Integers (a,b,c) whose squares form an A.P. – Determinant

 
 
A_0 = a \; = \; -1
B_0 = b \; = \; 5
C_0 = c \; = \; 7   …..   2(5^2) = 7^2 + 1^2

A_1 = c \; = \; 7
B_1 = 3 b - 2 a \; = \; 17
C_1 = 4 b - 3 a \; = \; 23 ….. 2(17^2) = 23^2 + 7^2

A_2 = 4 b - 3 a \; = \; 23
B_2 = -6 a + 9 b - 2 c \; = \; 37
C_2 = -8 a + 12 b - 3 c \; = \; 47 ….. 2(37^2) = 47^2 + 23^2

A_3 = -8 a + 12 b - 3 c \; = \; 47
B_3 = -12 a + 19 b - 6 c \; = \; 65
C_3 = -15 a + 24 b - 8 c \; = \; 79 ….. 2(65^2) = 79^2 + 47^2

A_4 = -15 a + 24 b - 8 c \; = \; 79
B_4 = -20 a + 33 b - 12 c \; = \; 101
C_4 = -24 a + 40 b - 15 c \; = \; 119 ….. 2(101^2) = 119^2 + 79^2

A_5 = -24 a + 40 b - 15 c \; = \; 119
B_5 = -30 a + 51 b - 20 c \; = \; 145
C_5 = -35 a + 60 b - 24 c \; = \; 167 ….. 2(145^2) = 167^2 + 119^2

A_{6} = -35 a + 60 b - 24 c \; = \; 167
B_{6} = -42 a + 73 b - 30 c \; = \; 197
C_{6} = -48 a + 84 b - 35 c \; = \; 223 ….. 2(197^2) = 223^2 + 167^2

A_{7} = -48 a + 84 b - 35 c \; = \; 223
B_{7} = -56 a + 99 b - 42 c \; = \; 257
C_{7} = -63 a + 112 b - 48 c \; = \; 287 ….. 2(257^2) = 287^2 + 223^2

A_{8} = -63 a + 112 b - 48 c \; = \; 287
B_{8} = -72 a + 129 b - 56 c \; = \; 325
C_{8} = -80 a + 144 b - 63 c \; = \; 359 ….. 2(325^2) = 359^2 + 287^2

A_{9} = -80 a + 144 b - 63 c \; = \; 359
B_{9} = -90 a + 163 b - 72 c \; = \; 401
C_{9} =  -99 a + 180 b - 80 c \; = \; 439 ….. 2(401^2) = 439^2 + 359^2

A_{10} = -99 a + 180 b - 80 c \; = \; 439
B_{10} = -110 a + 201 b - 90 c \; = \; 485
C_{10} = -120 a + 220 b - 99 c \; = \; 527 ….. 2(485^2) = 527^2 + 439^2

 

   ( \,A_0, \; B_0, \; C_0 \,) \; = \; ( \,-1, \; 5, \; 7 \,)
   ( \,A_1, \; B_1, \; C_1 \,) \; = \; ( \,7, \; 17, \; 23 \,)
   ( \,A_2, \; B_2, \; C_2 \,) \; = \; ( \,23, \; 37, \; 47 \,)
   ( \,A_3, \; B_3, \; C_3 \,) \; = \; ( \,47, \; 65, \; 79 \,)
   ( \,A_4, \; B_4, \; C_4 \,) \; = \; ( \,79, \; 101, \; 119 \,)
   ( \,A_5, \; B_5, \; C_5 \,) \; = \; ( \,119, \; 145, \; 167 \,)
   ( \,A_6, \; B_6, \; C_6 \,) \; = \; ( \,167, \; 197, \; 223 \,)
   ( \,A_7, \; B_7, \; C_7 \,) \; = \; ( \,223, \; 257, \; 287 \,)
   ( \,A_8, \; B_8, \; C_8 \,) \; = \; ( \,287, \; 325, \; 359 \,)
   ( \,A_9, \; B_9, \; C_9 \,) \; = \; ( \,359, \; 401, \; 439 \,)
( \,A_{10}, \; B_{10}, \; C_{10} \,) \; = \; ( \,439, \; 485, \; 527 \,)

 

Determinant = \begin{bmatrix}  \ \; A_0 \; & \; A_1 \; & \; A_2 \; \\  \ \; B_0 \; & \; B_1 \; & \; B_2 \; \\  \ \; C_0 \; & \; C_1 \; & \; C_2 \;   \end{bmatrix}

                            = \begin{bmatrix}  \ \; -1 \; & \; 7 \; & \; 23 \; \\  \ \; 5 \; & \; 17 \; & \; 37 \; \\  \ \; 7 \; & \; 23 \; & \; 47 \;  \end{bmatrix} \; = \; 128 \; = \; 2 \times 4^3
 

Determinant = \begin{bmatrix}  \ \; A_1 \; & \; A_2 \; & \; A_3 \; \\  \ \; B_1 \; & \; B_2 \; & \; B_3 \; \\  \ \; C_1 \; & \; C_2 \; & \; C_3 \;   \end{bmatrix}

                            = \begin{bmatrix}  \ \; 7 \; & \; 23 \; & \; 47 \; \\  \ \; 17 \; & \; 37 \; & \; 65 \; \\  \ \; 23 \; & \; 47 \; & \; 79  \;  \end{bmatrix} \; = \; 128
 

Determinant = \begin{bmatrix}  \ \; A_2 \; & \; A_3 \; & \; A_4 \; \\  \ \; B_2 \; & \; B_3 \; & \; B_4 \; \\  \ \; C_2 \; & \; C_3 \; & \; C_4 \;   \end{bmatrix}

                            = \begin{bmatrix}  \ \; 23 \; & \; 47 \; & \; 79 \; \\  \ \; 37 \; & \; 65 \; & \; 101 \; \\  \ \; 47 \; & \; 79 \; & \; 119 \;  \end{bmatrix} \; = \; 128
 

Determinant = \begin{bmatrix}  \ \; A_3 \; & \; A_4 \; & \; A_5 \; \\  \ \; B_3 \; & \; B_4 \; & \; B_5 \; \\  \ \; C_3 \; & \; C_4 \; & \; C_5 \;   \end{bmatrix}

                            = \begin{bmatrix}  \ \; 47 \; & \; 79 \; & \; 119 \; \\  \ \; 65 \; & \; 101 \; & \; 145 \; \\  \ \; 79 \; & \; 119 \; & \; 167 \;  \end{bmatrix} \; = \; 128

 

Check to see whether the pattern continues.

 
 
 
 
 
 
 
 
 
 
 

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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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