A007520 -id:A007520 - cp's OEIS frontend

A337145 a(n) is the determinant of the 2 X 2 matrix whose entries (when read by rows) are the n-th primes congruent to 1, 3, 5, 7 mod 8 respectively.

104, 800, 1712, 2592, 3760, 4840, 5728, 12848, 15664, 18424, 20888, 23520, 28232, 28560, 25320, 30280, 37248, 50520, 43680, 33664, 61560, 73920, 70544, 57696, 38696, 27408, 79280, 63392, 107328, 109536, 162608, 172296, 187352, 197040, 248064, 228320, 215912, 229152, 255480, 231304, 286408, 256320

Offset: 1

J. M. Bergot and Robert Israel, Jan 27 2021

The first negative term is a(20750) = -58207896.

All terms are divisible by 8.

			The first primes == 1, 3, 5, 7 (mod 8) are 17, 3, 5, 7 respectively, so a(1) = 17*7 - 3*5 = 104.
The second primes == 1, 3, 5, 7 (mod 8) are 41, 11, 13, 23 respectively, so a(2) = 41*23 - 11*13 = 800.
The third primes == 1, 3, 5, 7 (mod 8) are 73, 19, 29, 31 respectively, so a(3) = 73*31 - 19*29 = 1712.

Robert Israel, Table of n, a(n) for n = 1..30000

Cf. A335581, A335592, A337146, A337147.

Cf. A007519, A007520, A007521, A007522.

R:= NULL:
L:= [-7, -5, -3, -1]:
found:= false:
for k from 1 to 100 do
  for i from 1 to 4 do
   for x from L[i]+8 by 8 do until isprime(x);
   L[i]:= x;
  od;
  v:= L[1]*L[4]-L[2]*L[3];
  R:= R,v;
od:
R;

A339882 Fundamental positive solution y(n) of the Diophantine equation x^2 - A045339(n)*y^2 = -2, for n >= 1.

Original entry on oeis.org

1, 1, 1, 3, 9, 3, 27, 1, 3, 9, 747, 627, 153, 36321, 1, 121, 699, 537, 2900979, 43, 5843427, 803, 1, 59, 13809, 3, 12507, 541137, 11, 563210019, 57, 28906107, 55617, 3, 499, 212279001, 11516632737, 6633, 41281449, 33, 48957047673, 6284900361, 369485787, 3777

Offset: 1

Wolfdieter Lang, Dec 22 2020

nonn

The corresponding x = x(n) values are given in A339881.
The conjecture is that all A045339 entries have solutions.
For details and examples [x(n), y(n)] see also A339881.

Cf. A007520, A045339, A339881.

Generalized Pell equation. Fundamental solution a(n), with A339881(n)^2 - A045339(n)*a(n)^2 = -2, for n >= 1.

A106875 Primes of the form 3x^2+2xy+3y^2, with x and y nonnegative.

Original entry on oeis.org

3, 19, 59, 107, 163, 211, 251, 347, 419, 499, 571, 587, 619, 659, 883, 907, 947, 1051, 1171, 1259, 1283, 1459, 1483, 1499, 1579, 1627, 1747, 1907, 1931, 1987, 2003, 2083, 2179, 2203, 2347, 2411, 2459, 2467, 2531, 2539, 2707, 2803, 2819, 2963

Offset: 1

T. D. Noe, May 09 2005

Discriminant=-32.

Vincenzo Librandi and Ray Chandler, Table of n, a(n) for n = 1..10000 [First 1000 terms from Vincenzo Librandi]
N. J. A. Sloane et al., Binary Quadratic Forms and OEIS (Index to related sequences, programs, references)

Cf. A243177, A007520.

QuadPrimes2[3, 2, 3, 10000] (* see A106856 *)

A106876 Primes of the form 3x^2-2xy+3y^2, with x and y nonnegative.

