## To make {(x+y+z+u),(x+y+z-u),(x+y-z+u),(x-y+z+u),(-x+y+z+u)} squares

Find four positive integers   $x, y, z, u$   such that

$x \; + \; y \; + \; z \; + \; u \; = \; a^2$
$x \; + \; y \; + \; z \; - \; u \; = \; b^2$
$x \; + \; y \; - \; z \; + \; u \; = \; c^2$
$x \; - \; y \; + \; z \; + \; u \; = \; d^2$
$-x \; + \; y \; + \; z \; + \; u \; = \; e^2$

Hence,
$b^2 + c^2 + d^2 + e^2 \; = \; 2 x + 2 y + 2 z + 2 u$
$b^2 + c^2 + d^2 + e^2 = 2 a^2$

math grad - Interest: Number theory
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### 5 Responses to To make {(x+y+z+u),(x+y+z-u),(x+y-z+u),(x-y+z+u),(-x+y+z+u)} squares

1. paul says:

Here are a few.

192 + 170 + 80 + 42 = 22^2
192 + 170 + 80 – 42 = 20^2
192 + 170 – 80 + 42 = 18^2
192 – 170 + 80 – 42 = 12^2
– 192 + 170 + 80 + 42 = 10^2

320 + 210 + 96 + 50 = 26^2
320 + 210 + 96 – 50 = 24^2
320 + 210 – 96 + 50 = 22^2
320 – 210 + 96 – 50 = 16^2
– 320 + 210 + 96 + 50 = 6^2

266 + 176 + 138 + 96 = 26^2
266 + 176 + 138 – 96 = 22^2
266 + 176 – 138 + 96 = 20^2
266 – 176 + 138 – 96 = 18^2
– 266 + 176 + 138 + 96 = 12^2

240 + 210 + 176 + 50 = 26^2
240 + 210 + 176 – 50 = 24^2
240 + 210 – 176 + 50 = 18^2
240 – 210 + 176 – 50 = 16^2
– 240 + 210 + 176 + 50 = 14^2

442 + 288 + 112 + 58 = 30^2
442 + 288 + 112 – 58 = 28^2
442 + 288 – 112 + 58 = 26^2
442 – 288 + 112 – 58 = 18^2
– 442 + 288 + 112 + 58 = 4^2

418 + 208 + 162 + 112 = 30^2
418 + 208 + 162 – 112 = 26^2
418 + 208 – 162 + 112 = 24^2
418 – 208 + 162 – 112 = 22^2
– 418 + 208 + 162 + 112 = 8^2

378 + 352 + 112 + 58 = 30^2
378 + 352 + 112 – 58 = 28^2
378 + 352 – 112 + 58 = 26^2
378 – 352 + 112 – 58 = 14^2
– 378 + 352 + 112 + 58 = 12^2

560 + 290 + 240 + 66 = 34^2
560 + 290 + 240 – 66 = 32^2
560 + 290 – 240 + 66 = 26^2
560 – 290 + 240 – 66 = 24^2
– 560 + 290 + 240 + 66 = 6^2

546 + 416 + 128 + 66 = 34^2
546 + 416 + 128 – 66 = 32^2
546 + 416 – 128 + 66 = 30^2
546 – 416 + 128 – 66 = 18^2
– 546 + 416 + 128 + 66 = 8^2

506 + 336 + 186 + 128 = 34^2
506 + 336 + 186 – 128 = 30^2
506 + 336 – 186 + 128 = 28^2
506 – 336 + 186 – 128 = 22^2
– 506 + 336 + 186 + 128 = 12^2

Paul

• paul says:

Here is a larger list, the format is {x, y, z, u, sqrt1, sqrt2, sqrt3, sqrt4, sqrt5}.

