Rational numbers (x,y); each of x – 1 and x + 1 is a square

 
 

Solve,

x \; + \; 1 \; = \; p^2
x \; - \; 1 \; = \; q^2
 
 
Similarly for,

x \; + \; a \; = \; p^2
x \; - \; a \; = \; q^2

 
 
Solution:

Put
p \; + \; q \; = \; n
p \; - \; q \; = \; 2/n

we get,   p \; = \; (n + 2/n)/2   and   q \; = \; (n/2 - 1/n)

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

x \; = \; q^2 \; + \; 1
= \; (n/2 - 1/n)^2 \; + \; 1
= \; n^2/4 \; + \; 1/n^2

 
 
 

 
 
 
 
 
 
 
 
 
 
 
 
 
 
 

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About benvitalis

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

2 Responses to Rational numbers (x,y); each of x – 1 and x + 1 is a square

  1. pipo says:

    Here are some fractions, x, for which x -1 and x + 1 is a square:
    5 / 4
    65 / 16
    85 / 36
    325 / 36
    1025 / 64
    629 / 100
    2501 / 100
    145 / 144
    5185 / 144
    2405 / 196
    9605 / 196
    16385 / 256
    6565 / 324
    26245 / 324
    689 / 400
    40001 / 400
    14645 / 484
    58565 / 484
    1105 / 576
    82945 / 576
    28565 / 676
    2465 / 784
    949 / 900
    2581 / 900
    50629 / 900
    83525 / 1156
    6625 / 1296
    1649 / 1600
    2725 / 1764
    9685 / 1764
    14705 / 1936
    16465 / 2304
    28625 / 2704
    3425 / 3136
    5809 / 3600
    40081 / 3600
    50689 / 3600
    14965 / 4356
    58645 / 4356
    83585 / 4624
    4901 / 4900
    10229 / 4900
    7585 / 5184
    28885 / 6084
    17009 / 6400
    7585 / 7056
    15665 / 7744
    9061 / 8100
    26869 / 8100
    83845 / 10404
    29585 / 10816
    17141 / 12100
    59189 / 12100
    18785 / 12544
    51649 / 14400
    83569 / 14400
    16165 / 15876
    28645 / 15876
    31061 / 16900
    19825 / 17424
    84545 / 18496
    42401 / 19600
    22945 / 20736
    24245 / 23716
    60965 / 23716
    33745 / 24336
    85345 / 28224
    86021 / 28900
    31025 / 30976
    46561 / 32400
    38165 / 33124
    40885 / 39204
    65125 / 39204
    88705 / 41616
    44945 / 43264
    60229 / 44100
    54641 / 48400
    54805 / 54756
    93125 / 56644
    67009 / 57600
    68561 / 67600
    97585 / 69696
    99905 / 73984
    87125 / 81796

    pipo

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