 1 0 0 1 ADC 3 = 3
4: 1 1 1 0 0 0 ADC 4 = 2^2
5: 1 1 2 1 1 1 ADC 5 = 5
6: 1 1 2 0 0 1 ADC 6 = 2 * 3
6: 1 1 2 1 1 0 ADC 6 = 2 * 3
7: 1 1 2 0 1 0 ADC 7 = 7
8: 1 1 2 0 0 0 ADC 8 = 2^3
9: 1 1 3 0 0 1 ADC 9 = 3^2
10: 1 1 3 1 1 0 ADC 10 = 2 * 5
10: 1 2 2 2 1 1 ADC 10 = 2 * 5
11: 1 1 3 0 1 0 ADC 11 = 11
12: 1 1 3 0 0 0 ADC 12 = 2^2 * 3
12: 1 2 2 1 1 1 ADC 12 = 2^2 * 3
12: 1 2 2 2 0 0 ADC 12 = 2^2 * 3
13: 1 2 2 1 0 1 ADC 13 = 13
14: 1 1 5 1 1 1 ADC 14 = 2 * 7
15: 1 1 4 0 1 0 ADC 15 = 3 * 5
15: 1 2 2 1 0 0 ADC 15 = 3 * 5
16: 1 2 2 0 0 0 ADC 16 = 2^4
17: 1 2 3 2 1 1 ADC 17 = 17
18: 1 2 3 2 1 0 ADC 18 = 2 * 3^2
18: 2 2 2 1 2 2 ADC 18 = 2 * 3^2
20: 1 1 5 0 0 0 ADC 20 = 2^2 * 5
20: 1 2 3 1 0 1 ADC 20 = 2^2 * 5
20: 1 2 3 2 0 0 ADC 20 = 2^2 * 5
21: 1 2 3 0 0 1 ADC 21 = 3 * 7
21: 1 2 3 1 1 0 ADC 21 = 3 * 7
22: 1 2 3 0 1 0 ADC 22 = 2 * 11
24: 1 1 6 0 0 0 ADC 24 = 2^3 * 3
24: 1 2 3 0 0 0 ADC 24 = 2^3 * 3
24: 1 2 4 2 1 1 ADC 24 = 2^3 * 3
25: 2 2 2 -1 1 1 ADC 25 = 5^2
28: 2 2 3 2 2 2 ADC 28 = 2^2 * 7
30: 1 1 10 0 0 1 ADC 30 = 2 * 3 * 5
30: 1 3 3 1 1 1 ADC 30 = 2 * 3 * 5
32: 1 2 4 0 0 0 ADC 32 = 2^5
33: 1 2 5 1 1 1 ADC 33 = 3 * 11
36: 1 2 5 2 0 0 ADC 36 = 2^2 * 3^2
36: 1 3 3 0 0 0 ADC 36 = 2^2 * 3^2
36: 1 3 4 3 1 0 ADC 36 = 2^2 * 3^2
40: 1 2 5 0 0 0 ADC 40 = 2^3 * 5
42: 1 1 11 1 1 0 ADC 42 = 2 * 3 * 7
44: 1 2 6 2 0 0 ADC 44 = 2^2 * 11
45: 2 2 3 0 0 1 ADC 45 = 3^2 * 5
46: 1 3 5 3 1 1 ADC 46 = 2 * 23
48: 1 2 6 0 0 0 ADC 48 = 2^4 * 3
49: 1 2 7 0 0 1 ADC 49 = 7^2
50: 1 4 4 3 1 1 ADC 50 = 2 * 5^2
56: 1 3 5 2 0 0 ADC 56 = 2^3 * 7
60: 2 2 5 0 0 2 ADC 60 = 2^2 * 3 * 5
60: 2 3 3 0 0 2 ADC 60 = 2^2 * 3 * 5
63: 1 3 6 3 0 0 ADC 63 = 3^2 * 7
70: 1 2 9 0 1 0 ADC 70 = 2 * 5 * 7
72: 2 2 5 1 1 1 ADC 72 = 2^3 * 3^2
72: 2 3 3 0 0 0 ADC 72 = 2^3 * 3^2
75: 1 4 5 0 0 1 ADC 75 = 3 * 5^2
78: 1 5 5 4 1 1 ADC 78 = 2 * 3 * 13
