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 A002331 Values of x in the solution to p = x^2 + y^2, x <= y, with prime p = A002313(n). (Formerly M0096 N0033) 23

%I M0096 N0033

%S 1,1,2,1,2,1,4,2,5,3,5,4,1,3,7,4,7,6,2,9,7,1,2,8,4,1,10,9,5,2,12,11,9,

%T 5,8,7,10,6,1,3,14,12,7,4,10,5,11,10,14,13,1,8,5,17,16,4,13,6,12,1,5,

%U 15,2,9,19,12,17,11,5,14,10,18,4,6,16,20,19,10,13,4,6,15,22,11,3,5

%N Values of x in the solution to p = x^2 + y^2, x <= y, with prime p = A002313(n).

%C a(n+1) = MIN(A002972(n), 2*A002973(n)). [_Reinhard Zumkeller_, Feb 16 2010]

%D A. J. C. Cunningham, Quadratic Partitions. Hodgson, London, 1904, p. 1.

%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

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

%H T. D. Noe and Charles R Greathouse IV, <a href="/A002331/b002331.txt">Table of n, a(n) for n = 1..10000</a> (first 1000 terms from Noe)

%H John Brillhart, <a href="http://dx.doi.org/10.1090/S0025-5718-1972-0314745-6">Note on representing a prime as a sum of two squares</a>, Math. Comp. 26 (1972), pp. 1011-1013.

%H A. J. C. Cunningham, <a href="/A002330/a002330.pdf">Quadratic Partitions</a>, Hodgson, London, 1904 [Annotated scans of selected pages]

%H K. Matthews, <a href="http://www.numbertheory.org/php/serret.html">Serret's algorithm Server</a>

%H J. Todd, <a href="http://www.jstor.org/stable/2305526">A problem on arc tangent relations</a>, Amer. Math. Monthly, 56 (1949), 517-528.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Fermats4nPlus1Theorem.html">Fermat's 4n Plus 1 Theorem</a>

%F a(n) = A096029(n) - A096030(n) for n>1. - _Lekraj Beedassy_, Jul 16 2004

%e The following table shows the relationship

%e between several closely related sequences:

%e Here p = A002144 = primes == 1 mod 4, p = a^2+b^2 with a < b;

%e a = A002331, b = A002330, t_1 = ab/2 = A070151;

%e p^2 = c^2+d^2 with c < d; c = A002366, d = A002365,

%e t_2 = 2ab = A145046, t_3 = b^2-a^2 = A070079,

%e with {c,d} = {t_2, t_3}, t_4 = cd/2 = ab(b^2-a^2).

%e ---------------------------------

%e .p..a..b..t_1..c...d.t_2.t_3..t_4

%e ---------------------------------

%e .5..1..2...1...3...4...4...3....6

%e 13..2..3...3...5..12..12...5...30

%e 17..1..4...2...8..15...8..15...60

%e 29..2..5...5..20..21..20..21..210

%e 37..1..6...3..12..35..12..35..210

%e 41..4..5..10...9..40..40...9..180

%e 53..2..7...7..28..45..28..45..630

%e .................................

%p See A002330 for Maple program.

%t pmax = 1000; x[p_] := Module[{x, y}, x /. ToRules[Reduce[0 <= x <= y && x^2 + y^2 == p, {x, y}, Integers]]]; For[n=1; p=2, p<pmax, p = NextPrime[p], If[Mod[p, 4] == 1 || Mod[p, 4] == 2, a[n] = x[p]; Print["a(", n, ") = ", a[n]]; n++]]; Array[a, n-1] (* _Jean-François Alcover_, Feb 26 2016 *)

%o (PARI) f(p)=my(s=lift(sqrt(Mod(-1,p))),x=p,t);if(s>p/2,s=p-s); while(s^2>p,t=s;s=x%s;x=t);s

%o forprime(p=2,1e3,if(p%4-3,print1(sqrtint(p-f(p)^2)", ")))

%o \\ _Charles R Greathouse IV_, Apr 24 2012

%o (PARI) do(p)=qfbsolve(Qfb(1,0,1),p)[2]

%o forprime(p=2,1e3,if(p%4-3,print1(do(p)", "))) \\ _Charles R Greathouse IV_, Sep 26 2013

%Y Cf. A002330, A002313, A002144.

%K nonn

%O 1,3

%A _N. J. A. Sloane_.

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Last modified February 23 07:24 EST 2020. Contains 332159 sequences. (Running on oeis4.)