{192,170,80,42,10,12,18,20,22}
{240,210,176,50,14,16,18,24,26}
{266,176,138,96,12,18,20,22,26}
{320,210,96,50,6,16,22,24,26}
{378,352,112,58,12,14,26,28,30}
{418,208,162,112,8,22,24,26,30}
{442,288,112,58,4,18,26,28,30}
{506,336,186,128,12,22,28,30,34}
{546,416,128,66,8,18,30,32,34}
{560,290,240,66,6,24,26,32,34}
{560,480,330,74,18,22,28,36,38}
{594,434,272,144,16,24,30,34,38}
{704,522,144,74,6,20,34,36,38}
{714,384,272,74,4,26,30,36,38}
{640,594,370,160,22,24,32,38,42}
{682,490,432,160,20,28,30,38,42}
{682,544,304,234,20,26,34,36,42}
{784,594,304,82,14,24,34,40,42}
{810,490,304,160,12,28,34,38,42}
{880,432,370,82,2,30,32,40,42}
{880,490,234,160,2,28,36,38,42}
{768,680,320,168,20,24,36,40,44}
{770,608,480,258,24,30,34,40,46}
{896,720,410,90,18,26,36,44,46}
{1008,608,410,90,10,30,36,44,46}
{1050,480,410,176,4,34,36,42,46}
{858,672,602,368,28,34,36,42,50}
{962,912,528,98,24,26,38,48,50}
{1050,800,368,282,20,30,42,44,50}
{1088,602,528,282,18,36,38,44,50}
{1122,912,368,98,16,26,42,48,50}
{1122,1088,192,98,16,18,46,48,50}
{1152,800,450,98,14,30,40,48,50}
{1178,602,528,192,12,36,38,46,50}
{1178,672,368,282,12,34,42,44,50}
{1248,602,368,282,2,36,42,44,50}
{1248,962,192,98,2,24,46,48,50}
{960,840,704,200,28,32,36,48,52}
{1064,704,552,384,24,36,40,44,52}
{1280,840,384,200,12,32,44,48,52}
{880,810,736,490,34,36,38,44,54}
{946,736,658,576,32,38,40,42,54}
{1066,1008,736,106,28,30,38,52,54}
{1120,880,810,106,26,34,36,52,54}
{1170,880,658,208,24,34,40,50,54}
{1216,736,658,306,22,38,40,48,54}
{1216,810,490,400,22,36,44,46,54}
{1330,880,400,306,16,34,46,48,54}
{1386,1216,208,106,12,22,50,52,54}
{1408,810,490,208,10,36,44,50,54}
{1440,880,490,106,6,34,44,52,54}
{1450,576,490,400,4,42,44,46,54}
{1456,946,306,208,2,32,48,50,54}
{1232,1170,530,432,30,32,48,50,58}
{1394,1232,624,114,24,30,46,56,58}
{1440,1170,530,224,22,32,48,54,58}
{1482,1034,624,224,20,36,46,54,58}
{1520,800,714,330,18,42,44,52,58}
{1520,882,530,432,18,40,48,50,58}
{1520,1290,330,224,18,28,52,54,58}
{1610,800,624,330,12,42,46,52,58}
{1632,1394,224,114,10,24,54,56,58}
{1650,960,530,224,8,38,48,54,58}
{1674,1034,432,224,4,36,50,54,58}
{1512,1408,448,232,24,28,52,56,60}
{1672,832,648,448,16,44,48,52,60}
{1768,1152,448,232,8,36,52,56,60}
{1410,1200,770,464,32,38,48,54,62}
{1584,1274,864,122,26,36,46,60,62}
{1634,1200,770,240,24,38,48,58,62}
{1680,1040,770,354,22,42,48,56,62}
{1680,1472,570,122,22,30,52,60,62}
{1760,1274,570,240,18,36,52,58,62}
{1760,1722,240,122,18,20,58,60,62}
{1794,1040,770,240,16,42,48,58,62}
{1850,1200,672,122,12,38,50,60,62}
{1904,954,864,122,6,44,46,60,62}
{1904,1122,464,354,6,40,54,56,62}
{1914,1344,464,122,4,34,54,60,62}
{1456,1378,1026,496,38,40,48,58,66}
{1530,1210,1120,496,36,44,46,58,66}
{1600,1120,1026,610,34,46,48,56,66}
