which numbers in $m^2+4n^2$ form are prime

Question

For which natural numbers $m,n$

$m^2+4n^2$ is prime

I tried to check it by cases

case 1: if $m$ and $n$ are even then the expression will be divisible by two, thus it's not prime.

case 2: If $m$ and $n$ are perfect square then this expression isn't prime, since it can be factored.

any hint will be appreciated

Answer 1

Not a 'real' answer, but it was too big for a comment.

I wrote and ran some Mathematica code:

In[1]:=ParallelTable[
  If[PrimeQ[m^2 + 4*n^2], {m^2 + 4*n^2, m, n}, Nothing], {m, 0, 
   10^2}, {n, 0, 10^2}] //. {} -> Nothing

Running the code gives:

Out[1]={{{5, 1, 1}, {17, 1, 2}, {37, 1, 3}, {101, 1, 5}, {197, 1, 7}, {257, 
   1, 8}, {401, 1, 10}, {577, 1, 12}, {677, 1, 13}, {1297, 1, 
   18}, {1601, 1, 20}, {2917, 1, 27}, {3137, 1, 28}, {4357, 1, 
   33}, {5477, 1, 37}, {7057, 1, 42}, {8101, 1, 45}, {8837, 1, 
   47}, {12101, 1, 55}, {13457, 1, 58}, {14401, 1, 60}, {15377, 1, 
   62}, {15877, 1, 63}, {16901, 1, 65}, {17957, 1, 67}, {21317, 1, 
   73}, {22501, 1, 75}, {24337, 1, 78}, {25601, 1, 80}, {28901, 1, 
   85}, {30977, 1, 88}, {32401, 1, 90}, {33857, 1, 92}}, {{13, 3, 
   1}, {73, 3, 4}, {109, 3, 5}, {409, 3, 10}, {1033, 3, 16}, {1453, 3,
    19}, {1609, 3, 20}, {2713, 3, 26}, {3373, 3, 29}, {3853, 3, 
   31}, {4909, 3, 35}, {6733, 3, 41}, {7753, 3, 44}, {9613, 3, 
   49}, {10009, 3, 50}, {12109, 3, 55}, {12553, 3, 56}, {13933, 3, 
   59}, {19609, 3, 70}, {20173, 3, 71}, {25609, 3, 80}, {28909, 3, 
   85}, {35353, 3, 94}, {36109, 3, 95}, {40009, 3, 100}}, {{29, 5, 
   1}, {41, 5, 2}, {61, 5, 3}, {89, 5, 4}, {281, 5, 8}, {349, 5, 
   9}, {509, 5, 11}, {601, 5, 12}, {701, 5, 13}, {809, 5, 14}, {1049, 
   5, 16}, {1181, 5, 17}, {1321, 5, 18}, {1789, 5, 21}, {2141, 5, 
   23}, {2729, 5, 26}, {3389, 5, 29}, {4649, 5, 34}, {5209, 5, 
   36}, {5501, 5, 37}, {5801, 5, 38}, {8861, 5, 47}, {9241, 5, 
   48}, {9629, 5, 49}, {10429, 5, 51}, {11261, 5, 53}, {11689, 5, 
   54}, {12569, 5, 56}, {15401, 5, 62}, {15901, 5, 63}, {17449, 5, 
   66}, {17981, 5, 67}, {18521, 5, 68}, {19069, 5, 69}, {21341, 5, 
   73}, {21929, 5, 74}, {23741, 5, 77}, {24989, 5, 79}, {26921, 5, 
   82}, {27581, 5, 83}, {33149, 5, 91}, {39229, 5, 99}}, {{53, 7, 
   1}, {113, 7, 4}, {149, 7, 5}, {193, 7, 6}, {373, 7, 9}, {449, 7, 
   10}, {1493, 7, 19}, {2549, 7, 25}, {2753, 7, 26}, {3413, 7, 
   29}, {4673, 7, 34}, {5233, 7, 36}, {6133, 7, 39}, {6449, 7, 
   40}, {7793, 7, 44}, {8513, 7, 46}, {10453, 7, 51}, {12149, 7, 
   55}, {14449, 7, 60}, {16433, 7, 64}, {22549, 7, 75}, {25013, 7, 
   79}, {26293, 7, 81}, {28949, 7, 85}, {29633, 7, 86}, {35393, 7, 
   94}, {36913, 7, 96}}, {{97, 9, 2}, {181, 9, 5}, {277, 9, 7}, {337, 
   9, 8}, {757, 9, 13}, {1237, 9, 17}, {2017, 9, 22}, {3217, 9, 
   28}, {4177, 9, 32}, {5557, 9, 37}, {5857, 9, 38}, {6481, 9, 
   40}, {7477, 9, 43}, {11317, 9, 53}, {13537, 9, 58}, {16981, 9, 
   65}, {19681, 9, 70}, {21397, 9, 73}, {33937, 9, 92}, {37717, 9, 
   97}}, {{137, 11, 2}, {157, 11, 3}, {317, 11, 7}, {521, 11, 
   10}, {797, 11, 13}, {1021, 11, 15}, {1277, 11, 17}, {1721, 11, 
