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   $q \; = \; (n/2 - 1/n)$

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

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

math grad - Interest: Number theory
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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

• benvitalis says:

That’s right. I posted a general solution