

A053760


Smallest positive quadratic nonresidue modulo p, where p is the nth prime.


23



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
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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 nth 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 nth 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^((p1)/2) == 1 (mod p), where odd p = prime(n).  Thomas Ordowski, Apr 24 2019


REFERENCES

Steven R. Finch, Mathematical Constants, Cambridge, 2003, pp. 9498.
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., SpringerVerlag 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. 13911417, Math. Rev. 81j:10005, alternative link.
Paul Erdős, Remarks on number theory. I., Mat. Lapok, Vol. 12 (1961), pp. 1017; 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 nonresidue (mod p)
Enrique Treviño, The least kth power nonresidue, Journal of Number Theory, Vol. 149 (2015),pp. 201224, 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: 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