Original entry on oeis.org

3, 11, 43, 67, 83, 131, 139, 179, 227, 283, 307, 331, 379, 443, 467, 491, 523, 547, 563, 643, 683, 691, 739, 787, 811, 827, 859, 971, 1019, 1091, 1123, 1163, 1187, 1291, 1307, 1427, 1451, 1523, 1531, 1571, 1619, 1667, 1699, 1723, 1787, 1811

Offset: 1

T. D. Noe, May 09 2005

Discriminant=-32.

Vincenzo Librandi and Ray Chandler, Table of n, a(n) for n = 1..10000 [First 1000 terms from Vincenzo Librandi]
N. J. A. Sloane et al., Binary Quadratic Forms and OEIS (Index to related sequences, programs, references)

Cf. A243177, A007520.

QuadPrimes2[3, -2, 3, 10000] (* see A106856 *)

A269420 Record (maximal) gaps between primes of the form 8k + 3.

Original entry on oeis.org

8, 24, 32, 40, 48, 72, 120, 144, 152, 176, 216, 264, 320, 400, 520, 592, 600, 824, 856, 872, 936, 992, 1064, 1072, 1112, 1336, 1392, 1408, 1584, 1720, 2080

Offset: 1

Alexei Kourbatov, Feb 25 2016

nonn

Dirichlet's theorem on arithmetic progressions suggests that average gaps between primes of the form 8k + 3 below x are about phi(8)*log(x). This sequence shows that the record gap ending at p grows almost as fast as phi(8)*log^2(p). Here phi(n) is A000010, Euler's totient function; phi(8)=4.
Conjecture: a(n) < phi(8)*log^2(A269422(n)) almost always.
A269421 lists the primes preceding the maximal gaps.
A269422 lists the corresponding primes at the end of the maximal gaps.

			The first two primes of the form 8k + 3 are 3 and 11, so a(1)=11-3=8. The next prime of this form is 19; the gap 19-11 is not a record so nothing is added to the sequence. The next prime of this form is 43 and the gap 43-19=24 is a new record, so a(2)=24.

Alexei Kourbatov, On the distribution of maximal gaps between primes in residue classes, arXiv:1610.03340 [math.NT], 2016.
Alexei Kourbatov, On the nth record gap between primes in an arithmetic progression, arXiv:1709.05508 [math.NT], 2017; Int. Math. Forum, 13 (2018), 65-78.
Alexei Kourbatov and Marek Wolf, Predicting maximal gaps in sets of primes, arXiv preprint arXiv:1901.03785 [math.NT], 2019.

Cf. A007520, A269421, A269422.

re=0; s=3; forprime(p=11, 1e8, if(p%8!=3, next); g=p-s; if(g>re, re=g; print1(g", ")); s=p)

A269421 Primes 8k + 3 preceding the maximal gaps in A269420.

Original entry on oeis.org

3, 19, 179, 379, 691, 1187, 1307, 4787, 15467, 22307, 43067, 83579, 273323, 373459, 1543291, 5364091, 5768803, 20941259, 137649667, 251522291, 369352163, 426008699, 938378747, 1042908091, 1378014731, 1878780427, 11474650339, 12402606331, 15931938899, 51025309339, 144309631099

Offset: 1

Alexei Kourbatov, Feb 25 2016

nonn

Subsequence of A007520.

A269420 lists the corresponding record gap sizes. See more comments there.

			The first two primes of the form 8k + 3 are 3 and 11, so a(1)=3. The next prime of this form is 19; the gap 19-11 is not a record so nothing is added to the sequence. The next prime of this form is 43 and the gap 43-19=24 is a new record, so a(2)=19.

Alexei Kourbatov and Marek Wolf, Predicting maximal gaps in sets of primes, arXiv preprint arXiv:1901.03785 [math.NT], 2019.

