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A053760 Smallest positive quadratic nonresidue modulo p, where p is the n-th prime. 21
2, 2, 2, 3, 2, 2, 3, 2, 5, 2, 3, 2, 3, 2, 5, 2, 2, 2, 2, 7, 5, 3, 2, 3, 5, 2, 3, 2, 2, 3, 3, 2, 3, 2, 2, 3, 2, 2, 5, 2, 2, 2, 7, 5, 2, 3, 2, 3, 2, 2, 3, 7, 7, 2, 3, 5, 2, 3, 2, 3, 2, 2, 2, 11, 5, 2, 2, 5, 2, 2, 3, 7, 3, 2, 2, 5, 2, 2, 3, 7, 2, 2, 7, 5, 3, 2, 3, 5, 2, 3, 2, 13, 3, 2, 2, 5, 2, 3, 2, 2, 2, 2, 2 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

Assuming the Generalized Riemann Hypothesis, Montgomery proved a(n) << (log p(n))^2, meaning that there is a constant c such that |a(n)| <= c*(log p(n))^2. - Jonathan Vos Post, Jan 06 2007

a(n) < 1 + sqrt(p), where p is the n-th prime (Theorem 3.9 in Niven, Zuckerman, and Montgomery). - Jonathan Sondow, May 13 2010

Treviño proves that a(n) < 1.1 p^(1/4) log p for n > 2 where p is the n-th prime. - Charles R Greathouse IV, Dec 06 2012

a(n) is always a prime, because if x*y is a nonresidue, then x or y must also be a nonresidue. - Jonathan Sondow, May 02 2013

a(n) is the smallest prime q such that the congruence x^2 == q (mod p) has no solution 0 < x < p, where p = prime(n). For n > 1, a(n) is the smallest base b such that b^((p-1)/2) == -1 (mod p), where odd p = prime(n). - Thomas Ordowski, Apr 24 2019

REFERENCES

Steven R. Finch, Mathematical Constants, Cambridge, 2003, pp. 94-98.

H. L. Montgomery, Topics in Multiplicative Number Theory, 3rd ed., Lecture Notes in Mathematics, Vol. 227 (1971), MR 49:2616.

Ivan Niven, Herbert S. Zuckerman, and Hugh L. Montgomery, An Introduction to the Theory Of Numbers, Fifth Edition, John Wiley and Sons, Inc., NY 1991, p. 147.

P. Ribenboim, The New Book of Prime Number Records, 3rd ed., Springer-Verlag 1996; Math. Rev. 96k:11112.

LINKS

T. D. Noe, Table of n, a(n) for n = 1..10000

R. Baillie and S. S. Wagstaff, Lucas pseudoprimes, Math. Comp. 35 (1980) 1391-1417; Math. Rev. 81j:10005.

P. Erdős, Remarks on number theory. I., Mat. Lapok 12 (1961) 10-17; Math. Rev. 26 #2410.

Steven R. Finch, Quadratic Residues [Broken link]

Steven R. Finch, Quadratic Residues [From the Wayback machine]

K. Matthews, Finding n(p), the least quadratic non-residue (mod p)

Enrique Treviño, The least k-th power non-residue, 2011 preprint; J. Number Theory, 149, 201-224 (2015).

Eric Weisstein's World of Mathematics, Quadratic Nonresidue

FORMULA

a(n) = A020649(prime(n)) for n > 1. - Thomas Ordowski, Apr 24 2019

EXAMPLE

The 5th prime is 11, and the positive quadratic residues mod 11 are 1^2 = 1, 2^2 = 4, 3^2 = 9, 4^2 = 5 and 5^2 = 3. Since 2 is missing, a(5) = 2.

The only positive quadratic redidue mod 2 is 1, so a(1)=2.

MATHEMATICA

Table[ p = Prime[n]; First[ Select[ Range[p], JacobiSymbol[#, p] != 1 &]], {n, 1, 100}] (* Jonathan Sondow, Mar 03 2013 *)

PROG

(PARI) residue(n, m)={local(r); r=0; for(i=0, floor(m/2), if(i^2%m==n, r=1)); r

A053760(n)={local(r, m); r=0; m=0; while(r==0, m=m+1; if(!residue(m, prime(n)), r=1)); m} \\ Michael B. Porter, May 02 2010

(PARI) qnr(p)=my(m); while(1, if(!issquare(Mod(m++, p)), return(m)))

a(n)=if(n>1, qnr(prime(n)), 2) \\ Charles R Greathouse IV, Feb 27 2013

CROSSREFS

Cf. A000229, A020649.

Sequence in context: A258570 A257572 A160493 * A223942 A278597 A138789

Adjacent sequences:  A053757 A053758 A053759 * A053761 A053762 A053763

KEYWORD

nonn

AUTHOR

Steven Finch, Apr 05 2000

EXTENSIONS

More terms from James A. Sellers, Apr 08 2000

STATUS

approved

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Last modified November 17 16:03 EST 2019. Contains 329238 sequences. (Running on oeis4.)