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A091380
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Largest quadratic "mixed" residue modulo the n-th prime (LQxR(p_n)).
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5
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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
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OFFSET
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1,3
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COMMENTS
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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).
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REFERENCES
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H. Cohn, Advanced Number Theory, p. 19, Dover Publishing (1962)
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LINKS
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FORMULA
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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.
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PROG
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(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) }
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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Ferenc Adorjan (fadorjan(AT)freemail.hu)
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STATUS
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approved
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