Cf. A007520, A269420, A269422.

re=0; s=3; forprime(p=11, 1e8, if(p%8!=3, next); g=p-s; if(g>re, re=g; print1(s", ")); s=p)

A269422 Primes 8k + 3 at the end of the maximal gaps in A269420.

Original entry on oeis.org

11, 43, 211, 419, 739, 1259, 1427, 4931, 15619, 22483, 43283, 83843, 273643, 373859, 1543811, 5364683, 5769403, 20942083, 137650523, 251523163, 369353099, 426009691, 938379811, 1042909163, 1378015843, 1878781763, 11474651731, 12402607739, 15931940483, 51025311059, 144309633179

Offset: 1

Alexei Kourbatov, Feb 25 2016

nonn

Subsequence of A007520.

A269420 lists the corresponding record gap sizes. See more comments there.

			The first two primes of the form 8k + 3 are 3 and 11, so a(1)=11. The next prime of this form is 19; the gap 19-11 is not a record so nothing is added to the sequence. The next prime of this form is 43 and the gap 43-19=24 is a new record, so a(2)=43.

Alexei Kourbatov and Marek Wolf, Predicting maximal gaps in sets of primes, arXiv preprint arXiv:1901.03785 [math.NT], 2019.

Cf. A007520, A269420, A269421.

re=0; s=3; forprime(p=11, 1e8, if(p%8!=3, next); g=p-s; if(g>re, re=g; print1(p", ")); s=p)

A282722 Let p = n-th prime == 3 mod 8; a(n) = sum of quadratic residues mod p that are > p/2.

Original entry on oeis.org

0, 9, 44, 293, 461, 758, 1022, 1799, 2530, 3171, 4778, 5068, 7662, 8470, 9993, 14097, 16674, 19467, 20755, 25701, 29042, 34471, 37506, 40661, 45066, 48541, 54324, 54224, 58135, 60351, 68593, 75432, 74014, 83881, 85900, 98518, 112000, 117443, 122241, 132125, 143322, 151299, 163180, 161975, 181191

Offset: 1

N. J. A. Sloane, Feb 20 2017

nonn

Jean-François Alcover, Table of n, a(n) for n = 1..1000
Christian Aebi, Grant Cairns, Sums of Quadratic residues and nonresidues, arXiv:1512.00896 [math.NT], 2015.

Cf. A282035-A282043 and A282721-A282727.

with(numtheory):
Ql:=[]; Qu:=[]; Q:=[]; Nl:=[]; Nu:=[]; N:=[]; Th:=[];
for i1 from 1 to 300 do
p:=ithprime(i1);
if (p mod 8) = 3 then
ql:=0; qu:=0; q:=0; nl:=0; nu:=0; n:=0;
for j from 1 to p-1 do
if legendre(j,p)=1 then
q:=q+j;
if jA282721 - A282727
# 2nd program
A282722 := proc(n)
    local p,ar;
    p := A007520(n) ;
    a := 0 ;
    for r from (p+1)/2 to p do
        if numtheory[legendre](r,p) = 1 then
            a := a+r ;
        end if;
    end do:
    a ;
end proc:
seq(A282722(n),n=1..10) ; # R. J. Mathar, Apr 07 2017

b[1] = 3; b[n_] := b[n] = Module[{p}, p = NextPrime[b[n - 1]]; While[Mod[p, 8] != 3, p = NextPrime[p]]; p];
a[n_] := Module[{p, q, r}, p = b[n]; q = 0; For[r = (p + 1)/2, r <= p, r++, If[KroneckerSymbol[r, p] == 1, q = q + r]]; q];
Array[a, 45] (* Jean-François Alcover, Nov 27 2017, after R. J. Mathar *)

A282724 Let p = n-th prime == 3 mod 8; a(n) = sum of quadratic nonresidues mod p that are < p/2.

