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

Author

Charles R Greathouse IV, Aug 01 2013

Keywords

• nonn

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.

Crossrefs

Cf. A192453, A227781.

Programs

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);new=bitor(new,t));k++;cur=new);k

Formula

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

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