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.

Showing 1-3 of 3 results.

A002334 Least positive integer x such that prime A038873(n) = x^2 - 2y^2 for some y.

Original entry on oeis.org

2, 3, 5, 5, 7, 7, 7, 11, 9, 9, 11, 13, 11, 11, 15, 13, 13, 13, 17, 15, 19, 15, 19, 17, 21, 17, 19, 17, 17, 19, 21, 25, 19, 19, 23, 25, 23, 21, 23, 21, 21, 29, 23, 25, 23, 27, 29, 23, 31, 33, 25, 29, 27, 25, 25, 27, 29, 35, 31, 31, 27, 29, 33, 31, 29, 29, 29, 29, 37, 31, 41, 35
Offset: 1

Views

Author

N. J. A. Sloane

Keywords

Comments

A prime p is representable in the form x^2 - 2y^2 iff p is 2 or p == 1 or 7 (mod 8). - Pab Ter (pabrlos2(AT)yahoo.com), Oct 22 2005
From Wolfdieter Lang, Feb 17 2015: (Start)
For the corresponding y terms see A002335.
a(n), together with A002335(n), gives the fundamental positive solution of the first class of this (generalized) Pell equation. The prime 2 has only one class of proper solutions. The fundamental positive solutions of the second class for the primes from A001132 are given in A254930 and A254931. (End)

Examples

			The first solutions [x(n), y(n)] are (the prime is given as first entry): [2,[2,1]], [7,[3,1]], [17,[5,2]], [23,[5,1]], [31,[7,3]], [41,[7,2]], [47,[7,1]], [71,[11,5]], [73,[9,2]], [79,[9,1]], [89,[11,4]], [97,[13,6]], [103,[11,3]], [113,[11,2]], [127,[15,7]], [137,[13,4]], [151,[13,3]], [167,[13,1]], [191,[17,7]], [193,[15,4]], [199,[19,9]], [223,[15,1]], [233,[19,8]], [239,[17,5]], [241,[21,10]], [257,[17,4]], [263,[19,7]], [271,[17,3]], ... - _Wolfdieter Lang_, Feb 17 2015

References

  • A. J. C. Cunningham, Quadratic Partitions. Hodgson, London, 1904, p. 1.
  • D. H. Lehmer, Guide to Tables in the Theory of Numbers. Bulletin No. 105, National Research Council, Washington, DC, 1941, p. 55.
  • N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
  • N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Crossrefs

Cf. A002335, A035251.

Programs

  • Maple
    with(numtheory): readlib(issqr): for i from 1 to 250 do p:=ithprime(i): pmod8:=modp(p,8): if p=2 or pmod8=1 or pmod8=7 then for y from 1 do x2:=p+2*y^2: if issqr(x2) then printf("%d,",sqrt(x2)): break fi od fi od: # Pab Ter, May 08 2004
  • Mathematica
    maxPrimePi = 200;
Reap[Do[If[MatchQ[Mod[p, 8], 1|2|7], rp = Reduce[x > 0 && y > 0 && p == x^2 - 2*y^2, {x, y}, Integers]; If[rp =!= False, xy = {x, y} /. {ToRules[rp /. C[1] -> 1]}; x0 = xy[[All, 1]] // Min // Simplify; Print[{p, xy[[1]]} ]; Sow[x0]]], {p, Prime[Range[maxPrimePi]]}]][[2, 1]] (* Jean-François Alcover, Oct 27 2019 *)

Formula

a(n)^2 - 2*A002335(n)^2 = A038873(n), n >= 1, and a(n) is the least positive integer satisfying this Pell type equation. - Wolfdieter Lang, Feb 12 2015

Extensions

More terms from Pab Ter (pabrlos(AT)yahoo.com), May 08 2004
The name has been changed in order to be more precise and to conform with A002335. The offset has been changed to 1. - Wolfdieter Lang, Feb 12 2015

A255248 Fundamental positive solution y = y2(n) of the second class of the Pell equation x^2 - 2*y^2 = -A001132(n), n>=1 (primes congruent to {1,7} mod 8).

