A036134 a(n) = 3^n mod 79.

1, 3, 9, 27, 2, 6, 18, 54, 4, 12, 36, 29, 8, 24, 72, 58, 16, 48, 65, 37, 32, 17, 51, 74, 64, 34, 23, 69, 49, 68, 46, 59, 19, 57, 13, 39, 38, 35, 26, 78, 76, 70, 52, 77, 73, 61, 25, 75, 67, 43, 50, 71, 55, 7, 21, 63, 31, 14

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Comments

Because a(39) = 78, the Legendre symbol (3/79) = -1, meaning that 3 is not a quadratic residue of 79. Furthermore, it means that 3 is prime in Z[sqrt(79)]. - Alonso del Arte, Oct 01 2012

Examples

			a(4) = 2 because 3^4 = 81 and 81 - 79 = 2.
		

References

• I. M. Vinogradov, Elements of Number Theory, pp. 220 ff.

Links

• G. C. Greubel, Table of n, a(n) for n = 0..10000
• Index entries for linear recurrences with constant coefficients, signature (1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0, 0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1).

Crossrefs

Cf. A000244 (3^n).

Programs

GAP

List([0..60],n->PowerMod(3,n,79)); # Muniru A Asiru, Oct 17 2018

Magma

[Modexp(3, n, 79): n in [0..100]]; // G. C. Greubel, Oct 17 2018

Maple

[ seq(primroot(ithprime(i))^j mod ithprime(i),j=0..100) ];

Mathematica

Table[Mod[3^n, 79], {n, 0, 60}] (* Alonso del Arte, Oct 01 2012 *)
PowerMod[3,Range[0,100],79] (* Harvey P. Dale, Feb 21 2024 *)

PARI

a(n)=lift(Mod(3,79)^n) \\ Charles R Greathouse IV, Mar 22 2016

Python

for n in range(0, 100): print(int(pow(3, n, 79)), end=' ') # Stefano Spezia, Oct 17 2018

Formula

From G. C. Greubel, Oct 17 2018: (Start)

a(n) = a(n-1) - a(n-39) + a(n-40).

a(n+78) = a(n). (End)