<s>[INST] When the $a^2+b+c+d,b^2+a+c+d,c^2+a+b+d,d^2+a+b+c$ are all perfect squares? Find all $a,b,c,d\in \mathbb{Z}^+$, which $a^2+b+c+d,b^2+a+c+d,c^2+a+b+d,d^2+a+b+c$ are all perfect squares.
I found $(1,1,1,1)$, but I can't find more.
Is $a=b=c=d$ true?
 [/INST] Do brute force search. Here is a piece of Python code that should do the job
from math import sqrt, floor

_is_whole = lambda x: floor(x) == x
_is_ps = lambda n: _is_whole(sqrt(n))

min_int = 1
max_int = 150
for a in range(min_int, max_int + 1):
    for b in range(a, max_int + 1):
        for c in range(b, max_int + 1):
            for d in range(c, max_int + 1):
                for x in [a ** 2 + b + c + d, a + b ** 2 + c + d,
                          a + b + c ** 2 + d, a + b + c + d ** 2]:
                    if not _is_ps(x):
                        break
                else:
                    print a, b, c, d

When run, it should produce the following output:
1 1 1 1
1 5 5 5
1 8 8 8
1 16 16 16
1 21 21 21
1 33 33 33
1 40 40 40
1 56 56 56
1 65 65 65
1 85 85 85
1 96 96 96
1 120 120 120
1 133 133 133
6 6 11 11
40 57 96 96
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