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

[Maximilian Hasler]

You have the following additional solutions
of this Pell-type equation n*10^d+x^2=y^2
with 10^(d-1) <= x^2 < 10^d :

26*10^13 + 3109625^2 = 16421625^2
43*10^13 + 1242375^2 = 20773625^2
59*10^15 + 12765625^2 = 243234375^2
103e14 + 3907875^2 = 101564125^2
124e14 + 9346875^2 = 111746875^2
...

Maximilian

On Wed, Oct 28, 2009 at 5:29 AM, zak seidov <zakseidov at yahoo.com> wrote:
> Dear seqfans,
>
> 1. Sorry, in the previous post I missed 155
> (copy/paste typo :()
> correct (still preliminary!) list for n's
> with no solution looks like
>
> 25,26,37,38,40,41,43,59,61,62,64,65,97,100,
> 101,103,124,125,127,155,157,158,160,190,191,
> 193,194,196,197,199,200
>
> 2. Another sorry for the long list below.
>
> Prepending n to x^2 gives y^2.
> Here is the table for n<=200.
>  If x=0, there is no known solution for given n
> (for x<10^6)
> {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}
> {25,0,0}
> {26,0,0}
> {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}
> {37,0,0}
> {38,0,0}
> {39,1025,19775}
> {40,0,0}
> {41,0,0}
> {42,5,65}
> {43,0,0}
> {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}
> {59,0,0}
> {60,108125,2451875}
> {61,0,0}
> {62,0,0}
> {63,45,795}
> {64,0,0}
> {65,0,0}
> {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}
> {97,0,0}
> {98,98875,994875}
> {99,15,315}
> {100,0,0}
> {101,0,0}
> {102,2,32}
> {103,0,0}
> {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}
> {124,0,0}
> {125,0,0}
> {126,75,1125}
> {127,0,0}
> {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}
> {155,0,0}
> {156,5,125}
> {157,0,0}
> {158,0,0}
> {159,245,3995}
> {160,0,0}
> {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}
> {190,0,0}
> {191,0,0}
> {192,29575,439175}
> {193,0,0}
> {194,0,0}
> {195,26375,442375}
> {196,0,0}
> {197,0,0}
> {198,9,141}
> {199,0,0}
> {200,0,0}
>
> 3. Thanks for comments/yeas/nays,
> Zak
>
> --- On Wed, 10/28/09, zak seidov <zakseidov at yahoo.com> wrote:
> <...>
>
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