Pythagorean triples of the form (x^2 – 1, 2*x, x^2 + 1)

 
The integers   x^2 - 1,    2 \,x,    x^2 + 1   form a Pythagorean triple.

(x^2 - 1)^2 \; + \; (2 \,x)^2
= \; x^4 \; + \; 2 \, x^2 \; + \; 1
= \; (x^2 + 1)^2

where   x \; \geq \; 2
 

(3, 4, 5)   :   primitive

3^2 + 4^2 = 25 = 5^2

3 = 2^2 - 1,      4 = 2\times 2,      5 = 2^2 + 1

3^4 + 4^4 + 5^4 \; = \; 2 \,(2^8 + 14\times 2^4 + 1)
3^8 + 4^8 + 5^8 \; = \; 2 \,(2^8 + 14\times 2^4 + 1)^2

 

(8, 6, 10)   =   2[4, 3, 5]

8^2 + 6^2 = 100 = 10^2

8^4 + 6^4 + 10^4 \; = \; 2 \,(3^8 + 14\times 3^4 + 1)
8^8 + 6^8 + 10^8 \; = \; 2 \,(3^8 + 14\times 3^4 + 1)^2

 

(15, 8, 17)   :   primitive

15^2 + 8^2 = 289 = 17^2

15 = 4^2 - 1,      8 = 2\times 4,      17 = 4^2 + 1

8^4 + 15^4 + 17^4 \; = \; 2 \,(4^8 + 14\times 4^4 + 1)
8^8 + 15^8 + 17^8 \; = \; 2 \,(4^8 + 14\times 4^4 + 1)^2

 

(24, 10, 26)   =   2[12, 5, 13]

24^2 + 10^2 = 676 = 26^2

24^4 + 10^4 + 26^4 = 2(5^8 + 14\times 5^4 + 1)
24^8 + 10^8 + 26^8 = 2(5^8 + 14\times 5^4 + 1)^2

 

(35, 12, 37)   :   primitive

35^2 + 12^2 = 1369 = 37^2

35 = 6^2 - 1,    12 = 6\times 2,    37 = 6^2 + 1

35^4 + 12^4 + 37^4 = 2(6^8 + 14\times 6^4 + 1)
35^8 + 12^8 + 37^8 = 2(6^8 + 14\times 6^4 + 1)^2

 

(48, 14, 50)   =   2[24, 7, 25]

48^2 + 14^2 = 2500 = 50^2

48^4 + 14^4 + 50^4 = 2(7^8 + 14\times 7^4 + 1)
48^8 + 14^8 + 50^8 = 2(7^8 + 14\times 7^4 + 1)^2

 

(63, 16, 65)   :   primitive

63^2 + 16^2 = 4225 = 65^2

63 = 8^2 - 1,    16 = 8\times 2,    65 = 8^2 + 1

63^4 + 16^4 + 65^4 = 2 \,(8^8 + 14\times 8^4 + 1)
63^8 + 16^8 + 65^8 = 2 \,(8^8 + 14\times 8^4 + 1)^2

 

(80, 18, 82)   =   2[40, 9, 41]

80^2 + 18^2 = 6724 = 82^2

80^4 + 18^4 + 82^4 = 2 \,(9^8 + 14\times 9^4 + 1)
80^8 + 18^8 + 82^8 = 2 \,(9^8 + 14\times 8^4 + 1)^2

 

(99, 20, 101)   :   primitive

99^2 + 20^2 = 10201 = 101^2

99 = 10^2 - 1,    20 = 10\times 2,    101 = 10^2 + 1

99^4 + 20^4 + 101^4 = 2 \,(10^8 + 14\times 10^4 + 1)
99^8 + 20^8 + 101^8 = 2 \,(10^8 + 14\times 10^4 + 1)^2

 

(120, 22, 122)   =   2[60, 11, 61]

120^2 + 22^2 = 14884 = 122^2

120^4 + 22^4 + 122^4 = 2 \,(11^8 + 14\times 11^4 + 1)
120^8 + 22^8 + 122^8 = 2 \,(11^8 + 14\times 11^4 + 1)^2

 

(143, 24, 145)   :   primitive

143^2 + 24^2 = 21025 = 145^2

143 = 12^2 - 1,    24 = 12\times 2,    145 = 12^2 + 1

143^4 + 24^4 + 145^4 = 2 \,(12^8 + 14\times 12^4 + 1)
143^8 + 24^8 + 145^8 = 2 \,(12^8 + 14\times 12^4 + 1)^2

 

(168, 26, 170)   =   2[84, 13, 85]

168^2 + 26^2 = 28900 = 170^2

168^4 + 26^4 + 170^4 = 2 \,(13^8 + 14\times 13^4 + 1)
168^8 + 26^8 + 170^8 = 2 \,(13^8 + 14\times 13^4 + 1)^2

 

(195, 28, 197)   :   primitive

195^2 + 28^2 = 38809 = 197^2

195 = 14^2 - 1,    28 = 14\times 2,    197 = 14^2 + 1

195^4 + 28^4 + 197^4 = 2 \,(14^8 + 14*14^4 + 1)
195^8 + 28^8 + 197^8 = 2 \,(14^8 + 14*14^4 + 1)^2

 

(224, 30, 226)   =   2[112, 15, 113]

224^2 + 30^2 = 51076 = 226^2

224^4 + 30^4 + 226^4 = 2 \,(15^8 + 14\times 15^4 + 1)
224^8 + 30^8 + 226^8 = 2 \,(15^8 + 14\times 15^4 + 1)^2

 
 

 
 
 
 
 
 
 
 
 
 
 
 
 
 

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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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