   20}, {2237, 11, 23}, {2621, 11, 25}, {3037, 11, 27}, {3257, 11, 
   28}, {4217, 11, 32}, {5021, 11, 35}, {5897, 11, 38}, {6521, 11, 
   40}, {7177, 11, 42}, {7517, 11, 43}, {8221, 11, 45}, {9337, 11, 
   48}, {10937, 11, 52}, {13577, 11, 58}, {15497, 11, 62}, {17021, 11,
    65}, {18077, 11, 67}, {18617, 11, 68}, {20857, 11, 72}, {22621, 
   11, 75}, {27017, 11, 82}, {29021, 11, 85}}, {{173, 13, 1}, {233, 
   13, 4}, {269, 13, 5}, {313, 13, 6}, {569, 13, 10}, {653, 13, 
   11}, {953, 13, 14}, {1069, 13, 15}, {1193, 13, 16}, {1613, 13, 
   19}, {1933, 13, 21}, {2473, 13, 24}, {3533, 13, 29}, {3769, 13, 
   30}, {4013, 13, 31}, {4793, 13, 34}, {6569, 13, 40}, {8269, 13, 
   45}, {10169, 13, 50}, {11833, 13, 54}, {12269, 13, 55}, {12713, 13,
    56}, {15053, 13, 61}, {16553, 13, 64}, {19213, 13, 69}, {20333, 
   13, 71}, {22073, 13, 74}, {22669, 13, 75}, {28393, 13, 84}, {29753,
    13, 86}, {32569, 13, 90}, {36269, 13, 95}, {39373, 13, 
   99}, {40169, 13, 100}}, {{229, 15, 1}, {241, 15, 2}, {421, 15, 
   7}, {709, 15, 11}, {1009, 15, 14}, {1249, 15, 16}, {1381, 15, 
   17}, {1669, 15, 19}, {2161, 15, 22}, {2341, 15, 23}, {3361, 15, 
   28}, {5701, 15, 37}, {6949, 15, 41}, {7621, 15, 43}, {8689, 15, 
   46}, {9829, 15, 49}, {13681, 15, 58}, {14149, 15, 59}, {15601, 15, 
   62}, {18181, 15, 67}, {20389, 15, 71}, {22129, 15, 74}, {25189, 15,
    79}, {33349, 15, 91}, {35569, 15, 94}, {37861, 15, 97}}, {{293, 
   17, 1}, {353, 17, 4}, {389, 17, 5}, {433, 17, 6}, {613, 17, 
   9}, {773, 17, 11}, {1733, 17, 19}, {1889, 17, 20}, {2053, 17, 
   21}, {2593, 17, 24}, {2789, 17, 25}, {3889, 17, 30}, {4133, 17, 
   31}, {5189, 17, 35}, {6373, 17, 39}, {6689, 17, 40}, {7013, 17, 
   41}, {8389, 17, 45}, {8753, 17, 46}, {10289, 17, 50}, {11953, 17, 
   54}, {15173, 17, 61}, {16673, 17, 64}, {17189, 17, 65}, {17713, 17,
    66}, {19333, 17, 69}, {19889, 17, 70}, {22193, 17, 74}, {25253, 
   17, 79}, {25889, 17, 80}, {28513, 17, 84}, {29873, 17, 86}, {31973,
    17, 89}, {33413, 17, 91}, {36389, 17, 95}, {40289, 17, 
   100}}, {{397, 19, 3}, {461, 19, 5}, {557, 19, 7}, {617, 19, 
   8}, {761, 19, 10}, {937, 19, 12}, {1657, 19, 18}, {2297, 19, 
   22}, {2477, 19, 23}, {2861, 19, 25}, {4457, 19, 32}, {5261, 19, 
   35}, {6761, 19, 40}, {7417, 19, 42}, {7757, 19, 43}, {8461, 19, 
   45}, {11177, 19, 52}, {11597, 19, 53}, {15737, 19, 62}, {19961, 19,
    70}, {22861, 19, 75}, {24077, 19, 77}, {24697, 19, 78}, {27917, 
   19, 83}, {30637, 19, 87}, {31337, 19, 88}, {34217, 19, 92}, {37997,
    19, 97}, {40361, 19, 100}}, {{457, 21, 2}, {541, 21, 5}, {1117, 
   21, 13}, {1597, 21, 17}, {2377, 21, 22}, {2557, 21, 23}, {6217, 21,
    38}, {6841, 21, 40}, {9277, 21, 47}, {11257, 21, 52}, {11677, 21, 
   53}, {12541, 21, 55}, {15817, 21, 62}, {17341, 21, 65}, {18397, 21,
    67}, {21757, 21, 73}, {26041, 21, 80}, {27337, 21, 82}, {27997, 
   21, 83}, {34297, 21, 92}, {36541, 21, 95}}, {{593, 23, 4}, {673, 
   23, 6}, {853, 23, 9}, {929, 23, 10}, {1013, 23, 11}, {1429, 23, 
   15}, {1553, 23, 16}, {1973, 23, 19}, {2129, 23, 20}, {2293, 23, 
   21}, {2833, 23, 24}, {4129, 23, 30}, {4373, 23, 31}, {5153, 23, 
