A002330 Value of y in the solution to p = x^2 + y^2, x <= y, with prime p = A002313(n).
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, 14, 16, 17, 13, 14, 16, 18, 17, 18, 17, 19, 20, 20, 15, 17, 20, 21, 19, 22, 20, 21, 19, 20, 24, 23, 24, 18, 19, 25, 22, 25, 23, 26, 26, 22, 27, 26, 20, 25, 22, 26, 28, 25
Offset: 1
Keywords
Examples
The following table shows the relationship between several closely related sequences:
Here p = A002144 = primes == 1 (mod 4), p = a^2+b^2 with a < b;
a = A002331, b = A002330, t_1 = ab/2 = A070151;
p^2 = c^2+d^2 with c < d; c = A002366, d = A002365,
t_2 = 2ab = A145046, t_3 = b^2-a^2 = A070079,
with {c,d} = {t_2, t_3}, t_4 = cd/2 = ab(b^2-a^2).
---------------------------------
.p..a..b..t_1..c...d.t_2.t_3..t_4
---------------------------------
.5..1..2...1...3...4...4...3....6
13..2..3...3...5..12..12...5...30
17..1..4...2...8..15...8..15...60
29..2..5...5..20..21..20..21..210
37..1..6...3..12..35..12..35..210
41..4..5..10...9..40..40...9..180
53..2..7...7..28..45..28..45..630
.................................
References
- A. J. C. Cunningham, Quadratic Partitions. Hodgson, London, 1904, p. 1.
- 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
- Charles R Greathouse IV, Table of n, a(n) for n = 1..10000 (first 1000 terms from T. D. Noe)
- John Brillhart, Note on representing a prime as a sum of two squares, Math. Comp. 26 (1972), pp. 1011-1013.
- A. J. C. Cunningham, Quadratic Partitions, Hodgson, London, 1904. [Annotated scans of selected pages]
- K. Matthews, Serret's algorithm Server.
- J. Todd, A problem on arc tangent relations, Amer. Math. Monthly, 56 (1949), 517-528.
- Eric Weisstein's World of Mathematics, Fermat's 4n Plus 1 Theorem.
Programs
-
Maple
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"
-
Mathematica
Flatten[#, 1]&[Table[PowersRepresentations[Prime[k], 2, 2], {k, 1, 142}]][[All, 2]] (* Jean-François Alcover, Jul 05 2011 *) -
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 forprime(p=2,1e3,if(p%4-3,print1(f(p)", "))) \\ Charles R Greathouse IV, Apr 24 2012
-
PARI
do(p)=qfbsolve(Qfb(1,0,1),p)[1] forprime(p=2,1e3,if(p%4-3,print1(do(p)", "))) \\ Charles R Greathouse IV, Sep 26 2013
-
PARI
print1(1); forprimestep(p=5,1e3,4, print1(", "qfbcornacchia(1,p)[1])) \\ Charles R Greathouse IV, Sep 15 2021