x^2 + y^2 + z^2 = u^2 + v^2

 
 
Starting with

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

if   a^2 \; + \; b^2 \; = \; c^2,   then

a^2 \, d^2 \; + \; b^2 \, d^2 \; + \; c^2
= (a \, d)^2 \; + \; (b \, d)^2 \; + \; c^2

= \; (d^2 + 1) \, c^2

So,

(a \, d)^2 + (b \, d)^2 + c^2 = (a + b \, d)^2 + (a \, d - b)^2,

Using all 16 primitive Pythagorean triples with   c \; \leq \; 100 :
PPT (3,4,5) :

(3 \, d)^2 + (4 \, d)^2 + 5^2 = (3 + 4 \, d)^2 + (3 \, d - 4)^2   which is true

For   d = 1, 2

3^2 + 4^2 + 5^2 \; = \; 7^2 + 1^2 \; = \; 50
6^2 + 8^2 + 5^2 \; = \; 11^2 + 2^2 \; = \; 125

(5, 12, 13),    d = 1, 2
5^2 + 12^2 + 13^2 = 17^2 + 7^2 \; = \; 338
10^2 + 24^2 + 13^2 = 29^2 + 2^2 \; = \; 845

(8, 15, 17),    d = 1, 2
8^2 + 15^2 + 17^2 \; = \; 23^2 + 7^2 \; = \; 578
16^2 + 30^2 + 17^2 \; = \; 38^2 + 1^2 \; = \; 1445

(7, 24, 25),    d = 1, 2
7^2 + 24^2 + 25^2 \; = \; 31^2 + 17^2 \; = \; 1250
14^2 + 48^2 + 25^2 \; = \; 55^2 + 10^2 \; = \; 3125

(20, 21, 29),    d = 1, 2
20^2 + 21^2 + 29^2 \; = \; 41^2 + 1^2 \; = \; 1682
40^2 + 42^2 + 29^2 \; = \; 62^2 + 19^2 \; = \; 4205

(12, 35, 37),    d = 1, 2
12^2 + 35^2 + 37^2 \; = \; 47^2 + 23^2 \; = \; 2738
24^2 + 70^2 + 37^2 \; = \; 82^2 + 11^2 \; = \; 6845

(9, 40, 41),    d = 1, 2
9^2 + 40^2 + 41^2 \; = \; 49^2 + 31^2 \; = \; 3362
18^2 + 80^2 + 41^2 \; = \; 89^2 + 22^2 \; = \; 8405

(28, 45, 53),    d = 1, 2
28^2 + 45^2 + 53^2 \; = \; 73^2 + 17^2 \; = \; 5618
56^2 + 90^2 + 53^2 \; = \; 118^2 + 11^2 \; = \; 14045

(11, 60, 61),    d = 1, 2
11^2 + 60^2 + 61^2 \; = \; 71^2 + 49^2 \; = \; 7442
22^2 + 120^2 + 61^2 \; = \; 131^2 + 38^2 \; = \; 18605

(16, 63, 65),    d = 1, 2
16^2 + 63^2 + 65^2 \; = \; 79^2 + 47^2 \; = \; 8450
32^2 + 126^2 + 65^2 \; = \; 142^2 + 31^2 \; = \; 21125

(33, 56, 65),    d = 1, 2
33^2 + 56^2 + 65^2 \; = \; 89^2 + 23^2 \; = \; 8450
66^2 + 112^2 + 65^2 \; = \; 145^2 + 10^2 \; = \; 21125

(48, 55, 73),    d = 1, 2
48^2 + 55^2 + 73^2 \; = \; 103^2 + 7^2 \; = \; 10658
96^2 + 110^2 + 73^2 \; = \; 158^2 + 41^2 \; = \; 26645

(13, 84, 85),    d = 1, 2
13^2 + 84^2 + 85^2 \; = \; 97^2 + 71^2 \; = \; 14450
26^2 + 168^2 + 85^2 \; = \; 181^2 + 58^2 \; = \; 36125

(36, 77, 85),    d = 1, 2
36^2 + 77^2 + 85^2 \; = \; 113^2 + 41^2 \; = \; 14450
72^2 + 154^2 + 85^2 \; = \; 190^2 + 5^2 \; = \; 36125

(39, 80, 89),    d = 1, 2
39^2 + 80^2 + 89^2 \; = \; 119^2 + 41^2 \; = \; 15842
78^2 + 160^2 + 89^2 \; = \; 199^2 + 2^2 \; = \; 39605

(65, 72, 97),    d = 1, 2
65^2 + 72^2 + 97^2 \; = \; 137^2 + 7^2 \; = \; 18818
130^2 + 144^2 + 97^2 \; = \; 209^2 + 58^2 \; = \; 47045

 
 

 
 
 
 
 
 
 
 
 
 
 

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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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