   34}, {7253, 23, 41}, {8273, 23, 44}, {8629, 23, 45}, {10133, 23, 
   49}, {10529, 23, 50}, {14929, 23, 60}, {15413, 23, 61}, {20129, 23,
    70}, {20693, 23, 71}, {22433, 23, 74}, {23029, 23, 75}, {23633, 
   23, 76}, {28753, 23, 84}, {29429, 23, 85}, {30113, 23, 86}, {32213,
    23, 89}, {36629, 23, 95}, {39733, 23, 99}, {40529, 23, 
   100}}, {{641, 25, 2}, {661, 25, 3}, {769, 25, 6}, {821, 25, 
   7}, {881, 25, 8}, {1109, 25, 11}, {1201, 25, 12}, {1301, 25, 
   13}, {1409, 25, 14}, {2069, 25, 19}, {2389, 25, 21}, {2741, 25, 
   23}, {3329, 25, 26}, {3541, 25, 27}, {3761, 25, 28}, {3989, 25, 
   29}, {4721, 25, 32}, {6101, 25, 37}, {6709, 25, 39}, {7349, 25, 
   41}, {7681, 25, 42}, {8369, 25, 44}, {9461, 25, 47}, {12289, 25, 
   54}, {14081, 25, 58}, {14549, 25, 59}, {16001, 25, 62}, {18049, 25,
    66}, {19121, 25, 68}, {20789, 25, 71}, {25589, 25, 79}, {28181, 
   25, 83}, {31601, 25, 88}, {32309, 25, 89}, {33749, 25, 91}, {35221,
    25, 93}, {35969, 25, 94}, {37489, 25, 96}, {38261, 25, 
   97}, {39041, 25, 98}, {39829, 25, 99}}, {{733, 27, 1}, {829, 27, 
   5}, {1129, 27, 10}, {1213, 27, 11}, {1753, 27, 16}, {3229, 27, 
   25}, {3433, 27, 26}, {4093, 27, 29}, {7129, 27, 40}, {10333, 27, 
   49}, {10729, 27, 50}, {12829, 27, 55}, {14653, 27, 59}, {23833, 27,
    76}, {25693, 27, 79}, {29629, 27, 85}, {30313, 27, 86}, {32413, 
   27, 89}, {36073, 27, 94}}, {{857, 29, 2}, {877, 29, 3}, {941, 29, 
   5}, {1097, 29, 8}, {1741, 29, 15}, {1997, 29, 17}, {2137, 29, 
   18}, {2441, 29, 20}, {2777, 29, 22}, {2957, 29, 23}, {4441, 29, 
   30}, {4937, 29, 32}, {5197, 29, 33}, {5741, 29, 35}, {6317, 29, 
   37}, {8237, 29, 43}, {8941, 29, 45}, {9677, 29, 47}, {11657, 29, 
   52}, {12941, 29, 55}, {15241, 29, 60}, {16217, 29, 62}, {18797, 29,
    67}, {20441, 29, 70}, {21577, 29, 72}, {22157, 29, 73}, {27737, 
   29, 82}, {29741, 29, 85}, {31817, 29, 88}, {35437, 29, 93}, {40841,
    29, 100}}, {{977, 31, 2}, {997, 31, 3}, {1061, 31, 5}, {1217, 31, 
   8}, {1361, 31, 10}, {1637, 31, 13}, {1861, 31, 15}, {2897, 31, 
   22}, {3461, 31, 25}, {3877, 31, 27}, {4561, 31, 30}, {5861, 31, 
   35}, {6737, 31, 38}, {8017, 31, 42}, {10177, 31, 48}, {11777, 31, 
   52}, {12197, 31, 53}, {15361, 31, 60}, {18917, 31, 67}, {19457, 31,
    68}, {22277, 31, 73}, {24677, 31, 77}, {26561, 31, 80}, {28517, 
   31, 83}, {31237, 31, 87}, {37061, 31, 95}, {40961, 31, 
   100}}, {{1093, 33, 1}, {1153, 33, 4}, {1489, 33, 10}, {1873, 33, 
   14}, {2113, 33, 16}, {2689, 33, 20}, {3793, 33, 26}, {4933, 33, 
   31}, {7489, 33, 40}, {13633, 33, 56}, {15013, 33, 59}, {15973, 33, 
   61}, {17989, 33, 65}, {22993, 33, 74}, {26053, 33, 79}, {29989, 33,
    85}, {34213, 33, 91}, {36433, 33, 94}, {37189, 33, 95}}, {{1229, 
   35, 1}, {1289, 35, 4}, {1481, 35, 8}, {1549, 35, 9}, {1709, 35, 
   11}, {1801, 35, 12}, {1901, 35, 13}, {2381, 35, 17}, {2521, 35, 
   18}, {3529, 35, 24}, {3929, 35, 26}, {5581, 35, 33}, {5849, 35, 
   34}, {6701, 35, 37}, {7001, 35, 38}, {7309, 35, 39}, {7949, 35, 
   41}, {8969, 35, 44}, {9689, 35, 46}, {10061, 35, 47}, {12041, 35, 
