cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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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

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Author

Wolfdieter Lang, Dec 22 2020

Keywords

Comments

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.

Crossrefs

Cf. A007520, A045339, A339881.

Formula

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

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

Views

Author

Wolfdieter Lang, Dec 22 2020

Keywords

Comments

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)}.

Examples

			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]], ...

References

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

Links

Crossrefs

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

Formula

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

A007520 Primes == 3 (mod 8).

Original entry on oeis.org

3, 11, 19, 43, 59, 67, 83, 107, 131, 139, 163, 179, 211, 227, 251, 283, 307, 331, 347, 379, 419, 443, 467, 491, 499, 523, 547, 563, 571, 587, 619, 643, 659, 683, 691, 739, 787, 811, 827, 859, 883, 907, 947, 971, 1019, 1051, 1091, 1123, 1163, 1171, 1187
Offset: 1

Views

Author

N. J. A. Sloane, Robert G. Wilson v

Keywords

Comments

Primes of the form 3x^2 + 2xy + 3y^2 with x and y in Z. - T. D. Noe, May 07 2005
Also, primes of the form X^2 + 2Y^2, X=|x-y|, Y=x+y. - Zak Seidov, Dec 06 2011
Each term is the sum of no fewer than three positive squares. - T. D. Noe, Nov 15 2010
Smallest terms expressible as sum of three distinct positive squares: 59 = 1^2 + 3^2 + 7^2, 83 = 3^2 + 5^2 + 7^2, 107, 131, 139, 179, 211, 227, 251, 283, 307. - Zak Seidov, Dec 06 2011
Except for the first term it appears that the terms of the sequence are also primes of the form 2k+1 such that 3*(2k+1) divides 2^k+1. - Hilko Koning, Dec 06 2019
From Hilko Koning, Nov 24 2021: (Start)
Theorem (Legendre symbol): With p an odd prime and a an integer coprime to p the Legendre symbol L(a/p) = -1 if a is a quadratic non-residue (mod p) and L(2/p) = -1 if p == +-3 (mod 8).
Theorem (Euler's criterion): L(a/p) == a^((p-1)/2) (mod p) so with a = 2 and prime p = 2k + 1 then -1 == 2^k (mod (2k+1)). So prime numbers 2k+1 = +-3 (mod 8) are the prime numbers 2k+1 | 2^k+1.
If 2k+1 == -3 (mod 8) then k is even and 2^k+1 is not divisible by 3 and if 2k+1 == +3 (mod 8) then k is odd and 2^k+1 is divisible by 3.
Hence prime numbers 2k+1 == 3 (mod 8) are prime numbers such that 3*(2k+1) | 2^k+1. Or, including the first term of the sequence, prime numbers 2k+1 with k odd such that 2k+1 | 2^k+1.
(End)

References

  • N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Crossrefs

Cf. A294912, A045339 (for n >= 2).

Programs

  • Magma
    [p: p in PrimesUpTo(2000) | p mod 8 eq 3]; // Vincenzo Librandi, Aug 07 2012
  • Maple
    A007520 := proc(n)
    option remember;
    local a;
    if n = 1 then
        return 3;
    end if;
    a := nextprime(procname(n-1)) ;
    while modp(a,8) <> 3 do
        a := nextprime(a) ;
    end do:
    a ;
end proc:
seq(A007520(n),n=1..30) ; # R. J. Mathar, Apr 07 2017
  • Mathematica
    lst={};Do[p=8*n+3;If[PrimeQ[p], AppendTo[lst, p]], {n, 0, 10^3}];lst (* Vladimir Joseph Stephan Orlovsky, Aug 22 2008 *)
p=3;k=0;nn=1000;Reap[While[kZak Seidov, Dec 06 2011 *)
Select[Prime[Range[200]],Mod[#,8]==3&] (* Harvey P. Dale, Apr 05 2023 *)
  • PARI
    forprime(p=2,97,if(p%8==3,print1(p", "))) \\ Charles R Greathouse IV, Aug 17 2011
Showing 1-3 of 3 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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