{1600,1210,826,720,34,44,52,54,66}
{1728,1378,1120,130,30,40,46,64,66}
{1728,1530,720,378,30,36,54,60,66}
{1840,1210,928,378,26,44,50,60,66}
{1840,1666,720,130,26,32,54,64,66}
{1890,1600,610,256,24,34,56,62,66}
{1890,1840,496,130,24,26,58,64,66}
{1936,1786,378,256,22,28,60,62,66}
{1978,1296,826,256,20,42,52,62,66}
{2016,1600,610,130,18,34,56,64,66}
{2050,1456,720,130,16,38,54,64,66}
{2080,1120,1026,130,14,46,48,64,66}
{2080,1890,256,130,14,24,62,64,66}
{2106,928,826,496,12,50,52,58,66}
{2128,1378,720,130,10,40,54,64,66}
{2146,1026,928,256,8,48,50,62,66}
{2160,1456,610,130,6,38,56,64,66}
{2170,1210,720,256,4,44,54,62,66}
{2024,1344,744,512,24,44,56,60,68}
{2184,1664,512,264,16,36,60,64,68}
{2240,1160,960,264,12,48,52,64,68}
{1568,1482,1200,650,42,44,50,60,70}
{1728,1298,992,882,38,48,54,56,70}
{1802,1568,1392,138,36,42,46,68,70}
{1802,1728,1098,272,36,38,52,66,70}
{1938,1392,1298,272,32,46,48,66,70}
{1938,1568,992,402,32,42,54,64,70}
{2000,1298,1200,402,30,48,50,64,70}
{2000,1482,768,650,30,44,58,60,70}
{2058,1200,992,650,28,50,54,60,70}
{2058,1802,768,272,28,36,58,66,70}
{2112,2000,650,138,26,30,60,68,70}
{2162,1568,768,402,24,42,58,64,70}
{2162,1938,528,272,24,32,62,66,70}
{2208,1298,992,402,22,48,54,64,70}
{2250,1728,650,272,20,38,60,66,70}
{2288,1482,992,138,18,44,54,68,70}
{2288,1938,402,272,18,32,64,66,70}
{2378,1392,992,138,12,46,54,68,70}
{2378,1482,768,272,12,44,58,66,70}
{2378,2112,272,138,12,26,66,68,70}
{2432,1298,768,402,6,48,58,64,70}
{2432,1802,528,138,6,36,62,68,70}
{2432,2058,272,138,6,28,66,68,70}
{2448,1298,882,272,2,48,56,66,70}
{1770,1488,1280,938,44,50,54,60,74}
{2090,1680,1280,426,36,46,54,68,74}
{2090,1770,1056,560,36,44,58,66,74}
{2160,1586,1170,560,34,48,56,66,74}
{2160,2090,938,288,34,36,60,70,74}
{2288,1938,690,560,30,40,64,66,74}
{2288,2226,816,146,30,32,62,72,74}
{2400,2090,560,426,26,36,66,68,74}
{2450,1280,1056,690,24,54,58,64,74}
{2538,2090,560,288,20,36,66,70,74}
{2576,1938,816,146,18,40,62,72,74}
{2610,2160,560,146,16,34,66,72,74}
{2640,1586,690,560,14,48,64,66,74}
{2666,1056,938,816,12,58,60,62,74}
{2688,1586,1056,146,10,48,58,72,74}
{2720,1770,560,426,6,44,66,68,74}
{2240,1920,1320,296,36,44,56,72,76}
{2376,1736,1088,576,32,48,60,68,76}
{2816,2088,576,296,12,40,68,72,76}
{2856,1536,1088,296,8,52,60,72,76}
{1792,1690,1360,1242,50,52,58,60,78}
{2160,1890,1584,450,42,48,54,72,78}
{2160,2074,1120,730,42,44,62,68,78}
{2242,1890,1360,592,40,48,58,70,78}
{2242,1984,994,864,40,46,64,66,78}
{2320,1792,1242,730,38,50,60,68,78}
{2394,1584,1242,864,36,54,60,66,78}
{2464,2074,1242,304,34,44,60,74,78}
{2530,1890,1360,304,32,48,58,74,78}
{2530,1984,1120,450,32,46,62,72,78}
{2650,1584,1120,730,28,54,62,68,78}
{2650,2160,1120,154,28,42,62,76,78}
{2704,1984,1242,154,26,46,60,76,78}
{2800,1474,1360,450,22,56,58,72,78}