   52}, {12889, 35, 54}, {14221, 35, 57}, {15149, 35, 59}, {17609, 35,
    64}, {19181, 35, 67}, {20269, 35, 69}, {21961, 35, 72}, {22541, 
   35, 73}, {24329, 35, 76}, {25561, 35, 78}, {26189, 35, 79}, {30809,
    35, 86}, {32909, 35, 89}, {35081, 35, 92}, {38861, 35, 
   97}, {40429, 35, 99}}, {{1373, 37, 1}, {1433, 37, 4}, {1693, 37, 
   9}, {2153, 37, 14}, {2269, 37, 15}, {2393, 37, 16}, {2969, 37, 
   20}, {3673, 37, 24}, {4073, 37, 26}, {4733, 37, 29}, {4969, 37, 
   30}, {6269, 37, 35}, {6553, 37, 36}, {8093, 37, 41}, {9833, 37, 
   46}, {10973, 37, 49}, {11369, 37, 50}, {13033, 37, 54}, {13469, 37,
    55}, {13913, 37, 56}, {16253, 37, 61}, {18269, 37, 65}, {18793, 
   37, 66}, {23869, 37, 75}, {24473, 37, 76}, {30269, 37, 85}, {33053,
    37, 89}, {33769, 37, 90}, {36713, 37, 94}}, {{1621, 39, 5}, {1777,
    39, 8}, {2677, 39, 17}, {3121, 39, 20}, {3457, 39, 22}, {3637, 39,
    23}, {4021, 39, 25}, {4657, 39, 28}, {6421, 39, 35}, {6997, 39, 
   37}, {7297, 39, 38}, {10357, 39, 47}, {12757, 39, 53}, {19477, 39, 
   67}, {21121, 39, 70}, {25237, 39, 77}, {29077, 39, 83}, {32497, 39,
    88}, {39157, 39, 97}, {39937, 39, 98}, {41521, 39, 100}}, {{1697, 
   41, 2}, {1877, 41, 7}, {2081, 41, 10}, {2357, 41, 13}, {2837, 41, 
   17}, {3617, 41, 22}, {3797, 41, 23}, {4597, 41, 27}, {4817, 41, 
   28}, {5281, 41, 30}, {6037, 41, 33}, {6581, 41, 35}, {7457, 41, 
   38}, {8081, 41, 40}, {8737, 41, 42}, {9781, 41, 45}, {11681, 41, 
   50}, {12497, 41, 52}, {12917, 41, 53}, {13781, 41, 55}, {15137, 41,
    58}, {20177, 41, 68}, {24181, 41, 75}, {26017, 41, 78}, {27281, 
   41, 80}, {31957, 41, 87}, {35537, 41, 92}, {36277, 41, 93}, {37781,
    41, 95}, {39317, 41, 97}, {41681, 41, 100}}, {{1913, 43, 
   4}, {1949, 43, 5}, {1993, 43, 6}, {2333, 43, 11}, {2633, 43, 
   14}, {2749, 43, 15}, {3449, 43, 20}, {3613, 43, 21}, {4153, 43, 
   24}, {4349, 43, 25}, {5449, 43, 30}, {5693, 43, 31}, {6473, 43, 
   34}, {7933, 43, 39}, {8573, 43, 41}, {9949, 43, 45}, {10313, 43, 
   46}, {12253, 43, 51}, {13513, 43, 54}, {15773, 43, 59}, {16249, 43,
    60}, {18233, 43, 64}, {18749, 43, 65}, {19273, 43, 66}, {22013, 
   43, 71}, {23753, 43, 74}, {24953, 43, 76}, {26813, 43, 79}, {27449,
    43, 80}, {33533, 43, 89}, {38713, 43, 96}, {41849, 43, 
   100}}, {{2029, 45, 1}, {2089, 45, 4}, {2221, 45, 7}, {2281, 45, 
   8}, {3049, 45, 16}, {3181, 45, 17}, {3469, 45, 19}, {4729, 45, 
   26}, {5869, 45, 31}, {6121, 45, 32}, {9421, 45, 43}, {9769, 45, 
   44}, {10861, 45, 47}, {12841, 45, 52}, {17401, 45, 62}, {20521, 45,
    68}, {22189, 45, 71}, {23929, 45, 74}, {25741, 45, 77}, {28921, 
   45, 82}, {29581, 45, 83}, {35149, 45, 91}, {37369, 45, 
   94}}, {{2213, 47, 1}, {2273, 47, 4}, {2309, 47, 5}, {2609, 47, 
   10}, {2693, 47, 11}, {3109, 47, 15}, {4513, 47, 24}, {5573, 47, 
   29}, {6053, 47, 31}, {6833, 47, 34}, {7109, 47, 35}, {7393, 47, 
   36}, {8293, 47, 39}, {8609, 47, 40}, {8933, 47, 41}, {11813, 47, 
   49}, {12613, 47, 51}, {13873, 47, 54}, {14753, 47, 56}, {17093, 47,
    61}, {18593, 47, 64}, {24113, 47, 74}, {24709, 47, 75}, {27809, 
   47, 80}, {31793, 47, 86}, {33893, 47, 89}, {41413, 47, 99}, {42209,
