[seqfan] Re: Can more of these terms be found?

[israel at math.ubc.ca]

Continuing:

d = 10 has 2 solutions (1776695025 - 237050520)^2 = 2370505201776695025 
(2372316441 - 392118420)^2 = 3921184202372316441 d = 11 has 4 solutions 
(22183775236 - 3493328130)^2 = 349332813022183775236 (24272025796 - 
4077990582)^2 = 407799058224272025796 (32892249529 - 6805293006)^2 = 
680529300632892249529 (35202926161 - 7612381530)^2 = 761238153035202926161 
d = 12 has no solutions d = 13 has 2 solutions (2982643139584 - 
578155892256)^2 = 5781558922562982643139584 (3980343713961 - 
930285506742)^2 = 9302855067423980343713961 d = 14 has 2 solutions 
(34040242717696 - 7202593926432)^2 = 720259392643234040242717696 
(36957161062641 - 8244289775512)^2 = 824428977551236957161062641 d = 15 has 
26 solutions (122277119615769 - 12131956875156)^2 = 
12131956875156122277119615769 (129861643896900 - 13532475435030)^2 = 
13532475435030129861643896900 (143571110463129 - 16218649351252)^2 = 
16218649351252143571110463129 (158564886339600 - 19374080445460)^2 = 
19374080445460158564886339600 (163265306122449 - 20408163265306)^2 = 
20408163265306163265306122449 (166508647467609 - 21133835852212)^2 = 
21133835852212166508647467609 (180858471283216 - 24460367017812)^2 = 
24460367017812180858471283216 (199911675262276 - 29157118615602)^2 = 
29157118615602199911675262276 (201453353848900 - 29550567838230)^2 = 
29550567838230201453353848900 (208245817220761 - 31307254852830)^2 = 
31307254852830208245817220761 (239720319640401 - 39920119840200)^2 = 
39920119840200239720319640401 (244865168345616 - 41398631582412)^2 = 
41398631582412244865168345616 (248413698445764 - 42429492924306)^2 = 
42429492924306248413698445764 (280300407400324 - 52083127050342)^2 = 
52083127050342280300407400324 (286574093726361 - 54061913809630)^2 = 
54061913809630286574093726361 (317012472235536 - 64010166637780)^2 = 
64010166637780317012472235536 (323447135313081 - 66184211809572)^2 = 
66184211809572323447135313081 (326363855294169 - 67177543975156)^2 = 
67177543975156326363855294169 (337588580192256 - 71045313186240)^2 = 
71045313186240337588580192256 (347107438016529 - 74380165289256)^2 = 
74380165289256347107438016529 (361598994146089 - 79551045394806)^2 = 
79551045394806361598994146089 (371309614051600 - 83077659579060)^2 = 
83077659579060371309614051600 (372247522017849 - 83420847761806)^2 = 
83420847761806372247522017849 (382961713968849 - 87372803809306)^2 = 
87372803809306382961713968849 (392839808640000 - 91066892759200)^2 = 
91066892759200392839808640000 (411963997569489 - 98352298065672)^2 = 
98352298065672411963997569489 d = 16 has 8 solutions (1264874678201001 - 
129017217176502)^2 = 1290172171765021264874678201001 (1433891916460809 - 
161817366153862)^2 = 1618173661538621433891916460809 (1700791296589729 - 
219440128361202)^2 = 2194401283612021700791296589729 (1906391388260836 - 
268325865639930)^2 = 2683258656399301906391388260836 (2758015750830489 - 
506798093891422)^2 = 5067980938914222758015750830489 (2987745668450916 - 
579813657119470)^2 = 5798136571194702987745668450916 (3302186730156676 - 
684978100903650)^2 = 6849781009036503302186730156676 (3757031597333904 - 
846891524406052)^2 = 8468915244060523757031597333904 d = 17 has 1 solution 
(21487603305785124 - 3305785123966942)^2 = 
330578512396694221487603305785124 d = 18 has 1 solution (124802171127532944 
- 12591283338420732)^2 = 12591283338420732124802171127532944 d = 19 has no 
solution d = 20 has 6 solutions (11599440856401535044 - 
1101969127079213682)^2 = 110196912707921368211599440856401535044 
(14296916574953559561 - 1609663823733427980)^2 = 
160966382373342798014296916574953559561 (20979545881028335524 - 
3171326702946143742)^2 = 317132670294614374220979545881028335524 
(23997200319964004001 - 3999200119984002000)^2 = 
399920011998400200023997200319964004001 (36317812284859738449 - 
8012121377455225306)^2 = 801212137745522530636317812284859738449 
(39795210011210983044 - 9299738081908659682)^2 = 
929973808190865968239795210011210983044

