A147520 a(n) = Smallest number x such that Euler Polynomial x^2 + x + 41 is divisible by 41^n.

0, 40, 1721, 139563, 14268368, 1636255182, 6386359423, 1953929098233, 149759650255065, 1814531956108700, 243422399538851918, 9662500171353620019, 122479951673184550424, 12148820281768361731597, 177497315692809432279207, 14173382150616650630276616, 1225594969529024683212496795

Offset: 1

Author

Artur Jasinski, Nov 06 2008

Keywords

• nonn

Comments

For values of x^2 + x + 41, see A147521. For values (x^2 + x + 41)/(41^n), see A147522.

By Hensel's lemma, x^2 + x + 41 has two roots mod 41^n; their sum == -1 mod 41^n. Thus 0 <= a(n) < 41^n/2. - Robert Israel, Apr 09 2018

Links

• Robert Israel, Table of n, a(n) for n = 1..620

Crossrefs

Cf. A145292, A145293, A145294, A147520, A147521, A147522.

Programs

Maple

f:= n -> min(map(t -> rhs(op(t)), [msolve(x^2+x+41, 41^n)])):
map(f, [$1..30]); # Robert Israel, Apr 09 2018

Mathematica

a = {}; Do[x = 0; While[Mod[x^2 + x + 41, 41^n] != 0, x++ ]; AppendTo[a,x];Print[{n, x, x^2 + x + 41, (x^2 + x + 41)/41^n}], {n, 1, 6}];a (* Artur Jasinski *)

Extensions

More terms from Robert Israel, Apr 09 2018