A091380 Largest quadratic "mixed" residue modulo the n-th prime (LQxR(p_n)).

1, 1, 3, 4, 9, 11, 14, 17, 18, 27, 28, 35, 38, 41, 42, 51, 57, 59, 65, 76, 81, 86, 92, 99, 100, 105, 107, 110, 124, 129, 134, 137, 147, 148, 155, 161, 162, 171, 177, 179, 184, 188, 195, 196, 209, 220, 225, 227, 230, 232, 234, 249, 254, 258, 267, 268, 275, 278, 281

Offset

1,3

Comments

Due to the quadratic reciprocity (Euler's criterion), if a prime p is congruent to 1 mod 4, then (p-1) is a quadratic residue mod p (see A088190). Also, if p is congruent -1 mod 4 then p-1 is a quadratic non-residue mod p (see A088196). This sequence is created in such a way that when p is not congruent to 1 mod 4 then the largest quadratic residue is taken, otherwise the largest quadratic non-residue taken modulo p. Thus it is a merger of A088190 and A088196 by skipping the "trivial" terms. Important observations (tested up to 10^5 primes): - the sequence of largest "mixed" residues modulo the primes (denoted by LQxR(p_n)) is 'almost' monotonic, - for n>1, p_n-LQxR(p_n) is a prime value (see A091382) - if LQxR(p_n)<=LQxR(p_{n-1}) then p_n==+-1 mod 8 (when n>2) (see A091384) - if LQxR(p_n)<=LQxR(p_{n-1}) then p_n-LQxR(p_n) is a prime q>5 (see A091385).

References

H. Cohn, Advanced Number Theory, p. 19, Dover Publishing (1962)

Links

Table of n, a(n) for n=1..59.

Ferenc Adorjan, The sequence of largest quadratic residues modulo the primes

Formula

a(1)=1; a(n>1)=max{r<p_n | -(r/p_n)=|p_n|_4}, where (r/p_n) is the Legendre symbol and |x|_m is the least absolute residue of x modulo m.

Prog

(PARI) {/* Sequence of the largest "mixed" QR modulo the primes */ lqxr(to)=local(v=[1], k, r, q); for(i=2, to, k=prime(i)-1; r=prime(i)%4-2; while(kronecker(k, prime(i))<>r, k-=1); v=concat(v, k)); print(v) }

Crossrefs

Cf. A088190, A088196, A091381, A091382, A091383, A091384, A091385.

Sequence in context: A227939 A047459 A177943 * A321871 A050006 A176744

Adjacent sequences: A091377 A091378 A091379 * A091381 A091382 A091383

Keyword

nonn,easy

Author

Ferenc Adorjan (fadorjan(AT)freemail.hu)

Status

approved