A088190 -id:A088190 - cp's OEIS frontend

A088191 First differences of A088190.

0, 3, 0, 5, 3, 4, 1, 1, 10, 0, 8, 4, 1, 1, 10, 5, 3, 5, -1, 8, 4, 5, 7, 8, 4, 0, 5, 3, 4, 12, 5, 7, 1, 11, 0, 8, 5, 1, 10, 5, 3, 4, 8, 4, 0, 13, 11, 5, 3, 4, 0, 8, 9, 7, 2, 10, 0, 8, 4, 1, 11, 13, -5, 12, 4, 13, 7, 9, 3, 4, 0, 12, 8, 5, 1, 10, 8, 4, 8, 9, 3, 4, 8, 4, 5, 7, 8, 4, 0, 5, 1, 18, 5, 8, 1, 10, 12, 1, 19

Offset: 1

Ferenc Adorjan (fadorjan(AT)freemail.hu), Sep 22 2003

Seemingly the difference sequence is mostly positive. There are special characteristic features where it is nonpositive. (See A088193-A088195.)

Cf. A088190, A088192, A088193, A088194, A088195.

qrp_d(to)= {/* Difference sequence of the largest QR modulo the primes */ local(m,k=1,p,v=[]); for(i=2,to,m=1; p=prime(i); j=2; while((j<=(p-1)/2)&&(m

A088193 Prime numbers where the sequence of largest quadratic residues modulo the primes (A088190) is non-monotonic.

Original entry on oeis.org

3, 7, 31, 71, 103, 151, 199, 239, 271, 311, 359, 463, 599, 719, 823, 839, 911, 1063, 1231, 1279, 1303, 1439, 1559, 1871, 1879, 1951, 1999, 2143, 2239, 2311, 2351, 2383, 2399, 2551, 2711, 2791, 3191, 3391, 3463, 3559, 3583, 3823, 3911, 3919, 4079, 4159

Offset: 1

Ferenc Adorjan (fadorjan(AT)freemail.hu), Sep 22 2003

From the second term on, these primes are always ==7 mod 8. (Tested for the first 20000 primes)
From Robert Israel, Oct 31 2024: (Start)
This is true because if prime(n) == 1 mod 4, A088190(n) = prime(n) - 1 while if prime(n) == 3 mod 8, A088190(n) = prime(n) - 2.  In either case, A088190(n) > prime(n-1) - 1 >= A088190(n-1).
Primes prime(n) such that A088190(n) <= A088190(n-1). (End)

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A088190, A088191, A088192, A088194, A088195.

lqr:= proc(p) local k;
  for k from p-1 by -1 do if numtheory:-quadres(k,p) = 1 then return k fi od:
end proc:
p:= 2: v:= lqr(2): R:= NULL: count:= 0:
while count < 100 do
  q:= p; vq:= v; p:= nextprime(p); v:= lqr(p);
  if v <= vq then R:= R,p; count:= count+1;
  fi
od:
R; # Robert Israel, Oct 31 2024

qrp_p_nm(to)= {/* The primes where the sequence of the largest QR modulo the primes is non-monotonic */ local(m,k=1,p,v=[]); for(i=2,to,m=1; p=prime(i); j=2; while((j<=(p-1)/2)&&(m

A088195 Distance (A088192) of primes from the largest quadratic residues modulo the primes (A088190), where the latter is non-monotonic.

Original entry on oeis.org

3, 3, 3, 7, 3, 3, 3, 7, 3, 11, 7, 3, 7, 11, 3, 11, 7, 3, 3, 3, 3, 7, 17, 7, 3, 3, 3, 3, 3, 3, 13, 3, 11, 3, 7, 3, 11, 3, 3, 3, 3, 3, 13, 3, 11, 3, 3, 3, 3, 3, 11, 7, 11, 13, 3, 7, 7, 11, 7, 3, 3, 11, 19, 3, 11, 3, 3, 11, 17, 3, 11, 3, 7, 3, 13, 3, 3, 3, 3, 11, 11, 3, 3, 3, 3, 13, 19, 3, 3, 3, 7, 11

Offset: 1

Ferenc Adorjan (fadorjan(AT)freemail.hu), Sep 22 2003

The values are some odd primes, but never 5. The maximum value increases very slowly, it only reaches 31 for the first 20000 primes.
It is conjectured that if we denote the members of A088194 by D(n) and the member of this sequence by M(n) then if D(n)=-1 then M(n)=7, while if M(n)=3 then D(n)=0.
The values are odd primes, but never 5 (the primality is provable). The maximum value increases very slowly: it only reaches 43 for the first 10^5 primes.

