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A087427
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Number of lattice points on or inside the rectangle formed by [1 <= x <= (q-1)/2] and [1 <= y <= (p-1)/2], where p = n-th prime, q = (n-1)-st prime.
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2
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2, 6, 15, 30, 48, 72, 99, 154, 210, 270, 360, 420, 483, 598, 754, 870, 990, 1155, 1260, 1404, 1599, 1804, 2112, 2400, 2550, 2703, 2862, 3024, 3528, 4095, 4420, 4692, 5106, 5550, 5850, 6318, 6723, 7138, 7654, 8010, 8550, 9120, 9408, 9702, 10395, 11655
(list; graph; refs; listen; history; internal format)
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OFFSET
| 3,1
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COMMENTS
| Koshy, p. 499, states "We now employ this geometric approach to establish the lemma. It is due to the German mathematician Ferdinand Eisenstein, a student of Gauss at Berlin"., (where the geometric lemma applies to the Law of Quadratic Reciprocity, Koshy, p. 501): "let p and q be distinct odd primes. then (p/q)(q/p) = (-1)^[(p-1)/2 * (q-1)/2]."
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REFERENCES
| Thomas Koshy, "Elementary Number Theory with Applications", Harcourt Academic Press; 2002; p. 498-500.
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FORMULA
| (p - 1)/2 * (q - 1)/2, p = n-th prime, q = (n-1)th prime; starting with p = 5, q = 3. Sum[k=1, (p-1)/2]: floor[kq/p] + Sum[k=1, (q-1)/2]: floor[kp/q] = (p-1)/2 * (q-1)/2
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EXAMPLE
| Given the line y = (11/7)x, the number of lattice points on or inside the rectangle formed by (1 <= y <= 5), (1 <= x <= 3); where p = 11, q = 7; 5 = (p-1)/2, 3 = (q-1)/2 = (3)*(5) = 15.
The number of lattice points on or inside the rectangle, (below the line y = (11/7)x = 8, = Sum[k=1, (q-1)/2]:floor[k(11/7)] = floor[(11)(1)/7] + floor[(11)(2)/7] + floor[(11)(3)/7] = 1 + 3 + 4 = 8. The number of lattice points on or inside the rectangle above the line y = (11/7)x = Sum[k=1,(p-1)/2]:floor[k(7/11)] = floor[(7)(1)/11] + floor[(7)(2)/11] + floor[(7)(3)/11] + floor[(7)(4)/11] + floor[(7)/(5)/11] = 0 + 1 + 1 + 2 + 3 = 7.
Total number of lattice points inside or on the rectangle = 8 + 7 = 15.
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CROSSREFS
| Cf. A087428.
Sequence in context: A033286 A182724 A098651 * A141126 A056520 A078406
Adjacent sequences: A087424 A087425 A087426 * A087428 A087429 A087430
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KEYWORD
| nonn
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AUTHOR
| Gary W. Adamson (qntmpkt(AT)yahoo.com), Sep 01 2003
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EXTENSIONS
| Corrected and extended by Ray Chandler (rayjchandler(AT)sbcglobal.net), Sep 16 2003
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