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A169580
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Squares of the form x^2+y^2+z^2 with x,y,z positive integers.
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10
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9, 36, 49, 81, 121, 144, 169, 196, 225, 289, 324, 361, 441, 484, 529, 576, 625, 676, 729, 784, 841, 900, 961, 1089, 1156, 1225, 1296, 1369, 1444, 1521, 1681, 1764, 1849, 1936, 2025, 2116, 2209, 2304, 2401, 2500, 2601, 2704, 2809, 2916, 3025, 3136, 3249, 3364
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OFFSET
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1,1
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COMMENTS
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Integer solutions of a^2 = b^2 + c^2 + d^2, i.e., Pythagorean Quadruples. - Jon Perry, Oct 06 2012
Also null (or light-like, or isotropic) vectors in Minkowski 4-space. - Jon Perry, Oct 06 2012
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REFERENCES
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T. Nagell, Introduction to Number Theory, Wiley, 1951, p. 194.
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LINKS
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EXAMPLE
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9 = 1 + 4 + 4,
36 = 16 + 16 + 4,
49 = 36 + 9 + 4,
81 = 49 + 16 + 16,
so these are in the sequence.
16 cannot be written as the sum of 3 squares if zero is not allowed, therefore 16 is not in the sequence.
Also we can see that 49-36-9-4=0, so (7,6,3,2) is a null vector in the signatures (+,-,-,-) and (-,+,+,+). - Jon Perry, Oct 06 2012
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MAPLE
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M:= 10000: # to get all terms <= M
sort(convert(select(issqr, {seq(seq(seq(x^2 + y^2 + z^2,
z=y..floor(sqrt(M-x^2-y^2))), y=x..floor(sqrt((M-x^2)/2))),
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MATHEMATICA
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Select[Range[60]^2, Resolve@ Exists[{x, y, z}, Reduce[# == x^2 + y^2 + z^2, {x, y, z}, Integers], And[x > 0, y > 0, z > 0]] &] (* Michael De Vlieger, Jan 27 2016 *)
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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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