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

Jack Brennen jfb at brennen.net
Wed Oct 28 17:42:53 CET 2009

```25 appended to 29831431640625^2 gives 160903431640625^2
37 appended to 1206796875^2 gives 19273203125^2
38 appended to 377630078125^2 gives 6175969921875^2
40 appended to 37521096123046875^2 gives 633567543876953125^2
41 appended to 12111608359375^2 gives 202846471640625^2
61 appended to 37285546875^2 gives 781914453125^2
62 appended to 12816384765625^2 gives 249327615234375^2

Note that the search is probably most effectively conducted by
examining the factors of n*10^d.  Any solution will have
n*10^d = y^2 - x^2 = (y+x) * (y-x).  So for any factor of n*10^d,
you find values for y and x which solve the equation; you only
need to look until you find y and x which put y^2 very near to
n*10^d, then check that x^2 has the correct number of digits.

For example, the biggest solution above, for 40, was found by
noticing that 40*10^34 has a divisor 596046447753906250 with
co-divisor 671088640000000000.  This leads to solutions for
(y,x)=(633567543876953125,37521096123046875).  A simple check
shows that x^2 has 34 digits, so solution found...

It seems probable that solutions exist for all n, and it's
probably proveable; you just need to show that the plausible
factors of 10^d get dense enough as d increases.

Maximilian Hasler wrote:
> 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}
>>
>> Zak
>>
>> --- On Wed, 10/28/09, zak seidov <zakseidov at yahoo.com> wrote:
>> <...>
>>
>>
>>
>>
>>
>>
>>
>> _______________________________________________
>>
>> Seqfan Mailing list - http://list.seqfan.eu/
>>
>
>
> _______________________________________________
>
> Seqfan Mailing list - http://list.seqfan.eu/
>
>

```