A^2 + B^2 + C^2 = D^2 + E^2

 
 
a^2 \; + \; b^2 \; + \; (a^2+b^2) \; = \; (a+b)^2 \; + \; (a-b)^2

Also true is,

(a \, n)^2 + (b \, n)^2  \;+ (a^2+b^2) \; = \; (a + b \, n)^2 \; + \; (a \, n - b)^2

Let   a^2 \; + \; b^2 \; = \; c^2

(a \, n)^2 \; + \; (b \, n)^2 \; + \; c^2 \; = \; (a + b \, n)^2 \; + \; (a \, n - b)^2
 
 
Let’s take all primitive Pythagorean triples with   c \; \leq \; 300,   for   n = 1

  3^2 + 4^2 + 5^2 \; = \; 7^2 + 1^2 \; = \; 60
  5^2 + 12^2 + 13^2 \; = \; 17^2 + 7^2 \; = \; 338
  8^2 + 15^2 + 17^2 \; = \; 23^2 + 7^2 \; = \; 578
  7^2 + 24^2 + 25^2 \; = \; 31^2 + 17^2 \; = \; 1250
20^2 + 21^2 + 29^2 \; = \; 41^2 + 1^2 \; = \; 1682
12^2 + 35^2 + 37^2 \; = \; 47^2 + 23^2 \; = \; 2738
  9^2 + 40^2 + 41^2 \; = \; 49^2 + 31^2 \; = \; 3362
28^2 + 45^2 + 53^2 \; = \; 73^2 + 17^2 \; = \; 5618
11^2 + 60^2 + 61^2 \; = \; 71^2 + 49^2 \; = \; 7442
16^2 + 63^2 + 65^2 \; = \; 79^2 + 47^2 \; = \; 8450
33^2 + 56^2 + 65^2 \; = \; 89^2 + 23^2 \; = \; 8450
48^2 + 55^2 + 73^2 \; = \; 103^2 + 7^2 \; = \; 10658
13^2 + 84^2 + 85^2 \; = \; 97^2 + 71^2 \; = \; 14450
36^2 + 77^2 + 85^2 \; = \; 113^2 + 41^2 \; = \; 14450
39^2 + 80^2 + 89^2 \; = \; 119^2 + 41^2 \; = \; 15842
65^2 + 72^2 + 97^2 \; = \; 137^2 + 7^2 \; = \; 18818
20^2 + 99^2 + 101^2 \; = \; 119^2 + 79^2 \; = \; 20402
60^2 + 91^2 + 109^2 \; = \; 151^2 + 31^2 \; = \; 23762
15^2 + 112^2 + 113^2 \; = \; 127^2 + 97^2 \; = \; 25538
44^2 + 117^2 + 125^2 \; = \; 161^2 + 73^2 \; = \; 31250
88^2 + 105^2 + 137^2 \; = \; 193^2 + 17^2 \; = \; 37538
17^2 + 144^2 + 145^2 \; = \; 161^2 + 127^2 \; = \; 42050
24^2 + 143^2 + 145^2 \; = \; 167^2 + 119^2 \; = \; 42050
51^2 + 140^2 + 149^2 \; = \; 191^2 + 89^2 \; = \; 44402
85^2 + 132^2 + 157^2 \; = \; 217^2 + 47^2 \; = \; 49298
119^2 + 120^2 + 169^2 \; = \; 239^2 + 1^2 \; = \; 57122
52^2 + 165^2 + 173^2 \; = \; 217^2 + 113^2 \; = \; 59858
19^2 + 180^2 + 181^2 \; = \; 199^2 + 161^2 \; = \; 65522
57^2 + 176^2 + 185^2 \; = \; 233^2 + 119^2 \; = \; 68450
104^2 + 153^2 + 185^2 \; = \; 257^2 + 49^2 \; = \; 68450
95^2 + 168^2 + 193^2 \; = \; 263^2 + 73^2 \; = \; 74498
28^2 + 195^2 + 197^2 \; = \; 223^2 + 167^2 \; = \; 77618
84^2 + 187^2 + 205^2 \; = \; 271^2 + 103^2 \; = \; 84050
133^2 + 156^2 + 205^2 \; = \; 289^2 + 23^2 \; = \; 84050
21^2 + 220^2 + 221^2 \; = \; 241^2 + 199^2 \; = \; 97682
140^2 + 171^2 + 221^2 \; = \; 311^2 + 31^2 \; = \; 97682
60^2 + 221^2 + 229^2 \; = \; 281^2 + 161^2 \; = \; 104882
105^2 + 208^2 + 233^2 \; = \; 313^2 + 103^2 \; = \; 108578
120^2 + 209^2 + 241^2 \; = \; 329^2 + 89^2 \; = \; 116162
32^2 + 255^2 + 257^2 \; = \; 287^2 + 223^2 \; = \; 132098
23^2 + 264^2 + 265^2 \; = \; 287^2 + 241^2 \; = \; 140450
96^2 + 247^2 + 265^2 \; = \; 343^2 + 151^2 \; = \; 140450
69^2 + 260^2 + 269^2 \; = \; 329^2 + 191^2 \; = \; 144722
115^2 + 252^2 + 277^2 \; = \; 367^2 + 137^2 \; = \; 153458
160^2 + 231^2 + 281^2 \; = \; 391^2 + 71^2 \; = \; 157922
161^2 + 240^2 + 289^2 \; = \; 401^2 + 79^2 \; = \; 167042
68^2 + 285^2 + 293^2 \; = \; 353^2 + 217^2 \; = \; 171698

 
 
 
 
 

 
 
 
 
 
 
 
 

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About benvitalis

math grad - Interest: Number theory
This entry was posted in Number Puzzles and tagged . Bookmark the permalink.

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