A218710
a(n) is smallest number such that a(n)^2 + 1 is divisible by 17^n.
Original entry on oeis.org
0, 4, 38, 1985, 27493, 390112, 390112, 96940388, 3379649772, 24306922095, 450044583893, 5597937117454, 28673959190179, 3524407382568745, 13428985415474682, 13428985415474682, 5711417117604156904, 91610966633729580058, 6709533061724423693474
Offset: 0
a(4) = 27493 because 27493^2+1 = 2 * 5 ^ 2 * 17 ^ 4 * 181.
-
b=4;n17=17;jo=Join[{0,b},Table[n17=17*n17;b=PowerMod[b,17,n17];b=Min[b,n17-b],{99}]]
A218712
a(n) is the smallest number such that a(n)^2 + 1 is divisible by 29^n.
Original entry on oeis.org
0, 12, 41, 10133, 34522, 7745569, 253879357, 7986582530, 61012922706, 4563230639355, 67972499239990, 1330094199140593, 47471944863682723, 5000878909740249297, 5000878909740249297, 590115586441858677665, 77072583141941801290876, 423420364218752896284166
Offset: 0
a(4) = 34522 because 34522^2+1 = 5 * 29 ^ 4 * 337.
-
R:= 0,12: U:= [12,17]:
for n from 2 to 30 do
qs:= map(u -> (u^2+1)/29^(n-1), U);
ys:= [seq(-qs[i]/(2*U[i]) mod 29,i=1..2)];
U:= U + ys*29^(n-1) mod 29^n;
R:= R,min(U);
od:
R; # Robert Israel, Jan 13 2025
-
b=12;n29=29;jo=Join[{0,b},Table[n29=29*n29;b=PowerMod[b,29,n29];b=Min[b,n29-b],{99}]]
A218713
a(n) is smallest number such that a(n)^2 + 1 is divisible by 37^n.
Original entry on oeis.org
0, 6, 117, 9466, 800982, 6423465, 756360062, 24900904028, 1019349744435, 15069267560119, 794839706330581, 71333925879937154, 2419512779032508628, 116073623326088126430, 359642847542169431827, 144552623583462302226851, 3523356323886506075746572
Offset: 0
a(3) = 9466 because 9466^2+1 = 29 * 37 ^ 3 * 61.
-
b=6;n37=37;jo=Join[{0,b},Table[n37=37*n37;b=PowerMod[b,37,n37];b=Min[b,n37-b],{99}]]
A218714
a(n) is smallest number such that a(n)^2 + 1 is divisible by 41^n.
Original entry on oeis.org
0, 9, 378, 11389, 1251967, 46464143, 2363588163, 92615568742, 287369842623, 112076323050317, 2403749863808044, 56094387104417648, 1156752450536914530, 43970228150195457632, 10132163897314954464899, 503212117431217218892992, 19164391897329672149556204
Offset: 0
a(3) = 11389 because 11389^2+1 = 2 * 41 ^ 3 * 941.
-
b=9;n41=41;jo=Join[{0,b},Table[n41=41*n41;b=PowerMod[b,41,n41];b=Min[b,n41-b],{99}]]
A218715
a(n) is smallest number such that a(n)^2 + 1 is divisible by 53^n.
Original entry on oeis.org
0, 23, 500, 27590, 623098, 23048345, 5041394261, 416081467190, 11331029931180, 50928660480181, 6548598523124085, 2441875986594058601, 76594163421571591377, 7783548304686046882879, 252583670951378815076851, 4392422457122810120236558, 1165802007767335105471573954
Offset: 0
a(3) = 27590 because 27590^2+1 = 53 ^ 3 * 5113.
-
b=23;n53=53;jo=Join[{0,b},Table[n53=53*n53;b=PowerMod[b,53,n53];b=Min[b,n53-b],{99}]]
A218716
a(n) is smallest number such that a(n)^2 + 1 is divisible by 61^n.
Original entry on oeis.org
0, 11, 682, 51412, 6304056, 144762466, 9435321777, 988322434636, 71294762793847, 3138611770750343, 283798117998769727, 15409745938584647495, 320007169218635518122, 45443939732277600209579, 207359227164430355867160, 59053635973003478214807486
Offset: 0
a(3) = 51412 because 51412^2+1 = 5 * 17 * 61 ^ 3 * 137.
-
b=11;n61=61;jo=Join[{0,b},Table[n61=61*n61;b=PowerMod[b,61,n61];b=Min[b,n61-b],{99}]]
A218717
a(n) is smallest number such that a(n)^2 + 1 is divisible by 73^n.
Original entry on oeis.org
0, 27, 776, 153765, 6459524, 404034898, 41865466758, 3219884218827, 239822883201307, 9110883894036198, 991706090146518323, 142813358470363920740, 8641533837443707913816, 586811715371303018585730, 2756887299416274753296336, 729513196939063257288876118
Offset: 0
a(3) = 153765 because 153765^2+1 = 2 * 73 ^ 3 * 30389.
-
b=27;n73=73;jo=Join[{0,b},Table[n73=73*n73;b=PowerMod[b,73,n73];b=Min[b,n73-b],{99}]]
A034944
Successive approximations to 13-adic integer sqrt(-1).
Original entry on oeis.org
0, 5, 70, 239, 143044, 1999509, 6826318, 822557039, 85658552023, 1188526486815, 11941488851037, 291518510320809, 2108769149874327, 13920898306972194, 2675587335039691558, 63228498770709057089
Offset: 0
- K. Mahler, Introduction to p-Adic Numbers and Their Functions, Cambridge, 1973, p. 35.
-
seq(n)={my(v=vector(n), i=1, k=0); while(i<#v, k++; my(t=truncate(sqrt(-1 + O(13^k)))); if(t > v[i], i++; v[i]=t)); v} \\ Andrew Howroyd, Nov 10 2018
A218718
a(n) is smallest number such that a(n)^2 + 1 is divisible by 89^n.
Original entry on oeis.org
0, 34, 3861, 344464, 20099637, 2153335831, 102666405913, 4867146503697, 923990886302412, 50251663587824641, 5655954122907587985, 909925832091926912414, 85120439454684773642745, 2631773999763198769695986, 41332517834853462204330752
Offset: 0
a(3) = 344464 because 344464^2+1 = 37 * 89 ^ 3 * 4549.
Cf.
A002522,
A049532,
A034939,
A218709,
A218710,
A218712,
A218713,
A218714,
A218715,
A218716,
A218717.
-
b=34;n89=89;jo=Join[{0,b},Table[n89=89*n89;b=PowerMod[b, 89,n89];b=Min[b,n89-b],{99}]]
A218719
a(n) is smallest number such that a(n)^2 + 1 is divisible by 97^n.
Original entry on oeis.org
0, 22, 4052, 107551, 22709274, 331407850, 197177418061, 26457926739667, 2369608176604944, 76004727767164666, 25163629663367816827, 1965881512952938486496, 191165497320828772935835, 21700278688179406782082106, 560121950820639295011033922
Offset: 0
a(3) = 107551 because 107551^2+1 = 2 * 97 ^ 3 * 6337.
Cf.
A002522,
A049532,
A034939,
A218709,
A218710,
A218712,
A218713,
A218714,
A218715,
A218716,
A218717,
A218718.
-
b=22;n97=97;jo=Join[{0,b},Table[n97=97*n97;b=PowerMod[b, 97,n97];b=Min[b,n97-b],{99}]]
Showing 1-10 of 11 results.