A227784 Least number of fourth powers which add to -1 mod n.

0, 1, 2, 3, 4, 2, 2, 7, 2, 4, 2, 3, 2, 2, 4, 15, 1, 2, 2, 4, 2, 2, 2, 7, 4, 2, 2, 3, 3, 4, 2, 15, 2, 1, 4, 3, 2, 2, 2, 7, 1, 2, 2, 3, 4, 2, 2, 15, 2, 4, 2, 3, 2, 2, 4, 7, 2, 3, 2, 4, 2, 2, 2, 15, 4, 2, 2, 3, 2, 4, 2, 7, 1, 2, 4, 3, 2, 2, 2, 15, 2, 1, 2, 3, 4, 2, 3, 7, 1, 4, 2

Offset

1,3

Comments

Parnami, Agrawal, & Rajwade proved (1981, Theorem 1) that, for a prime p > 29, a(p) = 1 if p = 1 mod 8 and otherwise a(p) = 2.

Conjecture: a(n) = 15 if n = 9 mod 16 and a(n) = 7 if n = 8 mod 16, otherwise a(n) <= 4. (The associated lower bounds are obvious.)

References

J. C. Parnami, M. K. Agrawal, and A. R. Rajwade, On the 4-power Stufe of a field, Rendiconti del Circolo Matematico di Palermo (2) 30:2 (1981), pp. 245-254.

Links

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

J. C. Parnami, M. K. Agrawal, and A. R. Rajwade, On the 4-power Stufe of p-adic completions of algebraic number fields, Rendiconti del Seminario Matematico Università e Politecnico di Torino 44:1 (1986), pp. 141-153.

Formula

a(n) <= A002377(n-1) <= 19.

a(n) = 1 if and only if n > 1 is in A192453.

Prog

(PARI) a(n)=if(n==1, return(0)); if(n>29 && isprime(n), return(if(n%8>1, 2, 1))); my(N, cur, new, k=1); for(i=1, n\2, cur=N=bitor(1<<(i^4%n), N)); while(!bittest(cur, n-1), new=0; for(i=1, n\2, t=cur<<(i^4%n); t=bitor(bitand(t, (1<<n)-1), t>>n); new=bitor(new, t)); k++; cur=new); k

Crossrefs

Cf. A192453, A227781.

Sequence in context: A162247 A264809 A035578 * A256445 A275103 A359729

Adjacent sequences: A227781 A227782 A227783 * A227785 A227786 A227787

Keyword

nonn

Author

Charles R Greathouse IV, Aug 01 2013

Status

approved