0,2

This is the root congruent to 2 mod 5.

Or, residues modulo 5^n of a 5-adic solution of x^2+1=0.

The radix-5 expansion of a(n) is obtained from the n rightmost digits in the expansion of the following pentadic integer:

...422331102414131141421404340423140223032431212 = u

The residues modulo 5^n of the other 5-adic solution of x^2+1=0 are given by A048899 which corresponds to the pentadic integer -u:

...022113342030313303023040104021304221412013233 = -u

J. H. Conway, The Sensual Quadratic Form, p. 118.

K. Mahler, Introduction to p-Adic Numbers and Their Functions, Cambridge, 1973, p. 35.

G. P. Michon, On the witnesses of a composite integer

G. P. Michon, Introduction to p-adic integers

If n>0, a(n) = 5^n-A048899(n)

a(0)=0 because 0 satisfies any equation in integers modulo 1.

a(1)=2 because 2 is one solution of x^2+1=0 modulo 5. (The other solution is 3, which gives rise to A048899.)

a(2)=7 because the equation (5y+a(1))^2+1=0 modulo 25 means that y is 1 modulo 5.

(PARI) a(n)=if(n<2, 2, a(n-1)^5)%5^n

Cf. A000351 (powers of 5), A048899 (the other pentadic number whose square is -1), A034939(n)=Min(a(n), A048899(n)).

Cf. A034935. Different from A034935.

Sequence in context: A079410 A002658 A034939 this_sequence A034935 A121079 A105183

Adjacent sequences: A048895 A048896 A048897 this_sequence A048899 A048900 A048901

nonn,easy,nice

Michael Somos

Additional comments from Gerard P. Michon (g.michon(AT)att.net), Jul 15 2009

Edited by N. J. A. Sloane, Jul 25 2009