3,1

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]."

Thomas Koshy, "Elementary Number Theory with Applications", Harcourt Academic Press; 2002; p. 498-500.

(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

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.

Cf. A087428.

Adjacent sequences: A087424 A087425 A087426 this_sequence A087428 A087429 A087430

Sequence in context: A138621 A033286 A098651 this_sequence A141126 A056520 A078406

nonn

Gary W. Adamson (qntmpkt(AT)yahoo.com), Sep 01 2003

Corrected and extended by Ray Chandler (rayjchandler(AT)sbcglobal.net), Sep 16 2003