我们可以直接验证, 2019是所有可以用6种方式分解为三个素数平方和的最小正整数. Mathematica代码如下:

Solve[x Log[x] == 2019, x];
n = Floor[x /. %] // First
primes = Reap[For[k = 1, k <= n, k++, p = Prime[k];
    If[p < Sqrt[2019], Sow[p]]]][[-1, 1]]
LenPrimes = Length[primes];

ClearAll[i, j, k]
For[y = 1, y <= 2019, y++, county = 0;
 For[i = 1, i <= LenPrimes, i++, pi = primes[[i]];
  For[j = i, j <= LenPrimes, j++, pj = primes[[j]];
   For[k = j, k <=  LenPrimes, k++, pk = primes[[k]];
    If[pi^2 + pj^2 + pk^2 == y, county++;
     Print[Superscript[pi, 2], "+", Superscript[pj, 2], "+", 
      Superscript[pk, 2], "=", pi^2 + pj^2 + pk^2]]
    ]
   ]
  ];
 If[county != 0, Print[county]]
 ]

利用输出知
7² + 11² + 43² = 2019
7² + 17² + 41² = 2019
11² + 23² + 37² = 2019
13² + 13² + 41² = 2019
17² + 19² + 37² = 2019
23² + 23² + 31² = 2019