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Value of y in the solution to p = x^2 + y^2, x <= y, with prime p = A002313(n).
(Formerly M0462 N0169)
39

%I M0462 N0169 #52 Oct 16 2023 20:12:34

%S 1,2,3,4,5,6,5,7,6,8,8,9,10,10,8,11,10,11,13,10,12,14,15,13,15,16,13,

%T 14,16,17,13,14,16,18,17,18,17,19,20,20,15,17,20,21,19,22,20,21,19,20,

%U 24,23,24,18,19,25,22,25,23,26,26,22,27,26,20,25,22,26,28,25

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

%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 Charles R Greathouse IV, <a href="/A002330/b002330.txt">Table of n, a(n) for n = 1..10000</a> (first 1000 terms from T. D. Noe)

%H John Brillhart, <a href="https://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="https://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) + 1, for n>1. - _Lekraj Beedassy_, Jul 21 2004

%F a(n+1) = Max(A002972(n), 2*A002973(n)). - _Reinhard Zumkeller_, Feb 16 2010

%e The following table shows the relationship 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 a := []; for x from 0 to 50 do for y from x to 50 do p := x^2+y^2; if isprime(p) then a := [op(a),[p,x,y]]; fi; od: od: writeto(trans); for i from 1 to 158 do lprint(a[i]); od: # then sort the triples in "trans"

%t Flatten[#, 1]&[Table[PowersRepresentations[Prime[k], 2, 2], {k, 1, 142}]][[All, 2]] (* _Jean-François Alcover_, Jul 05 2011 *)

%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(f(p)", "))) \\ _Charles R Greathouse IV_, Apr 24 2012

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

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

%o (PARI) print1(1); forprimestep(p=5,1e3,4, print1(", "qfbcornacchia(1,p)[1])) \\ _Charles R Greathouse IV_, Sep 15 2021

%Y Cf. A002331, A002313, A002144.

%K nonn

%O 1,2

%A _N. J. A. Sloane_