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A092572
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Numbers of the form x^2 + 3y^2 where x and y are positive integers.
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34
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4, 7, 12, 13, 16, 19, 21, 28, 31, 36, 37, 39, 43, 48, 49, 52, 57, 61, 63, 64, 67, 73, 76, 79, 84, 91, 93, 97, 100, 103, 108, 109, 111, 112, 117, 124, 127, 129, 133, 139, 144, 147, 148, 151, 156, 157, 163, 169, 171, 172, 175, 181, 183, 189, 192, 193, 196, 199
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OFFSET
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1,1
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COMMENTS
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Superset of primes of the form 6n+1 (A002476).
It seems that all integer solutions of ((a+b)^3 - (a-b)^3) / (2*b) = c^3 have c = x^2 + 3*y^2. - Juergen Buchmueller (pullmoll(AT)t-online.de), Apr 04 2008
To prove the case of cubes in Fermat's last theorem, Euler considered numbers of the form a^2 + 3b^2. In the equation x^3 + y^3 = z^3, Euler specified that x = a - b and y = a + b. - Alonso del Arte, Jul 19 2012
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REFERENCES
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Paulo Ribenboim, 13 Lectures on Fermat's Last Theorem. New York: Springer-Verlag (1979): 4.
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LINKS
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EXAMPLE
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7 is of the specified form, since 2^2 + 3 * 1^2 = 7.
So is 12, since 3^2 + 3 * 1^2 = 12, and 13, with 1^2 + 3 * 2^2 = 13.
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MAPLE
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N:= 1000: # to get all terms <= N
S:= {seq(seq(x^2 + 3*y^2, x = 1 .. floor(sqrt(N - 3*y^2))),
y=1..floor(sqrt(N/3-1)))}:
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MATHEMATICA
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Union[Flatten[Table[a^2 + 3b^2, {a, 20}, {b, Ceiling[Sqrt[(400 - a^2)/3]]}]]] (* Alonso del Arte, Jul 19 2012 *)
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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