A053760 Smallest positive quadratic nonresidue modulo p, where p is the n-th prime.

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

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.

Hugh 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.

Paulo 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

Robert Baillie and Samuel S. Wagstaff, Lucas pseudoprimes, Mathematics of Computation, Vol. 35, No. 152 (1980), pp. 1391-1417, Math. Rev. 81j:10005, alternative link.

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

Steven R. Finch, Quadratic Residues [Broken link]

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

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

Enrique Treviño, The least k-th power non-residue, Journal of Number Theory, Vol. 149 (2015),pp. 201-224, alternative link.

Eric Weisstein's World of Mathematics, Quadratic Nonresidue.

Formula

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

Asymptotic mean: lim_{n->oo} (1/n) * Sum_{k=1..n} a(k) = A098990 (Erdős, 1961). - Amiram Eldar, Oct 29 2020

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, A098990.

Sequence in context: A257572 A343902 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