Pell equation : x^2 – 113*y^2 = ± 1

$x^2 \; - \; 113 \, y^2 \; = \; -1$

Here are the first few solutions:

$x \; = \; 776$
$y \; = \; 73$

$x \; = \; 1869156632$
$y \; = \; 175835465$

$x \; = \; 4502248794437416$
$y \; = \; 423535939558217$

$x \; = \; 10844593684652301387064$
$y \; = \; 1020173558829338801737$

$x \; = \; 26121437875779604016034941768$
$y \; = \; 2457298172193157811837164105$

$x^2 \; - \; 113 \, y^2 \; = \; +1$

$x \; = \; 1204353$
$y \; = \; 113296$

$x \; = \; 2900932297217$
$y \; = \; 272896754976$

$x \; = \; 6987493029899166849$
$y \; = \; 657328051091107760$

$x \; = \; 16830816386073401651890177$
$y \; = \; 1583310020631184911403584$

$x \; = \; 40540488414026331506287881514113$
$y \; = \; 3813728346553801555156190094544$

math grad - Interest: Number theory
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2 Responses to Pell equation : x^2 – 113*y^2 = ± 1

1. paul says:

These two are interesting using x^2 – n y^2 = 1

9801^2 – 725 * 364^2 = 1.
458^2 – 209763 *1^2 = 1

x, y and n are 0 – 9 pan digital.

and this one

30769208^2 – 5651487*12943^2 = 1

is 2 lots of 0 – 9 digits for x, y, and n

Paul.

• benvitalis says:

Yes, indeed. That’s very cool!