[seqfan] Re: Prepend k to square to get another square!

[zak seidov]

Many thanks to you two, 
Maximilian and Jack.

I calculated all 200 cases.

And I'm preparing to submit this table-
wouldn't it only fair to include 
you two as co-authors
and not forget me;))

Zak

You may wish check the table:

Prepending n to x^2 gives y^2. 
{n,x,y}
{1,15,35}
{2,5,15}
{3,65,185}
{4,3,7}
{5,25,75}
{6,2,8}
{7,15,85}
{8,1,9}
{9,55,305}
{10,75,325}
{11,95,345}
{12,1,11}
{13,15,115}
{14,2,12}
{15,25,125}
{16,3,13}
{17,8,42}
{18,7,43}
{19,6,44}
{20,5,45}
{21,4,46}
{22,75,475}
{23,295,1545}
{24,2975,15775}
{25,29831431640625,160903431640625}
{26,3109625,16421625}
{27,15,165}
{28,3,17}
{29,4,54}
{30,5,55}
{31,6,56}
{32,2,18}
{33,8,58}
{34,9,59}
{35,125,1875}
{36,1,19}
{37,1206796875,19273203125}
{38,377630078125,6175969921875}
{39,1025,19775}
{40,37521096123046875,633567543876953125}
{41,12111608359375,202846471640625}
{42,5,65}
{43,1242375,20773625}
43*10^13 + 1242375 ^2 = 20773625 ^2
{44,1,21}
{45,75,675}
{46,15,215}
{47,31,219}
{48,2,22}
{49,27,223}
{50,25,225}
{51,23,227}
{52,3,23}
{53,19,231}
{54,17,233}
{55,15,235}
{56,5,75}
{57,11,239}
{58,375,7625}
{59,12765625,243234375}
{60,108125,2451875}
{61,37285546875,781914453125}
{62,12816384765625,249327615234375}
{63,45,795}
{64,1500843844921875,25342701755078125}
{65,110496875,2551903125}
{66,65,815}
{67,38175,819425}
{68,11,261}
{69,13,263}
{70,15,265}
{71,17,267}
{72,3,27}
{73,21,271}
{74,23,273}
{75,25,275}
{76,27,277}
{77,29,279}
{78,2,28}
{79,2985,28265}
{80,2825,28425}
{81,15,285}
{82,9,91}
{83,85,915}
{84,1,29}
{85,75,925}
{86,7,93}
{87,65,935}
{88,6,94}
{89,55,945}
{90,5,95}
{91,45,955}
{92,4,96}
{93,35,965}
{94,13125,306875}
{95,115625,3084375}
{96,1,31}
{97,383671875,9856328125}
{98,98875,994875}
{99,15,315}
{100,11252109375,316427890625}
{101,12890509375,318066290625}
{102,2,32}
{103,3907875,101564125}
{104,1015,32265}
{105,25,325}
{106,1335,32585}
{107,35,1035}
{108,3,33}
{109,45,1045}
{110,5,105}
{111,55,1055}
{112,6,106}
{113,65,1065}
{114,7,107}
{115,75,1075}
{116,8,108}
{117,85,1085}
{118,9,109}
{119,95,1095}
{120,275,3475}
{121,290625,3490625}
{122,30625,350625}
{123,235,3515}
{124,9346875,111746875}
{125,983515625,11223515625}
{126,75,1125}
{127,901595625,11305435625}
{128,314725,3591525}
{129,155,3595}
{130,289125,3617125}
{131,276325,3629925}
{132,5,115}
{133,15,365}
{134,89,1161}
{135,85,1165}
{136,3,37}
{137,77,1173}
{138,73,1177}
{139,69,1181}
{140,25,375}
{141,61,1189}
{142,57,1193}
{143,53,1197}
{144,2,38}
{145,45,1205}
{146,41,1209}
{147,37,1213}
{148,15,385}
{149,1375,38625}
{150,125,3875}
{151,1125,38875}
{152,1,39}
{153,165,3915}
{154,384375,12415625}
{155,3453125,124546875}
{156,5,125}
{157,327204375,12534235625}
{158,368164375,12575195625}
{159,245,3995}
{160,450084375,12657115625}
{161,107675,4013925}
{162,75,1275}
{163,133275,4039525}
{164,146075,4052325}
{165,35,1285}
{166,39,1289}
{167,43,1293}
{168,1,41}
{169,51,1301}
{170,55,1305}
{171,59,1309}
{172,15,415}
{173,67,1317}
{174,71,1321}
{175,75,1325}
{176,2,42}
{177,83,1333}
{178,87,1337}
{179,91,1341}
{180,25,425}
{181,99,1349}
{182,5,135}
{183,2875,42875}
{184,3,43}
{185,3125,43125}
{186,865625,13665625}
{187,399375,13680625}
{188,94375,1374375}
{189,15,435}
{190,2713359375,43673359375}
{191,28354296875,437954296875}
{192,29575,439175}
{193,30795703125,440395703125}
{194,8469126953125,139541126953125}
{195,26375,442375}
{196,15752953125,442999046875}
{197,{96135361328125,1406855361328125}
{198,9,141}
{199,297519551015625,4470852031015625}
{200,287033791015625,4481337791015625}



--- On Wed, 10/28/09, Jack Brennen <jfb at brennen.net> wrote:

> From: Jack Brennen <jfb at brennen.net>
> Subject: [seqfan] Re: Prepend k to square to get another square!
> To: "Sequence Fanatics Discussion list" <seqfan at list.seqfan.eu>
> Date: Wednesday, October 28, 2009, 2:22 PM
> I found the first really tough nut to
> crack...  :)
> 
> 8693 prepended to
> 
> 7998581634584055861382823888216855160746222708835008603500682511366903781890869140625^2
> 
> 
> 
> gives
> 
> 932396899023245950513606405889579687094186222708835008603500682511366903781890869140625^2
> 
> So with that one, you end up with adding 8693 on the front
> of a
> 170 digit square to get a 174 digit square.  There are
> no
> solutions with fewer digits.
>