Original entry on oeis.org

0, 2, 13, 94, 129, 247, 306, 555, 745, 999, 1579, 1555, 2466, 2653, 3059, 4581, 5430, 6351, 6658, 8409, 9087, 11158, 11996, 12858, 14814, 15788, 17880, 17277, 18950, 19481, 22400, 24876, 23518, 27448, 28115, 32285, 36743, 38269, 39851, 43111, 47406, 50055, 53683, 51645, 58274, 66410, 65119, 76013, 80465

Offset: 1

N. J. A. Sloane, Feb 20 2017

nonn

Jean-François Alcover, Table of n, a(n) for n = 1..1000
Aebi, Christian, and Grant Cairns. Sums of Quadratic residues and nonresidues, arXiv preprint arXiv:1512.00896 (2015).

Cf. A282035-A282043 and A282721-A282727.

with(numtheory):
Ql:=[]; Qu:=[]; Q:=[]; Nl:=[]; Nu:=[]; N:=[]; Th:=[];
for i1 from 1 to 300 do
p:=ithprime(i1);
if (p mod 8) = 3 then
ql:=0; qu:=0; q:=0; nl:=0; nu:=0; n:=0;
for j from 1 to p-1 do
if legendre(j,p)=1 then
q:=q+j;
if jA282721 - A282727
# 2nd program
A282724 := proc(n)
    local p,a,r;
    p := A007520(n) ;
    a := 0 ;
    for r from 1 to (p-1)/2 do
        if numtheory[legendre](r,p) <> 1 then
            a := a+r ;
        end if;
    end do:
    a ;
end proc:
seq(A282724(n),n=1..10) ; # R. J. Mathar, Apr 07 2017

b[1] = 3; b[n_] := b[n] = Module[{p}, p = NextPrime[b[n - 1]]; While[Mod[p, 8] != 3, p = NextPrime[p]]; p];
a[n_] := Module[{p, q, r}, p = b[n]; q = 0; For[r = 1, r <= (p - 1)/2, r++, If[KroneckerSymbol[r, p] != 1, q = q + r]]; q];
Array[a, 50] (* Jean-François Alcover, Nov 27 2017, after R. J. Mathar *)

A339881 Fundamental nonnegative solution x(n) of the Diophantine equation x^2 - A045339(n)*y^2 = -2, for n >= 1.

Original entry on oeis.org

0, 1, 3, 13, 59, 23, 221, 9, 31, 103, 8807, 8005, 2047, 527593, 15, 1917, 11759, 9409, 52778687, 801, 113759383, 16437, 21, 1275, 305987, 67, 286025, 12656129, 261, 13458244873, 1381, 719175577, 1410305, 77, 13041, 5580152383, 313074529583, 186079, 1175615653, 949, 1434867510253, 186757799729, 11127596791, 116231

Offset: 1

Wolfdieter Lang, Dec 22 2020

nonn

The corresponding y values are given in A339882.
The Diophantine equation x^2 - p*y^2 = -2, of discriminant Disc = 4*p > 0 (indefinite binary quadratic form), with prime p can have proper solutions (gcd(x, y) = 1) only for primes p = 2 and p == 3 (mod 8) by parity arguments.
There are no improper solutions (with g >= 2, g^2 does not divide 2).
The prime p = 2 has just one infinite family of proper solutions with nonnegative x values. The fundamental proper solutions for p = 2 is (0, 1).
If a prime p congruent to 3 modulo 8, (p(n) = A007520(n)) has a solution then it can have only one infinite family of (proper) solutions with positive x value.
This family is self-conjugate (also called ambiguous, having with each solution (x, y) also (x, -y) as solution). This follows from the fact that there is only one representative parallel primitive form (rpapf), namely F_{pa(n)} = [-2, 2, -(p(n) - 1)/2].
The reduced principal form of Disc(n) = 4*p(n) is F_{p(n)} = [1, 2*s(n), -(p(n) - s(n)^2)], with s(n) = A000194(n'(n)), if p(n) = A000037(n'(n)). The corresponding (reduced) principal cycle has length L(n) = 2*A307372(n'(n)).
The number of all cycles, the class number, for Disc(n) is h(n'(n)) = A324252(n'(n)), Note that in the Buell reference, Table 2B in Appendix 2, p. 241, all Disc(n) <= 4*1051 = 4204 have class number 2, except for p = 443, 499, 659 (Disc = 1772, 1996, 26).
See the W. Lang link Table 1 for some principal reduced forms F_{p(n)} (there for p(n) in the D-column, and F_p is called FR(n)) with their t-tuples, giving the automporphic  matrix Auto(n) = R(t_1) R(t_1) ... R(t_{L(n)}), where R(t) := Matrix([[0, -1], [1, t]]), and the length of the principal cycle L(n) given above, and in Table 2 for CR(n).
To prove the existence of a solution one would have to show that the rpapf F{pa(n)} is properly equivalent to the principal form F_{pa(n)}.