    47, 100}}, {{2417, 49, 2}, {2437, 49, 3}, {2657, 49, 8}, {2801, 
   49, 10}, {3301, 49, 15}, {3557, 49, 17}, {3697, 49, 18}, {4001, 49,
    20}, {4337, 49, 22}, {4517, 49, 23}, {7877, 49, 37}, {10501, 49, 
   45}, {11617, 49, 48}, {12401, 49, 50}, {13217, 49, 52}, {19301, 49,
    65}, {20357, 49, 67}, {20897, 49, 68}, {26737, 49, 78}, {28001, 
   49, 80}, {29297, 49, 82}, {33377, 49, 88}, {36997, 49, 93}, {38501,
    49, 95}, {40037, 49, 97}}, {{2617, 51, 2}, {2797, 51, 7}, {2857, 
   51, 8}, {3001, 51, 10}, {4201, 51, 20}, {5101, 51, 25}, {5737, 51, 
   28}, {8377, 51, 38}, {9001, 51, 40}, {11437, 51, 47}, {12601, 51, 
   50}, {13417, 51, 52}, {16057, 51, 58}, {17977, 51, 62}, {19501, 51,
    65}, {23917, 51, 73}, {26317, 51, 77}, {28201, 51, 80}, {33577, 
   51, 88}, {36457, 51, 92}, {40237, 51, 97}, {41017, 51, 
   98}}, {{2909, 53, 5}, {2953, 53, 6}, {3209, 53, 10}, {3593, 53, 
   14}, {3709, 53, 15}, {3833, 53, 16}, {4253, 53, 19}, {4409, 53, 
   20}, {5113, 53, 24}, {5309, 53, 25}, {6173, 53, 29}, {6653, 53, 
   31}, {7433, 53, 34}, {7993, 53, 36}, {8893, 53, 39}, {9209, 53, 
   40}, {9533, 53, 41}, {10909, 53, 45}, {11273, 53, 46}, {12413, 53, 
   49}, {12809, 53, 50}, {17209, 53, 60}, {19709, 53, 65}, {20233, 53,
    66}, {22409, 53, 70}, {22973, 53, 71}, {25309, 53, 75}, {25913, 
   53, 76}, {27773, 53, 79}, {28409, 53, 80}, {31033, 53, 84}, {35933,
    53, 91}, {38153, 53, 94}, {42013, 53, 99}}, {{3041, 55, 2}, {3061,
    55, 3}, {3089, 55, 4}, {3169, 55, 6}, {3221, 55, 7}, {3701, 55, 
   13}, {4049, 55, 16}, {4789, 55, 21}, {6389, 55, 29}, {6869, 55, 
   31}, {7121, 55, 32}, {7649, 55, 34}, {8209, 55, 36}, {8501, 55, 
   37}, {9109, 55, 39}, {9749, 55, 41}, {11489, 55, 46}, {12241, 55, 
   48}, {13841, 55, 52}, {15569, 55, 56}, {16481, 55, 58}, {17909, 55,
    61}, {18401, 55, 62}, {20981, 55, 67}, {21521, 55, 68}, {23189, 
   55, 71}, {23761, 55, 72}, {27361, 55, 78}, {29269, 55, 81}, {29921,
    55, 82}, {31249, 55, 84}, {32609, 55, 86}, {33301, 55, 
   87}}, {{3253, 57, 1}, {3313, 57, 4}, {3733, 57, 11}, {4273, 57, 
   16}, {5749, 57, 25}, {5953, 57, 26}, {7873, 57, 34}, {9649, 57, 
   40}, {9973, 57, 41}, {10993, 57, 44}, {12853, 57, 49}, {13249, 57, 
   50}, {15349, 57, 55}, {18133, 57, 61}, {20149, 57, 65}, {25153, 57,
    74}, {32833, 57, 86}, {36373, 57, 91}, {38593, 57, 94}}, {{3517, 
   59, 3}, {3581, 59, 5}, {3677, 59, 7}, {3881, 59, 10}, {4057, 59, 
   12}, {4157, 59, 13}, {4637, 59, 17}, {5081, 59, 20}, {5417, 59, 
   22}, {5981, 59, 25}, {6397, 59, 27}, {7577, 59, 32}, {9257, 59, 
   38}, {12697, 59, 48}, {14717, 59, 53}, {15581, 59, 55}, {16477, 59,
    57}, {16937, 59, 58}, {17881, 59, 60}, {21977, 59, 68}, {23081, 
   59, 70}, {25981, 59, 75}, {27197, 59, 77}, {27817, 59, 78}, {32381,
    59, 85}, {33757, 59, 87}, {34457, 59, 88}, {37337, 59, 
   92}, {39581, 59, 95}, {41117, 59, 97}, {41897, 59, 98}, {43481, 59,
    100}}, {{3821, 61, 5}, {3917, 61, 7}, {4297, 61, 12}, {4397, 61, 
   13}, {4621, 61, 15}, {4877, 61, 17}, {5657, 61, 22}, {6221, 61, 
   25}, {6637, 61, 27}, {6857, 61, 28}, {7321, 61, 30}, {7817, 61, 
