# A082393 and A081232

Don Reble djr at nk.ca
Wed Apr 30 00:08:46 CEST 2003

```Seqfans:

%I A082393
%S A082393 4,180,8,1820,12,320,9100
%N A082393 Let p = n-th prime of the form 4k+1, take smallest solution
(x,y) to the Pellian equation x^2 - p*y^2 = 1 with x and y >= 1;
sequence gives value of y.
%Y A082393 Values of x are in A081232. Cf. A082394, A081233, A081234.
Equals A002349(p).
%A A082393 Cino Hilliard (hillcino368(AT)hotmail.com), Apr 14 2003
%E A082393 More terms from Robert G. Wilson v (rgwv(AT)rgwv.com),
Apr 15 2003
%E A082393 Should be rechecked, I have received conflicting versions of
this sequence. - njas, Apr 18, 2003

I wonder, in the continued fraction for square-root(p), does the
fundamental solution to x^2 - p*y^2 = 1 always appear among the
convergents (for each p)? If so, A082393 goes:

4 180 8 1820 12 320 9100 226153980 267000 53000 6377352 20
15140424455100 113296 519712 2113761020 3726964292220 190060
183567298683461940 448036604040 28 386460 70255304
649641205044600 32 820 9562401173878027020 134951575480 719780
1819380158564160 13967198980 112422913565764752 9081060
540608704 2712540 166660 42094239791738433660 40
1238789998647218582160 189073995951839020880499780706260
5025068784834899736 3388393513402120 326662411570389853632
55067617520620 13882400040814700 1444066532654320
159395869721270110077187138775196900 1180 702635588524014320 48
29629176560 1586878942101888360258625080
18741545784831997880308784340 135413180018248
103066257550962737720 410896226494013260
638728478116949861246791167518480580 183696788896587421699032600
52 10485980 25003993164540 7213860 136299971388 46400
19320788325040337217824455505160 129748968980 43276943002540
13235458622462202510640 314136625452886403879740 16665383182260
7377009103065498851032020 4502963741200 3933131148
761624136944072910800 433896111669844912840
15701968936415353889062192632 34845956052079180
488830275367615376 3481871275306470280 456624468 18360
9713562669309460 6219237759214762827187409503019432615976684540
2172772383489805842331071281808456 145754861076386671221320
5243626382156167474921782674260
22722526912283010072320240710785462723519145740 19639620
26678547908792 22362336179089374248695940740
2629972211566463612149241455626172190220 1680
618516849717028248841100388607776 8724388547388468773672620
42328623757159480 3578536376412875571438541467679723838177648640
286231896207756 43761444552128 26346740
66755254524262858038831090292970460

And here's A081232 (under the same assumption):

9 649 33 9801 73 2049 66249 1766319049 2281249 500001 62809633
201 158070671986249 1204353 6083073 25801741449 46698728731849
2499849 2469645423824185801 6224323426849 393 5848201 1072400673
10085143557001249 513 13449 159150073798980475849 2262200630049
12320649 32188120829134849 248678907849 2063810353129713793
169648201 10157115393 52387849 3287049 838721786045180184649 801
25052977273092427986049 3879474045914926879468217167061449
104564907854286695713 71798771299708449 6983244756398928218113
1182351890184201 313201220822405001 32961431500035201
3707453360023867028800645599667005001 27849 16760473211643448449
1153 721517598849 38902815462492318420311478049
464018873584078278910994299849 3363593612801313
2609429220845977814049 10499986568677299849
16421658242965910275055840472270471049
4765506835465395993032041249 1353 277631049 665782673992201
195307849 3750107388553 1280001
535781868388881310859702308423201 3607394696649 1221759532448649
376455160998025676163201 9000987377460935993101449
479835713751049 215454135724113414336120649 131822292741249
116476476553 22606256615916825861249 13224937103288377430049
480644425002415999597113107233 1068924905989944201
15090531843660371073 108832847723078562849 14418057673 583201
309159979019115849
198723867690977573219668252231077415636351801801
69833593956354694952807071952492833 4720747718766511296768801
170800615664635332564330517881801
742925865816843150858935268959512942700219559049 649296649
883619927128353 744702728637239515399291224201
87897747594260774254246835664214545379849 56449
21002267655460685387089654909788673 299819549856198391714823049
1462023556679567649
124015777380821763117401108566632174583648616449
9968927282677513 1526641059203073 923640201
2347849777171254083668416002496336649

--
Don Reble       djr at nk.ca

MIME-Version: 1.0

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