Cheers,
Robert Israel

On Aug 9 2013, israel at math.ubc.ca wrote:

>For an example with 2d+1 digits, you want an integer solution of
>
>(y-x)^2 = 10^d x + y
>
>where 0 <= y < 10^d and 10^(d-2) <= x < 10^(d-1).
>Now solving the quadratic equation for x, 
>
>x = y + 10^d/2 - 1/2 sqrt(10^(2d) + 4 y (10^d+1))
>  = y + 10^d/2 - z/2
>where z^2 = 10^(2d) + 4 y (10^d+1), and so
>y = (z^2 - 10^(2d))/(4 (10^d+1)).
>
> For y to be an integer, z must be even, and z^2 == 10^(2d) == 1 mod 
> (10^d+1). For 0 <= y < 10^d we need 10^(2d) <= z^2 < 5 * 10^(2d) + 4 * 
> 10^d. But z = 10^d and z = 10^d+2 produce x=0, so those are not allowed 
> In particular, if 10^d+1 is prime, the only solutions of z^2 == 1 mod 
> (10^d+1) are z == (+/-) 1 mod 10^(d+1), and the only even z in the 
> interval are 10^d and 10^d + 2. So we get no solutions for d = 1 or 2,
>
>d = 3: 10^3+1 = 7 * 11 * 13.  Candidates for z are 1156, 1574, 1728
>leading to (y,x) = [84, 6], [369, 82], [496, 132].  The one with x in
>the interval [100, 1000) is [369, 82] which is your solution.
>
>d = 4: Candidate for z is 12192 leading to your solution.
>
>d = 5: z =  136366, y = 21489, x = 3306
>(21489 - 3306)^2 = 330621489
>
>d = 6: z = 1673268, y = 449956, x = 113322 > 10^5
>
>d = 7: z = 15454548, y = 3471076, x = 743802
>(3471076 - 743802)^2 = 7438023471076
>
>d = 8: z = 147058824, y = 29065744, x = 5536332
>(29065744 - 5536332)^2 = 553633229065744
>
>d = 9: 20 candidates, of which 6 have x in the desired interval:
>(135490884 - 14611762)^2 = 14611762135490884
>(159622884 - 19605006)^2 = 19605006159622884
>(193545216 - 27553312)^2 = 27553312193545216
>(294293121 - 56530882)^2 = 56530882294293121
>(371816704 - 83263152)^2 = 83263152371816704
>(402366864 - 94674556)^2 = 94674556402366864
>
>Cheers,
>Robert Israel
>
>
>
>
>
>On Aug 8 2013, DAN_CYN_J wrote:
>
>>
>>
>>Hi all seqfans, 
>>
>>  
>>
>> So far only 2 terms are known that will always, I believe, be of odd 
>> length 
>>
>>and one integer per discrete length. 
>>
>> On the first term subtract the 2 high order digits from the 3 low order 
>> digits.
>>
>>Square the results which will give the original integer.   
>>
>>82369 ----369-82 =287---287^2 =82369 
>>
>>1201216 ---1216-120=1096---1096^2 =1201216 
>>
>>This was just posted on sci-math given the larger integer above. 
>>
>>I found the smaller one (82369) . 
>>
>>Is a (9),(11),(13)..  digit possible to calculate? 
>>
>>Dan   
>>
>> 
>>
>>_______________________________________________
>>
>>Seqfan Mailing list - http://list.seqfan.eu/
>>
>>
>
>
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