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 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 (list; graph; refs; listen; history; text; internal format)
 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], 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 *) 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 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

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Last modified February 23 16:23 EST 2020. Contains 332177 sequences. (Running on oeis4.)