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

 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 

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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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