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

Jack Brennen jfb at brennen.net
Wed Oct 28 19:22:13 CET 2009

```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.

Jack Brennen wrote:
> 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/
>>
>>
>
>
>
> _______________________________________________
>
> Seqfan Mailing list - http://list.seqfan.eu/
>
>

```