Original entry on oeis.org

4, 7, 6, 10, 9, 8, 16, 11, 10, 15, 19, 14, 13, 22, 17, 16, 14, 24, 19, 28, 16, 27, 22, 31, 21, 26, 20, 19, 24, 29, 37, 21, 20, 32, 36, 31, 25, 30, 23, 22, 43, 29, 34, 28, 38, 42, 25, 45, 49, 29, 40, 35, 28, 27, 34, 39, 52, 43, 42, 28, 36, 46, 41, 35, 33, 32
Offset: 1

Views

Author

Wolfdieter Lang, Feb 19 2015

Keywords

Comments

For the corresponding term x2(n) see A255247(n).
See the comments on A255247.

Examples

			See A255247.
a(4) = -(2*1 - 3*4) = 12 - 2 = 10.
n=4: 13^2 - 2*10^2 = 169 - 200 = -31 = -A001132(4).

Crossrefs

Cf. A001132, A255247, A255235, A255246, A254937, A255234, A254931.

Formula

A255247(n)^2 - 2*a(n)^2 = -A001132(n), n >= 1, gives the second smallest positive (proper) solution of this (generalized) Pell equation.
a(n) = -(2*A255235(n+1) - 3*A255246(n+1)), n >= 1.

Extensions

More terms from Colin Barker, Feb 26 2015

A254930 Fundamental positive solution x = x2(n) of the second class of the Pell equation x^2 - 2*y^2 = A001132(n), n >= 1 (primes congruent to 1 or 7 mod 8).

Original entry on oeis.org

5, 7, 11, 9, 13, 17, 13, 19, 23, 17, 15, 21, 25, 17, 23, 27, 35, 23, 29, 21, 41, 25, 31, 23, 35, 29, 39, 43, 37, 31, 27, 49, 53, 33, 31, 37, 47, 41, 55, 59, 31, 45, 39, 49, 37, 35, 61, 37, 35
Offset: 1

Views

Author

Wolfdieter Lang, Feb 12 2015

Keywords

Comments

The corresponding terms y = y2(n) are given in A254931(n).
There is only one fundamental solution for prime 2 (no second class exists), and this solution (x, y) has been included in (A002334(1), A002335(1)) = (2, 1).
The second class x sequence for the primes 1 (mod 8), which are given in A007519, is A254762, and for the primes 7 (mod 8), given in A007522, it is A254766.
The second class solutions give the second smallest positive integer solutions of this Pell equation.
For comments and the Nagell reference see A254760.

Examples

			n = 3: 11^2 - 2*7^2 = 23 = A001132(3) = A007522(2).
The first pairs of these second class solutions [x2(n), y2(n)] are (a star indicates primes congruent to 1 (mod 8)):
n  A001132(n)   a(n)  A254931(n)
1     7           5        3
2    17 *         7        4
3    23          11        7
4    31           9        5
5    41 *        13        8
6    47          17       11
7    71          13        7
8    73 *        19       12
9    89 *        17       10
10   97 *        15        8
11  103          21       13
12  113 *        25       16
13  127          17        9
14  137 *        23       14
15  151          27       17
16  167          35       23
17  191          23       13
18  193 *        29       18
19  199          21       11
20  223          41       27
...

Crossrefs

Cf. A001132, A254931, A002334, A002335, A007519, A254762, A007522, A254766, A254760.

Programs

  • Mathematica
    Reap[For[p = 2, p < 1000, p = NextPrime[p], If[MatchQ[Mod[p, 8], 1|7], rp = Reduce[x > 0 && y > 0 && x^2 - 2 y^2 == p, {x, y}, Integers]; If[rp =!= False, xy = {x, y} /. {ToRules[rp /. C[1] -> 1]}; x2 = xy[[-1, 1]] // Simplify; Print[x2]; Sow[x2]]]]][[2, 1]] (* Jean-François Alcover, Oct 28 2019 *)

Formula

a(n)^2 - 2*A254931(n)^2 = A001132(n), and a(n) is the second largest (proper) positive integer solving this (generalized) Pell equation.
a(n) = 3*A002334(n+1) - 4*A002335(n+1), n >= 1.
Showing 1-3 of 3 results.

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

Content is available under The OEIS End-User License Agreement.