{2800,1690,864,730,22,52,66,68,78}
{2800,2242,592,450,22,40,70,72,78}
{2800,2530,450,304,22,32,72,74,78}
{2842,2074,864,304,20,44,66,74,78}
{2880,1690,1360,154,18,52,58,76,78}
{2880,1890,864,450,18,48,66,72,78}
{2880,2320,730,154,18,38,68,76,78}
{2914,1584,994,592,16,54,64,70,78}
{2944,2394,592,154,14,36,70,76,78}
{2970,1690,1120,304,12,52,62,74,78}
{2970,1792,730,592,12,50,68,70,78}
{2992,1242,1120,730,10,60,62,68,78}
{2992,2074,864,154,10,44,66,76,78}
{3010,2320,450,304,8,38,72,74,78}
{3024,1474,994,592,6,56,64,70,78}
{3034,2592,304,154,4,30,74,76,78}
{3040,1474,1120,450,2,56,62,72,78}
{3040,2160,730,154,2,42,68,76,78}
{2304,2210,1440,770,46,48,62,72,82}
{2480,1794,1680,770,42,56,58,72,82}
{2480,2010,1184,1050,42,52,66,68,82}
{2562,2480,912,770,40,42,70,72,82}
{2714,2010,1680,320,36,52,58,78,82}
{2714,2640,1050,320,36,38,68,78,82}
{2784,1904,1562,474,34,54,60,76,82}
{2850,2480,770,624,32,42,72,74,82}
{2850,2784,770,320,32,34,72,78,82}
{2912,2210,1440,162,30,48,62,80,82}
{2912,2714,624,474,30,36,74,76,82}
{3074,2304,1184,162,24,46,66,80,82}
{3120,2210,770,624,22,48,72,74,82}
{3162,1680,1562,320,20,58,60,78,82}
{3162,1904,1184,474,20,54,66,76,82}
{3200,1440,1314,770,18,62,64,72,82}
{3200,1562,1050,912,18,60,68,70,82}
{3290,2640,474,320,12,38,76,78,82}
{3312,2480,770,162,10,42,72,80,82}
{3330,1440,1184,770,8,62,66,72,82}
{3330,2304,770,320,8,46,72,78,82}
{3330,2912,320,162,8,30,78,80,82}
{3344,1904,1314,162,6,54,64,80,82}
{3344,2010,1050,320,6,52,68,78,82}
{3354,1562,1184,624,4,60,66,74,82}
{2560,2376,1480,640,44,48,64,76,84}
{2728,1960,1728,640,40,56,60,76,84}
{2728,2176,1216,936,40,52,68,72,84}
{3136,2376,1216,328,28,48,68,80,84}
{3240,1960,1216,640,24,56,68,76,84}
{3520,1728,1480,328,4,60,64,80,84}
{3520,1960,936,640,4,56,72,76,84}
{2448,2240,1898,810,50,54,60,76,86}
{2546,2240,1650,960,48,54,64,74,86}
{2640,2130,1520,1106,46,56,66,72,86}
{2640,2346,2240,170,46,52,54,84,86}
{2730,1898,1520,1248,44,60,66,70,86}
{2816,2346,1898,336,42,52,60,82,86}
{2976,2730,1520,170,38,44,66,84,86}
{2976,2816,1106,498,38,42,72,80,86}
{3050,2016,1520,810,36,58,66,76,86}
{3050,2730,960,656,36,44,74,78,86}
{3120,1650,1520,1106,34,64,66,72,86}
{3186,2448,1106,656,32,50,72,78,86}
{3248,2130,1520,498,30,56,66,80,86}
{3248,2730,1248,170,30,44,70,84,86}
{3360,2346,1520,170,26,52,66,84,86}
{3410,2130,1520,336,24,56,66,82,86}
{3410,2240,1248,498,24,54,70,80,86}
{3456,1898,1386,656,22,60,68,78,86}
{3536,2730,960,170,18,44,74,84,86}
{3570,1650,1520,656,16,64,66,78,86}
{3600,2240,1386,170,14,54,68,84,86}
{3600,2816,810,170,14,42,76,84,86}
{3626,2640,960,170,12,46,74,84,86}
{3680,1520,1386,810,6,66,68,76,86}
{3680,1650,1106,960,6,64,72,74,86}
{3690,2016,1520,170,4,58,66,84,86}
{3690,2240,810,656,4,54,76,78,86}
