## Square with last digit (d=9) deleted remains a square

By solving the equation,   $x^2 \; = \; 10 \, y^2 \; + \; 9$
We obtain three families of solutions. Here are the first few ones:

(1)

$4 = 2^2$
$49 = 7^2$

$6400 = 80^2$
$64009 = 253^2$

$9229444 = 3038^2$
$92294449 = 9607^2$

$13308852496 = 115364^2$
$133088524969 = 364813^2$

$19191356070436 = 4380794^2$
$191913560704369 = 13853287^2$

$27673922144716864 = 166354808^2$
$276739221447168649 = 526060093^2$

$39905776541325648100 = 6317101910^2$
$399057765413256481009 = 19976430247^2$

$57544102098669439843984 = 239883517772^2$
$575441020986694398439849 = 758578289293^2$

$82978555320504790929377476 = 9109256573426^2$
$829785553205047909293774769 = 28805998562887^2$

$119655019228065809850722477056 = 345911866272416^2$
$1196550192280658098507224770569 = 1093869367100413^2$

$172542454748315577299950882537924 = 13135541661778382^2$
$1725424547483155772999508825379249 = 41538229951252807^2$
…………………………………………
…………………………………………

(2)

$16 = 4^2$
$169 = 13^2$

$23716 = 154^2$
$237169 = 487^2$

$34199104 = 5848^2$
$341991049 = 18493^2$

$49315084900 = 222070^2$
$493150849009 = 702247^2$

$71112318227344 = 8432812^2$
$711123182273449 = 26666893^2$

$102543913568745796 = 320224786^2$
$1025439135687457969 = 1012639687^2$

$147868252253813211136 = 12160109056^2$
$1478682522538132111369 = 38453641213^2$

$213225917206085081712964 = 461763919342^2$
$2132259172060850817129649 = 1460225726407^2$

$307471624742922434016883600 = 17534868825940^2$
$3074716247429224340168836009 = 55450123962253^2$

$443373869653376943767264438884 = 665863251466378^2$
$4433738696533769437672644388849 = 2105644484839207^2$
…………………………………………….
…………………………………………….

(3)

$324 = 18^2$
$3249 = 57^2$

$467856 = 684^2$
$4678569 = 2163^2$

$674648676 = 25974^2$
$6746486769 = 82137^2$

$972842923584 = 986328^2$
$9728429235849 = 3119043^2$

$14028388211601009 = 118441497^2$
$14028388211601009 = 118441497^2$

$2022892607269941264 = 1422284292^2$
$20228926072699412649 = 4497657843^2$

$2917009736844434143236 = 54009348606^2$
$29170097368444341432369 = 170792556537^2$

$4206326017637066764605696 = 2050932962736^2$
$42063260176370667646056969 = 6485619490563^2$

$6065519200422913430127271044 = 77881443235362^2$
$60655192004229134301272710449 = 246282748084857^2$

$8746474480683823529176760240400 = 2957443909981020^2$
$87464744806838235291767602404009 = 9352258807734003^2$