

A002334


Least positive integer x such that prime A038873(n) = x^2  2y^2 for some y.
(Formerly M0607 N0219)


7



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
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OFFSET

1,1


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)


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

Robert Israel, Table of n, a(n) for n = 1..10000
A. J. C. Cunningham, Quadratic Partitions, Hodgson, London, 1904 [Annotated scans of selected pages]


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


EXAMPLE

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


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], 127], 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]] (* JeanFrançois Alcover, Oct 27 2019 *)


CROSSREFS

Cf. A002335, A035251.
Sequence in context: A263721 A154290 A267259 * A115732 A246578 A048947
Adjacent sequences: A002331 A002332 A002333 * A002335 A002336 A002337


KEYWORD

nonn,easy


AUTHOR

N. J. A. Sloane


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


STATUS

approved