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

Cf. A088190, A088191, A088192, A088193, A088194.

qrp_pm_nm(to)= {/* The distance of LQR from the primes where the sequence of the largest QR modulo the primes is non-monotonic */ local(m,k=1,p,v=[]); for(i=2,to,m=1; p=prime(i); j=2; while((j<=(p-1)/2)&&(m

A088194 Members of the difference sequence (A088191) of the sequence of largest quadratic residues modulo the primes (A088190), where the latter is non-monotonic.

Original entry on oeis.org

0, 0, 0, -1, 0, 0, 0, 0, 0, -5, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, -10, -1, 0, 0, 0, 0, 0, 0, -7, 0, -4, 0, -1, 0, -5, 0, 0, 0, 0, 0, -7, 0, -4, 0, 0, 0, 0, 0, -4, 0, 0, -2, 0, -1, 0, -5, 0, 0, 0, 0, -8, 0, 0, 0, 0, -4, -11, 0, 0, 0, -1, 0, -2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -2, -5, 0, 0, 0, 0, -5, 0, 0, -5, 0, 0, 0, -1

Offset: 1

Ferenc Adorjan (fadorjan(AT)freemail.hu), Sep 22 2003

The negative values are either primes or composites (Cf. A088200).

Cf. A088190, A088191, A088192, A088193, A088195.

qrp_d_nm(to)= {/* The difference sequence values where the sequence of the largest QR modulo the primes is non-monotonic */ local(m,k=1,p,v=[]); for(i=2,to,m=1; p=prime(i); j=2; while((j<=(p-1)/2)&&(m

A088192 Distance between prime(n) and the largest quadratic residue modulo prime(n).

Original entry on oeis.org

1, 2, 1, 3, 2, 1, 1, 2, 5, 1, 3, 1, 1, 2, 5, 1, 2, 1, 2, 7, 1, 3, 2, 1, 1, 1, 3, 2, 1, 1, 3, 2, 1, 2, 1, 3, 1, 2, 5, 1, 2, 1, 7, 1, 1, 3, 2, 3, 2, 1, 1, 7, 1, 2, 1, 5, 1, 3, 1, 1, 2, 1, 2, 11, 1, 1, 2, 1, 2, 1, 1, 7, 3, 1, 2, 5, 1, 1, 1, 1, 2, 1, 7, 1, 3, 2, 1, 1, 1, 3, 2, 13, 3, 2, 2, 5, 1, 1, 2, 1

Offset: 1

Ferenc Adorjan (fadorjan(AT)freemail.hu), Sep 22 2003

a(n) = smallest m>0 such that -m is a quadratic residue modulo prime(n).
a(n) = smallest m>0 such that prime(n) either splits or ramifies in the imaginary quadratic field Q(sqrt(-m)). Equals -A220862(n) except when n = 1. Cf. A220861, A220863. - N. J. A. Sloane, Dec 26 2012
The values are 1 or a prime number (easily provable!). The maximum occurring prime values increase very slowly: up to 10^5 terms the largest prime is 43. The primes do not appear in order.

David A. Cox, "Primes of the Form x^2 + n y^2", Wiley, 1989, Cor. 5.17, p. 105. - From N. J. A. Sloane, Dec 26 2012

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
Ferenc Adorjan, The sequence of largest quadratic residues modulo the primes.
J. A. Bergstra, I. Bethke, A negative result on algebraic specifications of the meadow of rational numbers, arXiv preprint arXiv:1507.00548 [math.RA], 2015-2016.

Records are (essentially) given by A147971.

Cf. A088190, A088191, A088193, A088194, A088195, A220861, A220862, A220863.

a[n_] := With[{p = Prime[n]}, If[JacobiSymbol[-1, p] > 0, 1, For[d = 2, True, d = NextPrime[d], If[JacobiSymbol[-d, p] >= 0, Return[d]]]]]; Array[a, 100] (* Jean-François Alcover, Feb 16 2018, after Charles R Greathouse IV *)

qrp_pm(fr,to)= {/* The distance of largest QR modulo the primes from the primes */ local(m,p,v=[]); for(i=fr,to,m=1; p=prime(i); j=2; while((j<=(p-1)/2)&&(m

do(p)=if(kronecker(-1,p)>0, 1, forprime(d=2, p, if(kronecker(-d, p) >= 0, return(d))))
apply(do, primes(100)) \\ Charles R Greathouse IV, Oct 31 2012

a(n) = A053760(n) unless -1 is a quadratic residue mod prime(n). - Charles R Greathouse IV, Oct 31 2012

Edited by Max Alekseyev, Oct 29 2012

A088196 Largest number that is not a quadratic residue modulo prime(n).

