A088194 -id:A088194 - cp's OEIS frontend

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

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

A088190 Largest quadratic residue modulo prime(n).

Original entry on oeis.org

1, 1, 4, 4, 9, 12, 16, 17, 18, 28, 28, 36, 40, 41, 42, 52, 57, 60, 65, 64, 72, 76, 81, 88, 96, 100, 100, 105, 108, 112, 124, 129, 136, 137, 148, 148, 156, 161, 162, 172, 177, 180, 184, 192, 196, 196, 209, 220, 225, 228, 232, 232, 240, 249, 256, 258, 268, 268, 276

Offset: 1

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

nonn

Denote a(n) by LQR(p_n). Observations (tested up to 20000 primes): - the sequence of largest QR modulo the primes (LQR(p_n) is 'almost' monotonic, - p_n-LQR(p_n) is either 1 or a prime value (see A088192) - if LQR(p_n)<=LQR(p_{n-1}) then p_n==7 mod 8 (when n>2) (see A088194) - if LQR(p_n)<=LQR(p_{n-1}) then p_n-LQR(p_n) is an odd prime, but never 5 (see A088195) For a similar set of sequences, related to quadratic non-residues, see A088196-A088201.
From Robert Israel, Oct 31 2024: (Start)
a(n) = prime(n)-1 if and only if n is 1 or in A080147.
a(n) = prime(n)-2 if and only if prime(n) is in A007520.
a(n) = prime(n)-3 if and only if prime(n) is in A107006. (End)

Robert Israel, Table of n, a(n) for n = 1..10000
F. Adorjan, The sequence of largest quadratic residues modulo the primes.

Cf. A007520, A080147, A088191, A088192, A088193, A088194, A088195, A088196, A088197, A088198, A088199, A088200, A088201, A107006.

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:
seq(lqr(ithprime(i)),i=1..100); # Robert Israel, Oct 31 2024

a[n_] := With[{p = Prime[n]}, SelectFirst[Range[p - 1, 1, -1], JacobiSymbol[#, p] == 1&]]; Array[a, 100] (* Jean-François Alcover, Feb 16 2018 *)

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

a(n) = max(r, r==j^2 mod p(n)|j=1, 2, ...(p(n)-1)/2)

A088191 First differences of A088190.

Original entry on oeis.org

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

A088200 Members of the difference sequence (A088197) of LQnR(p_n) (A088196) where it is <= 0.

Original entry on oeis.org

-2, -2, -4, -2, -2, -4, -2, -4, -2, 0, -4, -4, -4, -2, -4, -2, -2, -10, -2, -4, 0, -4, -4, -8, -10, -2, -4, 0, -4, -2, -4, -4, -4, -2, -4, 0, -4, -6, -4, 0, -8, -4, -2, -2, 0, -4, -4, -4, -10, -2, -14, -2, 0, 0, -6, -4, -4, 0, -10, -2, 0, -4, -10, -4, -2, 0, -4, -2, -2, -6, -2, -4, 0, -2, -4, -4, -10, -8, -2, 0, 0, -8, -4, -8, -4, 0, -2

Offset: 1

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

sign

The members of the sequence are always even (conjectured!).

Cf. A088194, A088196, A088197, A088198, A088199, A088201.

qnrp_d_nm(n)= {/* The difference sequence of LQnR where the sequence of the largest QnR modulo the primes is nonmonotonic */ local(k=1,m,p,fl,jj,j,v=[]); for(i=2,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--); if(m-k<=0,v=concat(v,m-k)); k=m); print(v)}

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

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

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A088190 Largest quadratic residue modulo prime(n).

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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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A088200 Members of the difference sequence (A088197) of LQnR(p_n) (A088196) where it is <= 0.

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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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