Search: id:A087427
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%I A087427
%S A087427 2,6,15,30,48,72,99,154,210,270,360,420,483,598,754,870,990,1155,1260,
%T A087427 1404,1599,1804,2112,2400,2550,2703,2862,3024,3528,4095,4420,4692,5106,
%U A087427 5550,5850,6318,6723,7138,7654,8010,8550,9120,9408,9702,10395,11655
%N A087427 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.
%C A087427 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]."
%D A087427 Thomas Koshy, "Elementary Number Theory with Applications", Harcourt 
               Academic Press; 2002; p. 498-500.
%F A087427 (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
%e A087427 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.
%e A087427 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.
%e A087427 Total number of lattice points inside or on the rectangle = 8 + 7 = 15.
%Y A087427 Cf. A087428.
%Y A087427 Sequence in context: A163061 A033286 A098651 this_sequence A141126 A056520 
               A078406
%Y A087427 Adjacent sequences: A087424 A087425 A087426 this_sequence A087428 A087429 
               A087430
%K A087427 nonn
%O A087427 3,1
%A A087427 Gary W. Adamson (qntmpkt(AT)yahoo.com), Sep 01 2003
%E A087427 Corrected and extended by Ray Chandler (rayjchandler(AT)sbcglobal.net), 
               Sep 16 2003

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