Original entry on oeis.org

2, 3, 6, 10, 11, 14, 18, 22, 27, 30, 35, 38, 42, 46, 51, 58, 59, 66, 70, 68, 78, 82, 86, 92, 99, 102, 106, 107, 110, 126, 130, 134, 138, 147, 150, 155, 162, 166, 171, 178, 179, 190, 188, 195, 198, 210, 222, 226, 227, 230, 238, 234, 250, 254, 262, 267, 270, 275, 278

Offset: 2

Ferenc Adorjan (fadorjan(AT)freemail.hu), Sep 23 2003

These are sometimes called quadratic non-residues modulo p(n). Denote a(n) by LQnR(p_n).

Chris Caldwell: The Prime Pages Quadratic Residues.
F. Adorjan, The sequence of largest quadratic residues modulo the primes.

Cf. A088190, A088197, A088198, A088199, A088200, A088201.

qnrp(fr,n)= {/* The largest QnR modulo the primes */ local(m,p,fl,jj,j,v=[]); fr=max(fr,2); for(i=fr,n,m=0; p=prime(i); jj=0; fl=2^p-1; j=2; while((j<=(p-1)/2),jj=(j^2)%p; fl-=2^jj; j++); j=p-1; while(m==0,if(bitand(2^j,fl),m=j); j--); v=concat(v,m)); print(v)}

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

Original entry on oeis.org

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

Ferenc Adorjan (fadorjan(AT)freemail.hu)

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).

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

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

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

{/* 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) }

a(1)=1; a(n>1)=max{r

A091383 Prime numbers where the sequence of largest quadratic "mixed" residues modulo the primes (A091380) is non-monotonic.

Original entry on oeis.org

3, 7, 31, 71, 103, 151, 199, 239, 271, 311, 359, 463, 599, 719, 823, 839, 911, 1063, 1231, 1279, 1303, 1439, 1559, 1871, 1879, 1951, 1999, 2143, 2239, 2311, 2351, 2383, 2399, 2551, 2711, 2791, 3191, 3391, 3463, 3559, 3583, 3823, 3911, 3919, 4079, 4159

Offset: 1

Ferenc Adorjan (fadorjan(AT)freemail.hu)

All of these primes belong to the +-1 least absolute reside classes modulo 8. (Tested for 10^5 primes.)

Where does this first differ from A088193 (if at all)? - R. J. Mathar, Aug 27 2025

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

Cf. A091380, A091381, A091382, A091384, A091385, A088190, A088193, A088199.

{/* The primes where the sequence of the largest "mixed" QR modulo the primes is non-monotonic */ lqxr_nm_p(to)=local(v=[],k,r,q,p,e=1,n=0,i=1); while(nr,k-=1); if(k-e<=0,v=concat(v,p);n+=1);e=k); print(i);print(v) }

A091384 Members of the difference sequence (A091381) of the sequence of largest quadratic "mixed" residues modulo the primes (A091380) where the latter is non-monotonic.

Original entry on oeis.org

0, -1, -5, -1, -3, 0, 0, -8, 0, -1, 0, -7, -2, -6, -1, 0, 0, -5, -6, 0, 0, -7, -2, 0, -2, -3, -1, 0, -1, -5, -5, -7, -2, -11, 0, -1, 0, 0, -1, -2, -10, 0, 0, -6, -3, -1, -5, -5, -6, -5, 0, -1, -5, -7, -2, -5, -1, -5, 0, -2, -2, -7, 0, -7, -9, -4, -4, -8, -5, -13, 0, -4, -4, -7, -17, 0, -3, 0, -5, -1, -3, 0, -17, 0, -7, -6, -1, -2, -3, -3, 0

Offset: 1

Ferenc Adorjan (fadorjan(AT)freemail.hu)

The negative values are either primes or composites (Cf. A088200).