   32}, {9497, 61, 38}, {11117, 61, 43}, {11821, 61, 45}, {13721, 61, 
   50}, {14537, 61, 52}, {14957, 61, 53}, {18121, 61, 60}, {19597, 61,
    63}, {23321, 61, 70}, {25037, 61, 73}, {27437, 61, 77}, {28057, 
   61, 78}, {31277, 61, 83}, {32621, 61, 85}, {33997, 61, 87}, {38317,
    61, 93}, {39821, 61, 95}, {41357, 61, 97}, {43721, 61, 
   100}}, {{4993, 63, 16}, {5413, 63, 19}, {5569, 63, 20}, {6469, 63, 
   25}, {6673, 63, 26}, {7333, 63, 29}, {10369, 63, 40}, {12433, 63, 
   46}, {16069, 63, 55}, {20353, 63, 64}, {24133, 63, 71}, {25873, 63,
    74}, {27073, 63, 76}, {28933, 63, 79}, {29569, 63, 80}, {32869, 
   63, 85}, {39313, 63, 94}, {43969, 63, 100}}, {{4229, 65, 1}, {4241,
    65, 2}, {4261, 65, 3}, {4289, 65, 4}, {4421, 65, 7}, {4481, 65, 
   8}, {4549, 65, 9}, {4801, 65, 12}, {5009, 65, 14}, {5381, 65, 
   17}, {5521, 65, 18}, {5669, 65, 19}, {6529, 65, 24}, {7589, 65, 
   29}, {8069, 65, 31}, {8581, 65, 33}, {8849, 65, 34}, {10949, 65, 
   41}, {11621, 65, 43}, {11969, 65, 44}, {12689, 65, 46}, {13441, 65,
    48}, {13829, 65, 49}, {14629, 65, 51}, {15461, 65, 53}, {15889, 
   65, 54}, {17681, 65, 58}, {18149, 65, 59}, {20101, 65, 63}, {21649,
    65, 66}, {22721, 65, 68}, {23269, 65, 69}, {25541, 65, 
   73}, {27329, 65, 76}, {27941, 65, 77}, {30469, 65, 81}, {31121, 65,
    82}, {33809, 65, 86}, {34501, 65, 87}, {35201, 65, 88}, {38821, 
   65, 93}, {39569, 65, 94}, {42641, 65, 98}}, {{4493, 67, 1}, {4813, 
   67, 9}, {4889, 67, 10}, {4973, 67, 11}, {5273, 67, 14}, {6089, 67, 
   20}, {6793, 67, 24}, {7193, 67, 26}, {7853, 67, 29}, {8089, 67, 
   30}, {10889, 67, 40}, {11213, 67, 41}, {12589, 67, 45}, {12953, 67,
    46}, {14489, 67, 50}, {17033, 67, 56}, {18413, 67, 59}, {19373, 
   67, 61}, {20873, 67, 64}, {26393, 67, 74}, {29453, 67, 79}, {30089,
    67, 80}, {32713, 67, 84}}, {{4861, 69, 5}, {4957, 69, 7}, {5437, 
   69, 13}, {6361, 69, 20}, {9661, 69, 35}, {11161, 69, 40}, {12157, 
   69, 43}, {13597, 69, 47}, {18217, 69, 58}, {21661, 69, 65}, {22717,
    69, 67}, {28477, 69, 77}, {31657, 69, 82}, {42397, 69, 
   97}, {43177, 69, 98}}, {{5077, 71, 3}, {5237, 71, 7}, {5297, 71, 
   8}, {5441, 71, 10}, {5717, 71, 13}, {6197, 71, 17}, {6337, 71, 
   18}, {6977, 71, 22}, {7541, 71, 25}, {8641, 71, 30}, {9137, 71, 
   32}, {9397, 71, 33}, {9941, 71, 35}, {12097, 71, 42}, {12437, 71, 
   43}, {13877, 71, 47}, {19441, 71, 60}, {23537, 71, 68}, {26357, 71,
    73}, {27541, 71, 75}, {33941, 71, 85}, {35317, 71, 87}, {36017, 
   71, 88}, {37441, 71, 90}, {41141, 71, 95}, {42677, 71, 97}, {43457,
    71, 98}}, {{5333, 73, 1}, {5393, 73, 4}, {5653, 73, 9}, {5813, 73,
    11}, {6113, 73, 14}, {6229, 73, 15}, {6353, 73, 16}, {7829, 73, 
   25}, {8693, 73, 29}, {8929, 73, 30}, {9173, 73, 31}, {10513, 73, 
   36}, {15329, 73, 50}, {15733, 73, 51}, {16993, 73, 54}, {21713, 73,
    64}, {22229, 73, 65}, {24373, 73, 69}, {28433, 73, 76}, {30293, 
   73, 79}, {31573, 73, 81}, {34913, 73, 86}, {37013, 73, 89}, {38453,
    73, 91}, {42193, 73, 96}, {44533, 73, 99}, {45329, 73, 
   100}}, {{5641, 75, 2}, {5689, 75, 4}, {5821, 75, 7}, {5881, 75, 
