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Search: id:A089270
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| A089270 |
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Positive numbers represented by the integer binary quadratic form x^2 + x*y - y^2 with relative prime x and y. |
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+0
10
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| 1, 5, 11, 19, 29, 31, 41, 55, 59, 61, 71, 79, 89, 95, 101, 109, 121, 131, 139, 145, 149, 151, 155, 179, 181, 191, 199, 205, 209, 211, 229, 239, 241, 251, 269, 271, 281, 295, 305, 311, 319, 331, 341, 349, 355, 359, 361, 379, 389, 395, 401, 409, 419, 421, 431
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OFFSET
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1,2
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COMMENT
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The negative numbers represented by x^2 + x*y - y^2 with relative prime x and y are -a(n).
The discriminant of this binary form is D=5>0, hence this is an indefinite form.
It appears that these are also the n for which the equation x^2 = x+1 (mod n) has solutions. The number of solutions is 0 or a power of 2. It appears that n=5 is the only n for which x^2 = x+1 (mod n) has just one solution. The first n producing 4 solutions is 209. The first n producing 8 solutions is 6061. [From T. D. Noe (noe(AT)sspectra.com), Nov 04 2009]
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FORMULA
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a(n) = x^2 + x*y - y^2 with relative prime integers x and y (proper solutions of the Diophantine equation).
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EXAMPLE
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n=2: a(2)=5 with, for example, (x,y)= (2,1): 4+2-1=5 (there are infinitely many proper (x,y) solutions).
n=8: a(n)=55 with, for example, (x,y)=(7,6) or (7,1). In this case there exist two fundamental proper solutions.
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CROSSREFS
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Sequence in context: A051349 A048217 A132087 this_sequence A038872 A141158 A130828
Adjacent sequences: A089267 A089268 A089269 this_sequence A089271 A089272 A089273
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KEYWORD
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nonn
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AUTHOR
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Wolfdieter Lang (wolfdieter.lang_AT_physik_DOT_uni-karlsruhe_DOT_de), Nov 07 2003
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