			The fundamental solutions [A045339(n), [x = a(n), y = A339882(n)]] begin:
[2, [0, 1]], [3, [1, 1]], [11, [3, 1]], [19, [13, 3]], [43, [59, 9]], [59, [23, 3]], [67, [221, 27]], [83, [9, 1]], [107, [31, 3]], [131, [103, 9]], [139, [8807, 747]], [163, [8005, 627]], [179, [2047, 153]], [211, [527593, 36321]], [227, [15, 1]], [251, [1917, 121]], [283, [11759, 699]], [307, [9409, 537]], [331, [52778687, 2900979]], [347, [801, 43]], [379, [113759383, 5843427]], [419, [16437, 803]], [443, [21, 1]], [467, [1275, 59]], [491, [305987, 13809]], [499, [67, 3]], [523, [286025, 12507]], [547, [12656129, 541137]], [563, [261, 11]], [571, [13458244873, 563210019]], [587, [1381, 57]], [619, [719175577, 28906107]], [643, [1410305, 55617]], [659, [77, 3]], [683, [13041, 499]], [691, [5580152383, 212279001]], [739, [313074529583, 11516632737]], [787, [186079, 6633]], [811, [1175615653, 41281449]], [827, [949, 33]], [859, [1434867510253, 48957047673]], [883, [186757799729, 6284900361]], [907, [11127596791, 369485787]], [947, [116231, 3777]], ...

D. A. Buell, Binary Quadratic Forms, Springer, 1989.

Wolfdieter Lang, Cycles of reduced Pell forms, general Pell equations and Pell graphs

Cf. A000194, A000037, A000194, A007520, A045339, A307372, A324252, A339882 (y values), A336793 (record y values), A336792 (corresponding odd p numbers).

Generalized Pell equation: Positive fundamental a(n), with a(n)^2 - A045339(n)*A339882(n)^2 = -2, for n >= 1.

cp's OEIS Frontend

A337145 a(n) is the determinant of the 2 X 2 matrix whose entries (when read by rows) are the n-th primes congruent to 1, 3, 5, 7 mod 8 respectively.

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A339882 Fundamental positive solution y(n) of the Diophantine equation x^2 - A045339(n)*y^2 = -2, for n >= 1.

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A106875 Primes of the form 3x^2+2xy+3y^2, with x and y nonnegative.

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Mathematica

A106876 Primes of the form 3x^2-2xy+3y^2, with x and y nonnegative.

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Mathematica

A269420 Record (maximal) gaps between primes of the form 8k + 3.

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PARI

A269421 Primes 8k + 3 preceding the maximal gaps in A269420.

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PARI

A269422 Primes 8k + 3 at the end of the maximal gaps in A269420.

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PARI

A282722 Let p = n-th prime == 3 mod 8; a(n) = sum of quadratic residues mod p that are > p/2.

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Maple

Mathematica

A282724 Let p = n-th prime == 3 mod 8; a(n) = sum of quadratic nonresidues mod p that are < p/2.

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