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

Cf. A091380, A091381, A091382, A091383, A091385, A088190, A088196, A088200.

{/* The difference sequence values where the sequence of the largest "mixed" QR modulo the primes is non-monotonic */ lqxr_nm_d(to)=local(v=[],k,r,q,p,e=1,n=0,i=1); while(nr,k-=1); if(k-e<=0,v=concat(v,k-e);n+=1);e=k); print(i);print(v) }

A248222 Maximal gap between quadratic residues mod n.

Original entry on oeis.org

1, 1, 2, 3, 3, 2, 3, 4, 3, 3, 4, 5, 5, 3, 5, 7, 4, 3, 5, 7, 6, 4, 5, 8, 3, 5, 3, 7, 4, 5, 5, 8, 6, 4, 5, 9, 5, 5, 6, 11, 6, 6, 6, 8, 6, 5, 5, 12, 4, 3, 6, 8, 7, 3, 8, 9, 7, 4, 6, 11, 7, 5, 9, 8, 9, 6, 7, 13, 7, 5, 7, 12, 5, 5, 7, 8, 11, 6, 7, 15, 3, 6, 8, 12, 13, 6, 11, 16, 7, 6

Offset: 1

David W. Wilson and M. F. Hasler, Oct 04 2014

nonn

"Maximal gap between squares mod n" would be a less ambiguous definition.
The definition of quadratic residue modulo a nonprime varies from author to author. Sometimes, even when n is a prime, 0 is not counted as a quadratic residue. In this entry, an integer q is called a quadratic residue modulo n if it is congruent to a perfect square modulo n.
See A248376 for the variant with the additional restriction that the residue be coprime to the modulus. - M. F. Hasler, Oct 08 2014

			For n=7, the quadratic residues are all numbers congruent to 0, 1, 2, or 4 (mod 7), so the largest gap of 3 occurs for example between 4 = 2^2 (mod 7) and 7 = 0^2 (mod 7).
For n=16, the quadratic residues are the numbers congruent to 0, 1, 4 or 9 (mod 16), so the largest gap occurs between, e.g., 9 = 3^2 (mod 16) and 16 = 0^2 (mod 16).

K. Ireland and M. Rosen, A Classical Introduction to Modern Number Theory, Springer, 1982, p. 194. [Requires gcd(q,n)=1 for q to be a quadratic residue mod n.]
F. J. MacWilliams and N. J. A. Sloane, The Theory of Error-Correcting Codes, Elsevier-North Holland, 1978, p. 45.
G. B. Mathews, Theory of Numbers, 2nd edition. Chelsea, NY, p. 32. [Does not require gcd(q,n)=1.]
Ivan Niven and Herbert S. Zuckerman, An Introduction to the Theory of Numbers, New York: John Wiley, 2nd ed., 1966, p. 69. [Requires gcd(q,n)=1 for q to be a quadratic residue mod n.]
J. V. Uspensky and M. A. Heaslet, Elementary Number Theory, McGraw-Hill, NY, 1939, p. 270. [Does not require gcd(q,n)=1.]

Alois P. Heinz, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Quadratic Residue
Wikipedia, Quadratic residue

Cf. A063987, A130290, A088190, A088191, A088192.

(DD(v)=vecextract(v,"^1")-vecextract(v,"^-1")); a(n)=vecmax(DD(select(f->issquare(Mod(f,n)),vector(n*2,i,i))))

Comments and references added by N. J. A. Sloane, Oct 04 2014

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cp's OEIS Frontend

A088191 First differences of A088190.

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A088193 Prime numbers where the sequence of largest quadratic residues modulo the primes (A088190) is non-monotonic.

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A088195 Distance (A088192) of primes from the largest quadratic residues modulo the primes (A088190), where the latter is non-monotonic.

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A088194 Members of the difference sequence (A088191) of the sequence of largest quadratic residues modulo the primes (A088190), where the latter is non-monotonic.

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A088192 Distance between prime(n) and the largest quadratic residue modulo prime(n).

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Mathematica

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A088196 Largest number that is not a quadratic residue modulo prime(n).

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

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A091383 Prime numbers where the sequence of largest quadratic "mixed" residues modulo the primes (A091380) is non-monotonic.

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A091384 Members of the difference sequence (A091381) of the sequence of largest quadratic "mixed" residues modulo the primes (A091380) where the latter is non-monotonic.