   8}, {6301, 75, 13}, {6781, 75, 17}, {7069, 75, 19}, {7561, 75, 
   22}, {7741, 75, 23}, {8329, 75, 26}, {8761, 75, 28}, {9721, 75, 
   32}, {14461, 75, 47}, {18169, 75, 56}, {19081, 75, 58}, {20509, 75,
    61}, {21001, 75, 62}, {23581, 75, 67}, {24121, 75, 68}, {27529, 
   75, 74}, {28729, 75, 76}, {33181, 75, 83}, {37309, 75, 89}, {38749,
    75, 91}, {43261, 75, 97}, {44041, 75, 98}}, {{6029, 77, 5}, {6073,
    77, 6}, {6329, 77, 10}, {6829, 77, 15}, {7529, 77, 20}, {8233, 77,
    24}, {8429, 77, 25}, {9293, 77, 29}, {11113, 77, 36}, {12329, 77, 
   40}, {12653, 77, 41}, {14029, 77, 45}, {16333, 77, 51}, {19853, 77,
    59}, {28429, 77, 75}, {29033, 77, 76}, {30893, 77, 79}, {32173, 
   77, 81}, {38329, 77, 90}, {42793, 77, 96}}, {{6257, 79, 2}, {6277, 
   79, 3}, {6917, 79, 13}, {7537, 79, 18}, {7841, 79, 20}, {8741, 79, 
   25}, {9157, 79, 27}, {9377, 79, 28}, {10337, 79, 32}, {10597, 79, 
   33}, {11717, 79, 37}, {12641, 79, 40}, {13297, 79, 42}, {14341, 79,
    45}, {15077, 79, 47}, {17477, 79, 53}, {18341, 79, 55}, {19237, 
   79, 57}, {19697, 79, 58}, {20641, 79, 60}, {21617, 79, 62}, {24197,
    79, 67}, {25841, 79, 70}, {30577, 79, 78}, {33797, 79, 
   83}, {35141, 79, 85}, {37217, 79, 88}, {44657, 79, 98}}, {{6577, 
   81, 2}, {6661, 81, 5}, {6961, 81, 10}, {7237, 81, 13}, {7717, 81, 
   17}, {8161, 81, 20}, {8677, 81, 23}, {9697, 81, 28}, {10657, 81, 
   32}, {12037, 81, 37}, {16561, 81, 50}, {17377, 81, 52}, {18661, 81,
    55}, {21937, 81, 62}, {24517, 81, 67}, {25057, 81, 68}, {26161, 
   81, 70}, {33457, 81, 82}, {35461, 81, 85}, {37537, 81, 
   88}}, {{7213, 83, 9}, {7673, 83, 14}, {7789, 83, 15}, {10253, 83, 
   29}, {10733, 83, 31}, {11789, 83, 35}, {12073, 83, 36}, {12973, 83,
    39}, {13613, 83, 41}, {14633, 83, 44}, {16493, 83, 49}, {16889, 
   83, 50}, {17293, 83, 51}, {18553, 83, 54}, {19433, 83, 56}, {21773,
    83, 61}, {23789, 83, 65}, {25933, 83, 69}, {26489, 83, 
   70}, {28793, 83, 74}, {29389, 83, 75}, {36473, 83, 86}, {40013, 83,
    91}, {42989, 83, 95}, {43753, 83, 96}, {46093, 83, 99}, {46889, 
   83, 100}}, {{7229, 85, 1}, {7369, 85, 6}, {7481, 85, 8}, {7549, 85,
    9}, {7901, 85, 13}, {8009, 85, 14}, {8521, 85, 18}, {8669, 85, 
   19}, {9161, 85, 22}, {9341, 85, 23}, {9929, 85, 26}, {10141, 85, 
   27}, {10589, 85, 29}, {11069, 85, 31}, {11321, 85, 32}, {12409, 85,
    36}, {13001, 85, 38}, {13309, 85, 39}, {14281, 85, 42}, {14621, 
   85, 43}, {14969, 85, 44}, {16061, 85, 47}, {16829, 85, 49}, {18041,
    85, 52}, {18461, 85, 53}, {20681, 85, 58}, {21149, 85, 
   59}, {22109, 85, 61}, {23609, 85, 64}, {27961, 85, 72}, {28541, 85,
    73}, {29129, 85, 74}, {30941, 85, 77}, {32189, 85, 79}, {33469, 
   85, 81}, {34781, 85, 83}, {35449, 85, 84}, {36809, 85, 86}, {37501,
    85, 87}, {38201, 85, 88}, {41081, 85, 92}, {42569, 85, 
   94}, {44089, 85, 96}, {45641, 85, 98}}, {{7573, 87, 1}, {7669, 87, 
   5}, {8053, 87, 11}, {8353, 87, 14}, {9013, 87, 19}, {10069, 87, 
   25}, {10273, 87, 26}, {14293, 87, 41}, {15313, 87, 44}, {16033, 87,
    46}, {17569, 87, 50}, {20113, 87, 56}, {21493, 87, 59}, {22453, 
