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A103255
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Integers x > 0 such that x^3 + y^3 = z^2 for some y > 0, z > 0, and gcd(x,y) = 1.
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3
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1, 2, 11, 23, 37, 56, 57, 65, 112, 122, 193, 217, 242, 305, 312, 433, 592, 781, 851, 877, 889, 913, 1001, 1064, 1177, 1201, 1346, 1376, 1617, 1633, 1706, 1729, 1801, 1953, 1960, 1969, 2137, 2162, 2184, 2257, 2345, 2480, 2543, 2920, 3071, 3081, 3482, 3641, 3889, 4019
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OFFSET
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1,2
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LINKS
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EXAMPLE
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x=11, y=37, 11^3 + 37^3 = 228^2. 11 is the third entry in the list.
The pairs [x,y] = [a(n),a(?)] for the first few terms are [1, 2], [2, 1], [11, 37], [23, 1177], [37, 11], [56, 65], [57, 112], [65, 56], [112, 57], [122, 1201], [193, 3482], [217, 312], [242, 433]. [Joerg Arndt, Sep 30 2012]
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MATHEMATICA
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(* This program uses z-values from A099426 b-file. To get 50 terms, the first 200 z-values suffice, the result being the same with the whole b-file of 300 z-values. *)
terms = 50;
zz = Import["https://oeis.org/A099426/b099426.txt", "Table"][[1 ;; 4 terms, 2]];
r[z_] := {x, y, z} /. ToRules[Reduce[GCD[x, y] == 1 && 0<x<y && x^3 + y^3 == z^2, {x, y}, Integers]];
xyz = r /@ zz;
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PROG
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(Magma) [ k : k in [1..100] | exists{P : P in IntegralPoints(EllipticCurve([0, k^3])) | P[1] gt 0 and P[2] ne 0 and GCD(Integers()!P[1], k) eq 1} ]; // Geoff Bailey
(Sage) # apparently inefficient as of version 5.2
E = EllipticCurve([0, n^3])
E.gens(descent_second_limit=16);
for p in E.integral_points():
if p[0] > 0 and p[1] > 0 and gcd(p[1], n) == 1:
return true
return false
[n for n in (1..60) if is_A103255(n)]
(PARI)
{ /* Warning: just how big lim has to be is unclear */
my(x3=x^3);
for (y=1, lim,
if ( gcd(x, y) != 1, next() );
if ( issquare(x3+y^3), return(1) );
);
return(0);
}
/* Using lim=10^6 reproduces all terms <= 1000: */
for (n=1, 1000, if( is_A103255(n, 10^6), print1(n, ", ")) );
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CROSSREFS
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KEYWORD
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nonn,more
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AUTHOR
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EXTENSIONS
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Recomputed and extended by Geoff Bailey (geoff(AT)maths.usyd.edu.au) using MAGMA, Jan 28 2007
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STATUS
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approved
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