{3696,2546,656,498,2,48,78,80,86}
{3072,2720,1280,672,40,48,72,80,88}
{2698,2368,1872,1162,52,58,66,76,90}
{2800,2250,1738,1312,50,60,68,74,90}
{2800,2698,2250,352,50,52,60,86,90}
{2898,2482,2368,352,48,56,58,86,90}
{2992,2800,1458,850,46,50,72,80,90}
{3082,2368,2128,522,44,58,62,84,90}
{3082,2592,1738,688,44,54,68,82,90}
{3082,2698,1312,1008,44,52,74,78,90}
{3168,2482,1600,850,42,56,70,80,90}
{3168,3082,1162,688,42,44,76,82,90}
{3250,2128,1872,850,40,62,66,80,90}
{3250,2800,1872,178,40,50,66,88,90}
{3250,2898,1600,352,40,48,70,86,90}
{3250,2992,1008,850,40,46,78,80,90}
{3328,1872,1738,1162,38,66,68,76,90}
{3328,2002,1458,1312,38,64,72,74,90}
{3328,2592,2002,178,38,54,64,88,90}
{3402,2698,1312,688,36,52,74,82,90}
{3402,3168,1008,522,36,42,78,84,90}
{3472,2368,1738,522,34,58,68,84,90}
{3472,2482,1458,688,34,56,72,82,90}
{3472,2992,1458,178,34,46,72,88,90}
{3538,2002,1872,688,32,64,66,82,90}
{3538,2898,1312,352,32,48,74,86,90}
{3600,1738,1600,1162,30,68,70,76,90}
{3600,3472,850,178,30,34,80,88,90}
{3658,2592,1162,688,28,54,76,82,90}
{3658,3082,1008,352,28,44,78,86,90}
{3658,3402,688,352,28,36,82,86,90}
{3712,2128,1738,522,26,62,68,84,90}
{3712,2898,1312,178,26,48,74,88,90}
{3762,1872,1458,1008,24,66,72,78,90}
{3762,2800,850,688,24,50,80,82,90}
{3762,3472,688,178,24,34,82,88,90}
{3808,2482,1458,352,22,56,72,86,90}
{3808,3082,688,522,22,44,82,84,90}
{3808,3762,352,178,22,24,86,88,90}
{3850,2128,1600,522,20,62,70,84,90}
{3850,2250,1312,688,20,60,74,82,90}
{3888,1738,1312,1162,18,68,74,76,90}
{3888,2698,1162,352,18,52,76,86,90}
{3922,2128,1872,178,16,62,66,88,90}
{3922,2368,1458,352,16,58,72,86,90}
{3922,2482,1008,688,16,56,78,82,90}
{3922,2992,1008,178,16,46,78,88,90}
{3952,2002,1458,688,14,64,72,82,90}
{3978,2592,1008,522,12,54,78,84,90}
{3978,3082,688,352,12,44,82,86,90}
{4000,2250,1162,688,10,60,76,82,90}
{4000,2898,850,352,10,48,80,86,90}
{4018,2592,1312,178,8,54,74,88,90}
{4032,3538,352,178,6,32,86,88,90}
{4042,1738,1312,1008,4,68,74,78,90}
{4042,2698,1008,352,4,52,78,86,90}
{4048,2002,1872,178,2,64,66,88,90}
{4048,2368,1162,522,2,58,76,84,90}
{3080,2432,1920,1032,48,60,68,80,92}
{3584,2880,1640,360,36,52,72,88,92}
{4032,2432,1640,360,20,60,72,88,92}
{4200,1920,1640,704,8,68,72,84,92}
{2736,2618,2106,1376,58,60,68,78,94}
{2850,2370,2240,1376,56,64,66,78,94}
{2960,2240,2106,1530,54,66,68,76,94}
{2960,2370,1826,1680,54,64,72,74,94}
{3360,2618,1968,890,46,60,70,84,94}
{3450,2960,2240,186,44,54,66,92,94}
{3536,2618,2496,186,42,60,62,92,94}
{3770,2240,2106,720,36,66,68,86,94}
{3770,2496,1680,890,36,62,74,84,94}
{3770,3168,1530,368,36,50,76,90,94}
{3906,2736,1826,368,32,58,72,90,94}
{3968,2496,1826,546,30,62,72,88,94}
{3968,2618,1530,720,30,60,76,86,94}
{3968,3266,1056,546,30,48,82,88,94}
{4026,2240,1680,890,28,66,74,84,94}
