When the product of any two of (a,b,c,d) is one less an integer squared

 
 
a, \; b, \; c, \; d   are positive integers
 

a simple parameterization

a \; = \; n \; - \; 1
b \; = \; n \; + \; 1
c \; = \; 4 \, n
d \; = \; 4 \, n \, (4 \,n^2 - 1)

 

1 \; + \; a \, b \; = \; 1 \; + \; (n - 1) \,(n + 1) \; = \; n^2

1 \; + \; a \, c \; = \; 1 \; + \; 4 \, n \,(n - 1) \; = \; (2 n - 1)^2

1 \; + \; b \, c \; = \; 1 \; + \; 4 \, n \,(n + 1) \; = \; (2 n + 1)^2

1 \; + \; a \, d \; = \; 1 \; + \; (n - 1) \,(4 \, n) \,(4 \, n^2 - 1) \; = \; (4 \, n^2 - 2 \, n - 1)^2

1 \; + \; b \, d \; = \; 1 \; + \; (n + 1) \,(4 \, n) \,(4 \, n^2 - 1) \; = \; (4 \, n^2 + 2 \, n - 1)^2

1 \; + \; c \, d \; = \; 1 \; + \; (4 \, n) \,(4 \, n) \,(4 \, n^2 - 1) \; = \; (8 \, n^2 - 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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