84: 1 1 21 0 0 0 ADC 84 = 2^2 * 3 * 7
90: 1 1 30 0 0 1 ADC 90 = 2 * 3^2 * 5
92: 2 3 5 2 0 2 ADC 92 = 2^2 * 23
99: 2 3 5 3 1 0 ADC 99 = 3^2 * 11
100: 2 2 7 -1 1 1 ADC 100 = 2^2 * 5^2
100: 2 3 5 0 0 2 ADC 100 = 2^2 * 5^2
112: 2 3 5 2 0 0 ADC 112 = 2^4 * 7
120: 1 3 10 0 0 0 ADC 120 = 2^3 * 3 * 5
121: 1 3 11 0 0 1 ADC 121 = 11^2
126: 3 3 5 3 3 0 ADC 126 = 2 * 3^2 * 7
140: 1 2 18 2 0 0 ADC 140 = 2^2 * 5 * 7
147: 3 3 5 -2 2 1 ADC 147 = 3 * 7^2
150: 2 5 5 5 0 0 ADC 150 = 2 * 3 * 5^2
156: 2 3 7 0 2 0 ADC 156 = 2^2 * 3 * 13
169: 2 5 5 -3 1 1 ADC 169 = 13^2
180: 2 2 15 0 0 2 ADC 180 = 2^2 * 3^2 * 5
200: 1 5 10 0 0 0 ADC 200 = 2^3 * 5^2
234: 2 3 11 3 2 0 ADC 234 = 2 * 3^2 * 13
240: 2 5 6 0 0 0 ADC 240 = 2^4 * 3 * 5
252: 3 3 7 0 0 0 ADC 252 = 2^2 * 3^2 * 7
289: 3 5 6 1 2 3 ADC 289 = 17^2
294: 5 5 5 -3 3 4 ADC 294 = 2 * 3 * 7^2
300: 1 10 10 10 0 0 ADC 300 = 2^2 * 3 * 5^2
350: 3 3 10 0 0 1 ADC 350 = 2 * 5^2 * 7
360: 1 3 30 0 0 0 ADC 360 = 2^3 * 3^2 * 5
450: 5 5 6 0 0 5 ADC 450 = 2 * 3^2 * 5^2
468: 1 6 21 6 0 0 ADC 468 = 2^2 * 3^2 * 13
490: 3 3 14 0 0 1 ADC 490 = 2 * 5 * 7^2
588: 3 7 7 0 0 0 ADC 588 = 2^2 * 3 * 7^2
600: 2 5 15 0 0 0 ADC 600 = 2^3 * 3 * 5^2
700: 5 6 6 2 0 0 ADC 700 = 2^2 * 5^2 * 7
720: 2 6 15 0 0 0 ADC 720 = 2^4 * 3^2 * 5
882: 2 11 11 1 2 2 ADC 882 = 2 * 3^2 * 7^2
900: 3 10 10 10 0 0 ADC 900 = 2^2 * 3^2 * 5^2
980: 6 6 7 0 0 2 ADC 980 = 2^2 * 5 * 7^2
1014: 1 13 23 13 1 0 ADC 1014 = 2 * 3 * 13^2
1200: 1 10 30 0 0 0 ADC 1200 = 2^4 * 3 * 5^2
1764: 1 21 21 0 0 0 ADC 1764 = 2^2 * 3^2 * 7^2
1800: 5 6 15 0 0 0 ADC 1800 = 2^3 * 3^2 * 5^2
2028: 2 7 39 0 0 2 ADC 2028 = 2^2 * 3 * 13^2
2450: 1 9 70 0 0 1 ADC 2450 = 2 * 5^2 * 7^2
3042: 3 17 17 8 3 3 ADC 3042 = 2 * 3^2 * 13^2
3600: 3 10 30 0 0 0 ADC 3600 = 2^4 * 3^2 * 5^2
4900: 2 18 35 0 0 2 ADC 4900 = 2^2 * 5^2 * 7^2
6084: 6 13 21 0 6 0 ADC 6084 = 2^2 * 3^2 * 13^2
