A181539 Smallest number m > 1 such that m^2 == 1 (mod 10^n).

9, 49, 249, 1249, 18751, 218751, 781249, 24218751, 74218751, 1425781249, 13574218751, 163574218751, 163574218751, 19836425781249, 19836425781249, 2480163574218751, 12519836425781249, 12519836425781249, 487480163574218751, 15487480163574218751, 215487480163574218751, 215487480163574218751

Offset

1,1

Comments

a(n) > 10^floor(n/2).

All terms have last digit 1 or 9.

Squares of terms are listed in A085877.

Decimal representation of each term is formed by the reverse concatenation of initial terms of either A063006 or A091661.

Except for 3, there are no solutions for n>1 and m^2 == -1 (mod 10^n). See comment in A063006 under extensions. - Robert G. Wilson v, Jan 26 2013

If a(n)<(10^n)/2 then (10^n-a(n))^2 is also congruent (modulo 10^n), its just not the least. - Robert G. Wilson v, Jan 26 2013

Links

Max Alekseyev, Table of n, a(n) for n = 1..1000

Formula

Let b(n) = A224474(n) (or equivalently b(n) = A224473(n)), then for n >= 3, there are eight solutions in [0,10^n) to x^2 == 1 (mod 10^n), namely x = 1, 5*10^(n-1) - 1, 5*10^(n-1) + 1, 10^n - 1, b(n), 10^n - b(n), |b(n) - 5*10^(n-1)|, and 10^n - |b(n) - 5*10^(n-1)|, so a(n) = min{b(n), |b(n) - 5*10^(n-1)|, 10^n - b(n)} < 25*10^(n-2). - Jianing Song, Sep 23 2024

Example

1249^2 = 1560001 == 1 (mod 10^4), and there is no smaller m > 1 such that m^2 == 1 (mod 10^4). Hence a(4) = 1249.

Prog

(PARI) install(Zn_quad_roots, GGG);
a181539(n) = vecsort(Zn_quad_roots(10^n, 0, -1)[2])[2]; \\ Max Alekseyev, Oct 13 2024

Crossrefs

Cf. A085610, A181607.

Cf. A063006, A091661 (the two nontrivial 10-adic square roots of 1).

Cf. A224473, A224474 (approximation of the two nontrivial 10-adic square roots of 1 up to powers of 10).

Sequence in context: A192814 A228018 A081655 * A224473 A146798 A055428

Adjacent sequences: A181536 A181537 A181538 * A181540 A181541 A181542

Keyword

nonn

Author

Kevin Batista (kevin762401(AT)yahoo.com), Oct 29 2010

Extensions

a(2) through a(4), a(7) through a(11) corrected, comment added, example replaced by Klaus Brockhaus, Nov 01 2010

Edited by N. J. A. Sloane, Oct 29 2010, Nov 09 2010

Definition to avoid the constant sequence a(n)=1 constrained by R. J. Mathar, Nov 18 2010

a(1) corrected, terms a(13) onward added by Max Alekseyev, Dec 10 2012

Status

approved