{4026,3066,1376,368,28,52,78,90,94}
{4130,2370,1968,368,24,64,70,90,94}
{4130,3618,720,368,24,40,86,90,94}
{4176,2240,1530,890,22,66,76,84,94}
{4218,3360,890,368,20,46,84,90,94}
{4256,4026,368,186,18,28,90,92,94}
{4290,2960,1218,368,16,54,80,90,94}
{4320,2618,1530,368,14,60,76,90,94}
{4346,2240,1530,720,12,66,76,86,94}
{4346,3066,1056,368,12,52,82,90,94}
{4386,3536,546,368,8,42,88,90,94}
{4400,2850,1218,368,6,56,80,90,94}
{4400,3360,890,186,6,46,84,92,94}
{4410,3168,890,368,4,50,84,90,94}
{4416,1826,1376,1218,2,72,78,80,94}
{2880,2754,2210,1760,62,64,72,78,98}
{3002,2490,2352,1760,60,68,70,78,98}
{3002,2624,2064,1914,60,66,74,76,98}
{3344,3120,2210,930,54,58,72,88,98}
{3450,3120,1760,1274,52,58,78,84,98}
{3552,3234,2624,194,50,56,66,96,98}
{3650,2754,1760,1440,48,64,78,82,98}
{3834,3744,1274,752,44,46,84,90,98}
{3920,2624,2490,570,42,66,68,92,98}
{3920,3650,1104,930,42,48,86,88,98}
{4002,3344,2064,194,40,54,74,96,98}
{4002,3920,930,752,40,42,88,90,98}
{4080,2490,1760,1274,38,68,78,84,98}
{4080,3120,2210,194,38,58,72,96,98}
{4154,3002,2064,384,36,60,74,94,98}
{4154,3120,1760,570,36,58,78,92,98}
{4224,3002,1274,1104,34,60,84,86,98}
{4290,2352,2210,752,32,70,72,90,98}
{4290,2624,1760,930,32,66,78,88,98}
{4352,2064,1914,1274,30,74,76,84,98}
{4352,2210,1602,1440,30,72,80,82,98}
{4410,3002,1440,752,28,60,82,90,98}
{4464,3344,1602,194,26,54,80,96,98}
{4514,3234,1104,752,24,56,86,90,98}
{4560,3920,930,194,22,42,88,96,98}
{4602,2490,1760,752,20,68,78,90,98}
{4602,2624,1274,1104,20,66,84,86,98}
{4602,3344,1274,384,20,54,84,94,98}
{4640,3120,1274,570,18,58,84,92,98}
{4640,3650,930,384,18,48,88,94,98}
{4674,4352,384,194,16,30,94,96,98}
{4704,2210,1760,930,14,72,78,88,98}
{4730,3552,752,570,12,50,90,92,98}
{4730,3920,570,384,12,42,92,94,98}
{4770,2880,1760,194,8,62,78,96,98}
{4784,2490,1760,570,6,68,78,92,98}
{3432,2688,2408,1472,56,68,72,84,100}
{3848,3648,2112,392,48,52,76,96,100}
{4200,3200,1472,1128,40,60,84,88,100}
{4352,2408,2112,1128,36,72,76,88,100}
{4488,3648,1472,392,32,52,84,96,100}
{4488,4352,768,392,32,36,92,96,100}
{4608,3200,1800,392,28,60,80,96,100}
{4712,2408,2112,768,24,72,76,92,100}
{4712,2688,1472,1128,24,68,84,88,100}
{4992,2408,1472,1128,4,72,84,88,100}
{4992,3848,768,392,4,48,92,96,100}

Paul.

• paul says:

here is a link to dropbox with a bigger list

2. benvitalis says:

Nice work. Thanks for the bigger list

• paul says:

I worked backwards from a set of squares to make my lists, generating the required squares was the longest time, however, using the equality you gave above makes it a lot quicker to find the candidates, using the solve function in MMa. In fact only the solutions are the requited candidates, still have to work backwards to find the individual terms, but that’s trivial.

Paul.