   87, 61}, {24469, 87, 65}, {27733, 87, 71}, {29473, 87, 74}, {32533,
    87, 79}, {36469, 87, 85}, {40693, 87, 91}, {43669, 87, 
   95}, {47569, 87, 100}}, {{7937, 89, 2}, {8117, 89, 7}, {8597, 89, 
   13}, {8821, 89, 15}, {9521, 89, 20}, {9857, 89, 22}, {10037, 89, 
   23}, {10837, 89, 27}, {11057, 89, 28}, {12277, 89, 33}, {12821, 89,
    35}, {13397, 89, 37}, {13697, 89, 38}, {14321, 89, 40}, {17137, 
   89, 48}, {17921, 89, 50}, {19157, 89, 53}, {20021, 89, 55}, {21377,
    89, 58}, {23297, 89, 62}, {24821, 89, 65}, {26417, 89, 
   68}, {28657, 89, 72}, {32257, 89, 78}, {33521, 89, 80}, {36821, 89,
    85}, {38197, 89, 87}, {41777, 89, 92}, {44021, 89, 95}, {45557, 
   89, 97}, {46337, 89, 98}}, {{8297, 91, 2}, {8317, 91, 3}, {8537, 
   91, 8}, {8681, 91, 10}, {9181, 91, 15}, {9437, 91, 17}, {10781, 91,
    25}, {11197, 91, 27}, {12377, 91, 32}, {12637, 91, 33}, {13757, 
   91, 37}, {14057, 91, 38}, {16381, 91, 45}, {17117, 91, 47}, {17497,
    91, 48}, {21277, 91, 57}, {21737, 91, 58}, {26237, 91, 
   67}, {26777, 91, 68}, {29017, 91, 72}, {30781, 91, 75}, {35837, 91,
    83}, {37181, 91, 85}, {38557, 91, 87}, {44381, 91, 95}, {48281, 
   91, 100}}, {{8713, 93, 4}, {9049, 93, 10}, {9133, 93, 11}, {9433, 
   93, 14}, {10093, 93, 19}, {11149, 93, 25}, {11353, 93, 26}, {15373,
    93, 41}, {18253, 93, 49}, {20749, 93, 55}, {21193, 93, 
   56}, {22573, 93, 59}, {25033, 93, 64}, {28813, 93, 71}, {30553, 93,
    74}, {33613, 93, 79}, {37549, 93, 85}, {48649, 93, 100}}, {{9029, 
   95, 1}, {9041, 95, 2}, {9221, 95, 7}, {9281, 95, 8}, {9349, 95, 
   9}, {9601, 95, 12}, {10181, 95, 17}, {10321, 95, 18}, {10789, 95, 
   21}, {11329, 95, 24}, {11941, 95, 27}, {12161, 95, 28}, {13121, 95,
    32}, {13381, 95, 33}, {13649, 95, 34}, {15749, 95, 41}, {16421, 
   95, 43}, {17489, 95, 46}, {19429, 95, 51}, {19841, 95, 52}, {20261,
    95, 53}, {21569, 95, 56}, {22481, 95, 58}, {23909, 95, 
   61}, {25409, 95, 64}, {26449, 95, 66}, {26981, 95, 67}, {28069, 95,
    69}, {29761, 95, 72}, {30341, 95, 73}, {38609, 95, 86}, {39301, 
   95, 87}, {40709, 95, 89}, {47441, 95, 98}}, {{9413, 97, 1}, {9473, 
   97, 4}, {9733, 97, 9}, {10193, 97, 14}, {10433, 97, 16}, {10853, 
   97, 19}, {11173, 97, 21}, {11909, 97, 25}, {12113, 97, 26}, {13009,
    97, 30}, {14033, 97, 34}, {14593, 97, 36}, {15493, 97, 
   39}, {15809, 97, 40}, {17509, 97, 45}, {19013, 97, 49}, {19813, 97,
    51}, {23333, 97, 59}, {25793, 97, 64}, {26309, 97, 65}, {26833, 
   97, 66}, {29009, 97, 70}, {29573, 97, 71}, {37633, 97, 84}, {38993,
    97, 86}, {41809, 97, 90}, {42533, 97, 91}, {44753, 97, 
   94}, {46273, 97, 96}, {49409, 97, 100}}, {{9817, 99, 2}, {9901, 99,
    5}, {10477, 99, 13}, {10957, 99, 17}, {12301, 99, 25}, {15277, 99,
    37}, {18637, 99, 47}, {19801, 99, 50}, {26701, 99, 65}, {28297, 
   99, 68}, {29401, 99, 70}, {35401, 99, 80}, {36697, 99, 82}, {37357,
    99, 83}, {49801, 99, 100}}}

So, we can see that when we have a lot of solutions given in the form $\left(\text{m}^2+4\text{n}^2,\text{m},\text{n}\right)$ in between the following boundaries $0\le\text{m}\le10^2$ and $0\le\text{n}\le10^2$.